University of Alberta
Many-Sorted Free Logic
by
Aristotle George Hadjiantoniou
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Arts.
Department of Philosophy
Edmonton, Alberta
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1+1 Canada DEDICATION
I dedicate this thesis to the sacred memory of my father George and my mother Helen. ABSTRACT
After a brief discussion of some pertinent concepts from classical logic, a comprehensive outline of standard free logic is presented, including proof theory, model theory and its implications for Russell's theory of descriptions. It is then shown that standard free logic, for all the good intentions of its founders and its early promises, is a thoroughly unsatisfactory system in need of emendation. To this end, many-sorted logic is introduced and subsequently utilized to develop a hybrid new branch of logic, many-sorted free logic. It is then demonstrated that many-sorted free logic enjoys all the advantages and none of the disadvantages of standard free logic. Finally, many-sorted free logic is applied to the thorny problem of non existents in literature, with sufficiently many examples to illustrate various facets of this problem. ACKNOWLEDGEMENTS
Throughout the writing of this thesis two people helped me directly and their contribution was truly enormous. The first was my thesis supervisor, Dr. Adam Morton. From the very beginning to the very end, Dr. Morton was a beacon whose immense learning, subtlety of thinking, and intimate understanding of all matters pertaining to both logic, and more generally in philosophy, were indispensable to me in researching my topic, in developing my ideas, and in the actual writing of my thesis. The second person was Wilfred Kozub, who not only typed the thesis but went through it with a fine tooth comb, corrected my grammar, and in endless discussions did his utmost to help me out. To both Dr. Morton and to Wilf I owe a very sincere thank you. But there were two more people who helped me, albeit indirectly, and whose names deserve honourable mention. The first is Fred Willard. Fred is a retired conjuror with whom I had many enlightening discussions on the role of misdirection in the art of conjuring. And what I got from these discussions is that nobody can deceive us, but instead we willingly deceive ourselves by our own prejudices even if we are unaware of the fact. Whenever I met with a dead end in my research, I reexamined my own hidden assumptions and usually it soon became clear where I had gone astray. Finally, I owe a debt of gratitude to Lee Sellers. Lee has been treated unbelievably harshly by life; she has faced untold hardship that would have broken most people, and yet her spirit remains undaunted. Whenever I felt like abandoning my project as a lost cause, Lee's sad but kindly face would come to mind and I would immediately find the strength to go on with my thesis. TABLE OF CONTENTS
I. LOGICAL PRELIMINARIES A. FORMAL LOGIC B. METALOGIC C. OBJECT LANGUAGE AND METALANGUAGE D. PROOF THEORY E. MODEL THEORY F. SUBSTITUTIONAL AND OBJECTUAL SEMANTICS
II. MYTHOLOGICAL PRELIMINARIES
III. STANDARD FREE LOGIC A. THE WHY OF FREE LOGIC B. THE ESSENCE OF FREE LOGIC C. MISCONCEPTIONS WITH REGARDS TO FREE LOGIC D. THE PROOF THEORY OF FREE LOGIC E. THE MODEL THEORY OF FREE LOGIC • OBJECTUAL SEMANTICS FOR FREE LOGIC F. FREE LOGIC AND THE THEORY OF DEFINITE DESCRIPTIONS 1. RUSSELL'S THEORY OF DEFINITE DESCRIPTIONS 2. THE DERIVATION OF THE ALLEGED PARADOXES 3. THE ALLEGED PARADOXES OF RUSSELL'S THEORY 4. THE ALLEGED INCOMPATIBILITY BETWEEN FREE LOGIC AND THE THEORY OF DEFINITE DESCRIPTIONS G. INFIRMITIES OF FREE LOGIC IV. MANY-SORTED CLASSICAL LOGIC A. INTRODUCTION B. UNI-SORTED LOGIC C MANY-SORTED FREE LOGIC - INFORMAL DISCUSSION D. THE LANGUAGE AND CONVENTIONS OF MANY- SORTED FREE LOGIC E. RELATIONS WITHIN AND BETWEEN SORTS
V. MANY-SORTED FREE LOGIC A. INTRODUCTION B. THE LANGUAGE OF MANY-SORTED FREE LOGIC C. AN ASCENDING HIERARCHY AND THE ROLE OF CONTEXT D. PROMISES KEPT E. THE SEMANTIC PRINCIPLES OF MANY-SORTED FREE LOGIC F. PROOF THEORY FOR MANY-SORTED FREE LOGIC G. THE MODEL THEORY OF MANY-SORTED FREE LOGIC
VI. THE ROLE OF MANY-SORTED FREE LOGIC IN LITERATURE
VII. ADDENDUM: THE EMPIRICAL DIMENSION OF SCIENCE IN CONFLICT WITH LOGIC
• ENDNOTES • BIBLIOGRAPHY 1
I. LOGICAL PRELIMINARIES
In this part I will present some very rudimentary ideas from formal logic, just those that will be needed later on in this thesis. Only the bare bones will be offered and no attempt at presenting the full system will be made.
A. FORMAL LOGIC
Formal logic has as its object of study the rules and procedures that underlie correct reasoning, by using techniques that originated in the field of Boolean Algebra. So far as its methodology is concerned it is a branch of abstract mathematics, but it is mostly philosophers that make the greatest use of its results and conclusions. It is my assumption that anybody reading this thesis is already familiar with the basic symbols of formal logic, as well as two of its techniques for proving theorems. These are the technique of natural deduction and that of tree diagrams, some examples of each of which will be given a little later on.1
B. METALOGIC
Metalogic is an advanced and rather esoteric branch of formal logic which is not concerned with the study of results within a theory, but rather with the study of results about a theory. Typical theorems proved in metalogic have to do with whether a given set of axioms is consistent, which means that no contradictory results can be derived from them, or whether these axioms are all independent one from the other and so none of them can be derived from the rest, and the like.2 In this thesis we will not be proving any metalogical theorems, still certain ideas from metalogic will be seen to be of immense value in developing many-sorted free logic, and none more so than the distinction between the object language and the metalanguage of a system. The reason is that every result in the object language of many-sorted 2 logic has its counterpart in the metalanguage of classical logic, and so the latter can be used to prove results that hold good for the former.
C. OBJECT LANGUAGE AND METALANGUAGE
These two terms in spite of being somewhat intimidating actually refer to rather pedestrian aspects of day to day talk and not just profound matters of logic and allied mathematical disciplines. Let us see what they are. Given any subject, its object language includes all the sentences one can utter or write about the objects, physical or abstract, of that subject. An example of this is the sentence "snow is white". This is a sentence of the subject-predicate kind which says something about snow. In the meta language, however, one does not encounter sentences about objects. Rather one encounters sentences about sentences in the object language. Thus, returning to the previous example, the new sentence "the sentence 'snow is white' is true" is part of the metalanguage of the study of snow.3 Consider a more sophisticated example, that of axiomatic set theory as developed by Zermelo. It consists of ten axioms, not all of which are independent as it turned out, from which one can derive the various theorems of set theory. Some of these axioms have such sonorous names as extensionality, pairing, union, foundation, replacement, etc. Now consider any of the many theorems of set theory such as De Morgan's law, C - (A D B) - (C - A) U (C - B), where A, B and C are all sets. Since this theorem is about sets, it is part of the object language of set theory whose objects of study are sets. But consider the following theorem: The axiom of pairing of set theory is derivable from the axioms of union and that of replacement.4 This is not a theorem in the object language of set theory as it makes no direct claim about sets. It makes a claim about axioms whose objects are set, i.e. about sentences about sets, and as such it belongs to the metalanguage of set theory. Finally, the statement "the sentence 'the pairing axiom of set theory is derivable from the union and the replacement axioms' is true" belongs to the meta metalanguage of set theory. Metalogic has grown 3 into a vast study, one with many branches; the two original ones on which rests all the rest are proof theory and model theory.
D. PROOF THEORY
Proof theory, as the name implies, has as its object of study the proofs of theorems and what we can say about them. Come to think of it, all mathematicians are, at some very real although informal level, proof theorists because they construct proofs and they evaluate claims that such and such an argument constitutes a proof. Formal proof theory considers the proofs of theorems themselves to be mathematical objects in their own right and studies them by means of the methods of formal logic. Now what is a proof? A proof is a finite sequence of symbols such that the very last one follows from the rest via the rule of detachment, or modus ponens. The initial such symbols which can not be derived this way, and hence we must accept them on faith, are called axioms. Symbols that come after the axioms and can be derived from them by modus ponens are called theorems. Any such set of symbols is called a theory. In proof theory we introduce a new symbol, that of derivability, and we represent it by means of the turnstile |- . Thus if T is some theory and A is some result that can be formally derived from T then we write r |- A. Here is a trivial example. If P implies Q, and if Q implies R, then P implies R. This examples brings to the forefront a basic fact about proof theory. And that is that proof theory, while itself being an interpreted system, the turnstile symbol meaning derivability, treats the various symbols which are the building blocks of theories to be uninterpreted. Thus in the above example, it matters little what the contents of P, Q and R may be, or how these are to be interpreted. They may have to do with geometry, or set theory or abstract algebra; it makes no difference whatever, as the derivation is sound in all these cases. It is the form that matters and not the content. Two of the most frequently used theorems of proof theory are the Deduction Theorem and its inverse. Since we will have occasion to use both 4 in the part of the thesis devoted to the proof theory of free logic, I will state both of them without proof, which can be found in any basic textbook on metalogic. The Deduction Theorem, abbreviated as D. T., is if r U { A} |- B then r|-ADB. The inverse of the above does not seem to have a standard name so we will call it the Inverse Deduction Theorem, which we abbreviate as I. D. T. This is if T |- A D B then r U { A } |- B.5
E. MODEL THEORY
In a very real sense proof theory and model theory are polar opposites. While proof theory completely ignores interpretations, model theory is the study of interpretations. These two branches of metalogic complement each other, and this complementation is deeper than meets the eye. They are both indispensable in the study of metalogic. And what is an interpretation, a.k.a. a model of a theory r ? A theory, as we saw, contains certain axioms without any a priori content. When we assign some content to the various symbols in such a way that the resulting structure satisfies all the axioms, then we say we have an interpretation or a model of the theory. For example, the Peano axioms of arithmetic constitute a theory which is ab initio uninterpreted, and its symbols are meaningless. But this theory has an interpretation, several in fact. The standard model (interpretation) for Peano arithmetic is the natural numbers 0,1, 2,... with the successor relation which holds between any n and n + 1. This verifies the ordinary arithmetic we first learn in grade school. All this is very vague and even simplistic, but what I try to do here is just give only an intuitive feeling for the subject. Later on the discussion will become somewhat more technical. We have introduced a new word, the word "satisfiability". It plays an entirely analogous role that the word "derivability" plays in proof theory. 5
A sentence A is satisfiable in a model ^ if when the elements of A are given meanings in ^ then A is true. For satisfiability we use the double turnstile symbol "|= " and we write **• |» A. Let us introduce two very simple theories using the language of arithmetic and that of the predicate calculus. Let us call these two theories respectively I\ and r2. Let I\ be given by a single axiom 3x Vy : x s y. Does I\ have a model? The answer is yes and, in fact, several models. One such is *-*• whose domain is the set {1, 2, 3 }. The axiom is satisfied in *•*• seeing that one of the elements of the domain, the number 3, is greater or equal to every member of the domain. In fact, every finite set of integers can serve as the domain of a model for I\. Let now T2 be given by the single axiom 3x Vy : x = y. T2 also has many models, infinitely many in fact. The models for T2 are just those whose domains are singletons, whose domains contain just one number. One example is {8}. It is immediately obvious that for I\ the two models having domains of the same cardinality, such as {1, 2, 3,4} and {2,4, 6, 8} have the same properties. We call such models isomorphic. But the models whose domains are, say, {1, 2} and {7, 8. 9} do not have the same properties and are not isomorphic. But let us go back to T2. Since all models T2 are just those whose domains are singletons, all the models of r2 are isomorphic. Such a theory all whose models are isomorphic is called a categorical theory.7 A theory with models may also have models whose domains are subdomains of the original models. We call the latter submodels of the theory. A model stands to its submodels the same way as a theory stands to its models. I.e., submodels turn models into theories in their own right. Submodels will play a part, a small one, when we develop the model theory of many-sorted free logic. The single and most glorious theory is the very first theory that was shown to have finitely many nonisomorphic models. And this long before formal logic came into existence! And that theory is of course none other than projective geometry. This theory has exactly three 6 nonisomorphic models: the geometry of Euclid, the geometry of Riemann, and, finally, the geometry of Lobachewsky and Bolyai, each of which is a huge field. It is noteworthy that although the single and the double turnstiles are two distinct things, they do share an uncannily large number of similarities. The fact did not escape the attention of Kurt Godel who showed in his 1931 doctoral dissertation that in the case of first order logic, the concept of derivability and that of satisfiability are equivalent. Or that T |- A iff T |= A.8 This is Godel's celebrated completeness theorem for first order logic, not to be confused with his even better known, and indeed almost notorious, incompleteness theorems of arithmetic.
F. SUBSTITUTIONAL AND OBJECTUAL SEMANTICS
As we saw in model theory, as contrasted with proof theory, the concepts of truth and falsity are of paramount importance. We are therefore led to the inevitable question of when is a quantified sentence true or false. For example, under what conditions is the existentially quantified sentence 3xFx true? The answer is immediately obvious. 3xFx is true just in case there is "something" predicated by F. But just what is this "something"? There are two ways of looking at this question, one of them being intrinsically superior to the other, and yet for many purposes the second is preferable to the first on purely pragmatic grounds. In the first case we take this "something" to be some object, either physical or abstract, that has the property F. In this case we eliminate the existential quantifier by replacing the bound variable "x" with the free variable "u". We have 3xFx |« Fu. Since we are in the domain of the first order logic, we also have 3xFx |- Fu. The free variable "u" stands for some object, if there is one, having property F. In this case, and as the name implies, we are using objectual semantics. In substitutional semantics, on the other hand, what we have is 3xFx |- Fa. Here "a" is a constant rather than a free variable. A constant which stands for a name and not an object. Of course when an object having 7 property "F" exists and whose name is "a", it makes little difference which method we use, that of objectual semantics or that of substitutional semantics as the latter is called.9 Now which of the two is better? That depends entirely on the nature of the problem we are dealing with. For many instances when there exist objects having some property "F", and when each and every one of these objects can be named, it is a lot more convenient to use substitutional semantics. Imagine, for example, a telephone directory which instead of having names gave a short description of all the salient properties of each entry! In such cases substitutional semantics must be preferred on purely pragmatic grounds. When we are working, however, in the realm of theory, objectual semantics is almost always the better choice, and this because, as we shall see later when we study free logic, substitutional semantics raises some very thorny problems. Incidentally, when free logicians developed the model theory of the subject, they invariably used substitutional semantics. The reason is not at all hard to guess. In free logic, in addition to referring names, such as Albert Einstein, we also have non referring names, such as Sherlock Holmes. That is to say names for which there is no object to which they can be attached. And this is one of the sources, but by no means the only one, for the many infirmities of free logic. These problems, however, will occupy us when we come to the study of free logic. For now, I will present one reason which holds good for any kind of logic you care to mention, which illustrates why substitutional semantics is a most unsatisfactory way of doing things. First of all, what is a name? The answer is that a name is a sequence of symbols each one of which is a member of some finite set. Thus the name "John" is a sequence of four symbols each of which is a member of the English alphabet which is a finite set containing just 26 elements. Every such sequence, however, is a countable set. Therefore the set of all names consists of all countable sequences whose members are symbols belonging to some finite set. It is thus obvious that the set of all names is countable. And this is the reason that we can name any number belonging to some countable set, such as the set of all rational numbers. Uncountable sets of numbers, 8 however, such as the set of all real numbers, lack this property and so most of their members can not be named. There are more objects than names.10 And such "names" as "n" are names by convention only. The symbol "n", far from being a name, is only an abbreviation, that being the unique real number representing the ratio of the circumference of the circle to its diameter. Thus when faced for the first time with some rational number, no matter how many digits it may have, such as 0.739203465, we can derive all its many properties. Not so when we first meet ft, and this because n is not a name.
II. MYTHOLOGICAL PRELIMINARIES
Here I will present a short description of certain beings from Greek mythology which will be used at various points of the thesis as examples illustrating some pertinent concepts therein.
GORGONS: These were three "women" who could fly and had live snakes growing on their heads instead of hair. They struck terror in the human heart as their mere gaze would turn living beings into stone. Two of them, whose names have not come down to us, were immortal. The third one was mortal and she was called Medusa. She was slain by the hero Perseus. FURIES: Three immortal female deities who had huge bat like wings and were sent by the Gods to torment all humans who had transgressed divine law. Their names were Alecto, Tisiphone and Megaera. CENTAURS: Hybrid creatures, half men from the waist upwards and half horses from the waist downwards. We know the names of very few of them, the most conspicuous being Chiron. He was blessed with immense learning, was a highly skilled physician but also a superb archer. He served as tutor to many of the heroes of mythology. CYCLOPES: Uncouth and ungainly one eyed giants possessed of enormous physical strength but correspondingly low intelligence who formed a primitive agrarian society. On the positive side they were renowned and discerning gourmets with an especial fondness for human flesh. This was to 9
them a rare and much sought after delicacy, which, when available, was consumed raw, pretty much like our steak tartare.
III. STANDARD FREE LOGIC
A. THE WHY OF FREE LOGIC
Let us begin with two basic laws of classical first order logic. First, the law of universal instantiation, VxFx |- Fu (U.I.) What the above tells is is that if everything that is has a certain property, then so does any particular you care to mention. The second is the law of existential generalization, Fu |- 3xFx (E.G.) which tells us that if some particular has a certain property, then there exists something that has that property. Taken together these two laws imply that VxFx |- 3xFx (1) In making the transition from (U.I.) and (E.G.) to the above, Fu acts pretty much as the middle term in a traditional syllogism. Hence in order for the above derivation to hold good, there must be at least one object in the universe having property F, and if there are more than one objects in the universe, then all must share in the above property. While quantification over atomic sentences is seen to be quite straightforward, quantification over compound sentences is less so. We would like to have a law, one strictly analogous to Vx Fx |- 3xFx, which holds for atomic sentences, that holds good for compound sentences. The most likely candidate would have to be the following: Vx (Fx D Gx) |- 3x (Fx A Gx). The only problem with the above proposed law is that it is invalid. Here is the reason why. The existential quantifier, together with the conjunction within its scope assert a matter of fact. That 10 something having two properties exists. The universal quantifier, however, together with the material conditional within its scope, fail to make any such matter of fact assertion. What they do is to make a hypothetical statement. That if something - anything, in fact - has a certain property, then it also has a second property. But it fails to establish that there is something having the first property to begin with. Thus the theorem that every equilateral triangle is equiangular, fails to inform us if there be any such triangles. A much better candidate to act the part of (1) for compound sentences is the following: { Vx (Fx D Gx), 3xFx } |- 3x (Fx A GX) (2) There is a sound derivation of the latter as can easily be seen from the following natural deduction:
(1) Vx (Fx D Gx) Assumption (2) 3xFx Assumption (3) Fu (2), Existential Instantiation, u/x (4) Fu D Gu (1) Universal Instantiation, u/x (5) Gu (3), (4), Modus Ponens (6) Fu A Gu (3), (6), Conjunction Introduction (7) 3x (Fx A Gx) (6), Existential Generalization
It is easily seen that the presence of 3xFx plays a dual role. Positively, when there is something predicated by F, it makes possible the derivation of 3x (Fx A Gx). Negatively, when there is nothing predicated by F, it blocks this derivation without making it necessary to negate Vx (Fx D Gx). Consider this example: Let "F ..." stand for "... is a Gorgon," and "G ..." stand for "... can turn humans into stone." Then we have Vx (Fx D Gx) since, according to Greek mythology, all Gorgons were possessed of this ghastly ability, but since we do not have 3xFx, there being thankfully no Gorgons, we cannot derive the, in this case, fallacious 3x (Fx A GX). When, however, we go from objectual to substititutional semantics, and we start instantiating quantified sentences by means of constants, i.e., 11 names, instead of variables, it is then that our problems begin. Let "a" be the name of the one mortal Gorgon, Medusa, and, this time, let "H ..." stand for " ... = a", or, whatever is predicated by H is identical to "Medusa". Then we have Vx (Hx D Gx) j- 3x (Hx A GX), or from the universally quantified conditional, "Everything which is identical to 'Medusa' can turn humans into stone" we can apparently derive "there exists something identical to 'Medusa' and it can turn humans into stone", dispensing with the condition that "there exists something identical with 'Medusa'," as the following natural deduction shows:
(1) Vx ((x=a) D Gx) Assumption (2) a = a D Ga (1), Universal Instantiation, a/x (3) a = a Reflexive Law of Identity (4) Ga (2), (3), Modus Ponens (5) a = a A Ga (3), (4), Conjunction Introduction (6) 3x ((x=a) A Gx) (5), Existential Generalization
Since "Medusa", being a Gorgon, could turn humans into stone, then this derivation represents an argument which is both sound and valid. A most disquieting conclusion!11 It is not hard to find out what went wrong. To do so, let us examine in reverse order the natural deductions we used to derive the two laws of logic: { Vx (Fx D Gx), 3xFx } |- 3x (Fx A GX)
Vx (x = a D Gx) |- 3x (x = a A GX). Both the above have as a conclusion the derivation of an existential claim within whose scope is a conjunction. The scope of each is a conjunction of two conjuncts, each of which is an atomic sentence predicating something about some entity "x". In both cases, one of the conjuncts is the same, that the entity in question can turn humans into stone. What the two conjunctions differ in is the other 12 conjunct. Now, this other conjunct is in both cases the antecedent in a material conditional. And they differ widely from each other. In the first case, the predicate attributes to the entity in question a certain substantive property, one that allows us to place that entity in a class containing all similar entities, the property of being a Gorgon. Whaf s more, this is an entirely contingent property, not one shared in by everything. If so, then we have no a priori guarantee that such beings actually exist. For this reason, in order to arrive at a sound derivation we need the existential assumption 3xFx. In the second case, however, the predicate "H" in the antecedent does something entirely different. It does not attribute to the entity in question any one property that can be either ascertained or refuted. What it does is give a name to the entity in question, the name "Medusa", here represented by the lower case letter "a". By giving it a name it stipulates that the entity in question and the one named "Medusa" share in all the same properties, that they are the same. And since Medusa, according to mythology, could turn people into stone, so does the entity that answers to the name "Medusa". Let us return to the natural deductions. Since in both cases we wish to derive a conjunction of two conjuncts, we must have in both cases the two appropriate conjuncts. The conjunct "Gx", however, the one common in both cases, is the consequent of a material conditional. This means that in order to establish Gx, we must first detach Gx from the conditional by means of modus ponens. In order to perform modus ponens, however, we must first establish the two antecedents. In the first case the antecedent is established by the instantiation of the existential condition 3xFx. In the second case things are a lot trickier. Here, too, usually, we need an existential clause, but for one notable exception. When we instantiate Vx (Hx D Gx) for some object-variable "u" we derive Hu D Gu, which gives u = a D Gu, and finally, by substituting "a" for "u" we get a = a D Ga. This ordinarily implies that there is some object represented by "u" which answers to the name "Medusa". So, obviously, an existential clause, one affirming the reality of such an object, is essential. The exception 13 arises when we instantiate directly for the name "a", skipping the step of using some object - variable. In this instance what we get is a = a D Ga. Here the antecedent is a logical axiom, that of the reflexivity of identity, and as such it always holds good. If so, then we can use the law "a = a" in order to perform a successful modus ponens which engenders absolutely disastrous results. What happened here is that we used a valid logical principle as a ruse, something which would have not happened had we used objectual semantics instead of the substitutional semantics that we employed. The trouble with names is that they are arbitrary stipulations not dictated by either fact or reason. That we called "two" the immediate successor of the number we called "one" is an arbitrary stipulation having nothing to do with mathematical truth. Provided that we have an adequate supply of names we can attach them to anything we choose, whether real or just a figment of the imagination. When we attach a name to something real, we call it a "referring name", because it refers to something actual. When we attach a name to a non existent, we call it a "non referring name", or, simply, an "empty name". Now "Medusa", represented here by "a", is an empty name. There is nothing in the world which answers to this name. The fallacy we committed consists of us having treated an empty name as if it were referring to something real, something "out there", which it does not. The inescapable conclusion, and one which is crucial for the subsequent development of our subject, is that referring names and empty names can not be treated on par. And how does classical logic deal with the problem posed by empty names? Simply by banishing them altogether. In classical logic, instantiation by empty names is totally inadmissible. Regarding this matter, one can perhaps do no better than to quote from Bertrand Russell's seminal article "Descriptions": "Logic, I should maintain, must no more admit a unicorn than zoology can; for logic is concerned with the real world just as truly as zoology, though with its more abstract and general features. To say that unicorns have an existence in heraldry, or in literature, or in imagination is a most pitiful and paltry evasion. What exists in heraldry is not an animal, made of flesh and blood, 14 moving and breathing of its own initiative. What exists is a picture, or a description in words." 12 What Russell proposed here, in the most uncompromising terms, is to perform radical surgery, one that severs the gangrenous limb, in this case empty names, in the interest of saving the rest of the body, or, at any rate, what remains of it. I will call it the "Russell Therapy". Now Russell never mentioned "empty names" as in "empty proper names," but rather "empty predicates" such as unicorns. Empty predicates however have empty extensions and any name given to a member of such an extension is ipso facto an empty name, as there is no object bearing that name. Unhappily, though, there are occasions when the proposed therapy may be more harmful than the disease it purports to cure, and this is a case in point. Let us see why. First, the comparison made by Russell between logic and zoology, and the claim made that logic studies the same world as zoology does, albeit on a more abstract level, is most infelicitous. And this because logic differs form zoology in kind, and not just in degree of abstraction. Zoology, and all other sciences, studies certain denizens of this world, and the various phenomena that obtain thereof. Consequently, its methods are largely empirical. Logic, on the other hand, studies the statements that can be written or uttered in the various sciences, and how the truth value of atomic sentences determines that of compound sentences. Consequently, its methods are entirely rational, i.e. mathematical. Scientific claims are established or refuted by the facts that give rise to such claims. The proposed theorems of logic, on the other hand, are proved or disproved entirely by the form they take. Thus the derivation from A D B and BDC to ADC is a sound one by virtue of the form it has, the content being entirely irrelevant. To assert that the difference between logic, on the one hand, and zoology on the other, is one of degree rather than one of kind is to make a category mistake of the most egregious kind. Zoology deals with matters of fact; logic deals exclusively with form. But if so, then logic, by its very nature, lacks the capacity to adjudicate on the existence or non existence of things, whether these things exist in reality or in picture only. 15
Second, the adoption of the Russell therapy will prove to be excessively wasteful as it cuts too much. And this because human speech is replete with empty names. Whaf s more, these names arise in connection with intellectual disciplines worthy of the most serious study. The banishment of empty names, as dictated by Russell's therapy, makes it impossible to apply to the study of these fields the marvelously subtle tools that have been developed in logic. Three such disciplines are literature, the empirical sciences, and mathematics, all of which illustrate my point. Let us have a brief look at each of these in turn. In literature, and here I include mythology also, we are dealing, for the most part, with fictitious beings. Consider the Conan Doyle story "The Adventure of the Speckled Band". Here, Sherlock Holmes (an empty name) caused the death of Dr. Grimesby Roylott of Stoke Moran (another empty name) by snake bite. Is this a true or a false statement? So far as reality goes this statement is patently false, the persons as well as the events described therein being but figments of the imagination. In the realm of fiction, however, the above statement is true as it corresponds with the events described in the story. What about the statement that Sherlock Holmes caused the death of Dr. Grimesby Roylott of Stoke Moran by gun wound? This statement is false both in reality, as the persons and the event described are fictitious, and in literature, as what the statement asserts is not in correspondence with the story. Now, classical logic, being held in thrall to Russell's Therapy, can offer us no criterion for distinguishing between the truth values of these two statements, qua pieces of fiction, nor yet a criterion for distinguishing between the truth values of the first statement when considered as literature and when considered as a description of real life. In the empirical sciences we face the very same problem, as they contain terms that do not refer to anything existent. The reason is that as a science develops, she supplements her role of a mere collector of data with that of a theory erector. And abstract theories deal with universals as opposed to particulars, the one over the many. Now, the way theories are constructed is a process of abstraction, by means of which some feature common in many distinct things is considered in isolation, all particular characteristics that allow us to differentiate between these individuals, being ignored. This leads to the study of abstract objects which do not exist in reality. Physics, the most mature of all sciences, is the paradigm case. In physics we talk about the frictionless plane, the absolute vacuum, the perfect conductor as well as the perfect insulator; the list is really endless. And none of these theoretical entities exists; they are all simply conceptual fictions. Consider the statement, "all frictionless planes allow unrestricted movement over them". This is written Vx (Fx D Gx) which is perfectly reasonable, and yet, since the frictionless plane does not exist, such a statement is beyond the purview of classical logic. This takes care of "empty predicates" as encountered in the theoretical aspect of science. But there still remains the problem of "empty names" as encountered in the empirical aspect of science. This takes quite a bit longer to explain and, as I do not wish to interrupt the continuity of the thesis, I will deal with it at the very end in an addendum. Finally, mathematics suffers a most grievous wrong at the hands of the Russell therapy. And to think, Russell and Whitehead developed their system in order to place, among others, the axioms of arithmetic on a firm footing! The axiom of extensionality asserts that two sets are equal just in case they have the same members. A set X is a subset of another set Y just in case every member of X is also a member of Y. The empty set, i.e. the set with no members, is vacuously a subset of every set, since its "members" are members of every set. But observe. The members of the empty set being non existent are objects inadmissible in classical logic. What is clearly needed is an emended system of logic, one which does not stand in direct opposition to classical Russellian logic, but instead expands the range of applicability of the latter by admitting empty names as legitimate objects of study. Free logic is one such.
THE ESSENCE OF FREE LOGIC
Free logic experienced its birth in a seminal paper written in 1956 by Henry Leonard. It subsequently grew into a full fledged branch of logic in the hands 17 of Karel Lambert, who incidentally coined the name. What exactly is free logic is not easy to say unequivocally. One can do no better than to quote the following informal definition given by Ermano Bencivenga: A free logic is a formal system of quantification theory, with or without identity, which allows for some singular terms in some instances to be thought of denoting no existing objects, and in which quantifiers are invariably thought of as having existential import.13 There are several noteworthy things about this definition. First, the indefinite article "a", which is the very first word in this definition, indicates that fee logic developed sufficiently so that it branched out in several distinct systems, each one motivated by mutually opposing Weltanschaunungs. Whaf s more, each of these free logics can be developed by means of predicate calculi with or without identity. Third, what distinguishes free from classical logic is that the former admits of non-denoting terms, i.e. empty names, while the latter eschews them. Fourth, and very significantly, while empty names are admitted in free logic, this admission is reluctant and very restricted by means of an ironclad condition. And that is that while non- existents can occur in unquantified sentences, they can never lie within the scope of quantifiers, which therefore always have existential import. As we shall see, this one condition will dictate the way that both the proof theory and the model theory for free logic will take. And it should be obvious that the intent of the founders was to deviate as little as possible from classical logic. Since the distinguishing characteristic of free logic is the restricted admissions of empty names and non denoting terms, it may be that "Logic of Singular Terms" would have been a better choice than the oftentimes misleading "Free Logic". In fact, the former is the title of the chapter introducing the subject, in a logic textbook coauthored by Lambert and Van Fraassen.14 Speaking of names, since they are the source of the difficulties that prompted the development of the subject, it should come as no surprise that the model theory of free logic as worked out by Lambert, uses substitutional semantics, with all the attending infirmities that these inflict on classical logic, 18 and then some. We have already noted that a thorny problem with substitutional semantics is that there are more objects than names which can be attached to them, there being only countably many names, while the sets of some kind of objects, e.g. real numbers, are uncountable. Free logic admits of empty names and, whaf s more, by adopting substitutional semantics, use names exclusively to instantiate quantified sentences. This leads to the highly paradoxical result that, while in reality there are more objects than names, in free logic there are more names than objects. Worse is yet to come. It seems to me that the only good reason for developing free logic is to enable us to deal with non existents in a way that does justice to the contexts in which they occur. If so, then, given some context in which nonexistents occur, any logical system that purports to deal in true and just measure with that context, must be able to deal with each and every nonexistent that occurs in it. But standard free logic, as developed mainly by Lambert, and tied up to the substitutional semantics that the latter used for that purpose, is incapable of accomplishing this task. And this because not every nonexistent has a name. Consider two trios of mythological creatures, the three Furies and the three Gorgons. We know the names of all of the three Furies but not of all of the three Gorgons. "Medusa" is the name of the one Gorgon we know. She is the one mortal Gorgon and was slain by Perseus, but her two immortal sisters are nameless. This causes severe problemSi In both our private cogitations and in our linguistic communications we need ways and means of distinguishing between various distinct entities, both those that exist and those that do not, that we think or we talk about. For existents the problem is, at least in theory, relatively easy, and its solution is provided by Leibniz' law of the identity of indiscernibles. Two distinct objects must be differentiated by at least one characteristic, even if it only be that they are in different places at one time. And such a distinguishing characteristic is, at least in theory, always detectable by some empirical process. But, for two fictional objects, e.g. Gorgons, such a directly detectable empirical criterion is never available. All that is available is indirect criteria, if any, provided to us by the narrative. And in many cases the narrative fails completely to offer us 19 any distinguishing characteristics. All we know about the two nameless Gorgons is that they shared in every Gorgon like property, such as the capacity to turn men into stone, which allows us to distinguish them from, say, Centaurs, and that they were both immortal, which allows us to distinguish them form "Medusa", but nothing that allows us to distinguish them for each other. But then the law of the identity of indiscernibles forces on us the unwelcome conclusion that the two immortal Gorgons were just one. And, just in case this seems to be too far fetched, there are systems of free logic which identify all nonexistents, using the asterisk symbol " * " as a dummy name. If so, every Gorgon is a Centaur! Such a conclusion does gross injustice to mythology, literature, etc. Finally, we must say a few things about distinct subsystems, within free logic. The latter has branched out in three distinct branches each of which has its own subvariations. All three accept empty names, in contrast to what classical logic does, and they all differ in one key point, this being the truth value of atomic sentences containing such. These are positive free logic, negative free logic, and Fregean free logic. Positive free logic accepts that atomic sentences containing empty names can sometimes be true and sometimes be false. It has the advantage of respecting the context within which such sentences are written or uttered. Thus, "Medusa can turn humans into stone," would be considered to be true in the context of mythology, but false in the context of everyday life, by positive free logic. On the other hand, "Medusa can argue cases in court" would be false in every context that is of some significance. Negative free logic, while admitting of empty names, declares each and every atomic sentence containing them to be false. Thus the sentence, "Medusa can turn humans into stone" is always false, and so negative free logic does not respect the context in which such sentences are sometimes uttered, as for example when we meet them in mythology. The advantage that negative free logic has over positive such is that the former respects a very intuitive idea we have about nonexistents, and this is that they can not have properties. Since "Medusa", being an empty name, refers to nothing, 20
how can it be predicated by anything? Having said this I must hasten to add that this stipulation has some very curious consequences. Consider the atomic sentence Fa, where "a" stands for "Medusa", and "F ..." stand for "... = a". Thus "Fa" stands for "Medusa is Medusa". Now in negative free logic, by what we have said, this sentence is false. The price that negative free logic has to pay, though, is a steep one, as she has to accept, as a consequence of her stipulation, that one of the most basic laws of logic and mathematics, the law of reflexivity of identity, may not always hold good. Finally, Fregean free logic regards sentences containing empty names to lack truth value.15
C. MISCONCEPTIONS WITH REGARDS TO FREE LOGIC
Misconceptions that tend to becloud an area of logic which is already murky enough as it is, unfortunately abound. First and foremost is the explanation offered in many books regarding the predicate "free" occurring in "free logic". Many such works maintain that "free logic" is logic free from existential assumptions. What they most probably mean here is that free logic is "free" from the stipulation of classical Russellian logic banning empty names. As it stands, however, such a definition is both highly misleading and factually erroneous. It is misleading as it fosters the mistaken view that free logic treats both referring names and non referring ones on par. Nothing could be further from the truth. Bencivenga's definition to the effect that free logic, while admitting empty names in atomic sentences, requires that quantifiers always have existential import, gives the lie to this view. Referring and non referring names are not treated equally in free logic in the case they occur in quantified sentences. So much for empty names. How about existential assumptions? When people claim that free logic is "free" from existential assumptions such an assertion is more than just misleading; it is factually erroneous, as free logic is replete with existential assumptions, and, superficially speaking, more so 21 than classical logic. Of course this last is more apparent than real, and, in fact, both systems use the very same existential assumptions, but they do so in different ways. Classical logic does so by making empty names inadmissible. Every name used must refer to some existent, and thus there is no need for postulating any additional existential assumptions, as everything named in classical logic has already been assumed to exist. Free logic, by admitting empty names, does not enjoy this luxury. Free logic must set forth rules that disallow nonexistents to play a role in quantified sentences. To demonstrate the above I will compare a widely used logical law, the form it takes, first in classical logic, and then in free logic:
{ Vx (Fx D Gx), Fa } |- Ga
{ Vx (Fx D Gx), 3x (x=a A Fx)} |- Ga.
The distinguishing difference between classical and free logic is that in deriving Ga, the former uses Fa, while the latter uses 3x (x=a A FX). The difference is not vast, but it is revealing of how the two systems differ. The one uses "Fa" to indicate that something, in this case "a", has some property, and there is no need to indicate that "a" denotes an existent, as it is a blanket assumption of classical logic that all names refer. Free logic, on the other hand, while using the very same existential assumption as classical logic does, must be a bit more explicit in its formalism about this existential assumption than classical logic is. This because free logic admits empty names and allows them a strictly prescribed role in its system. Thus it is imperative for the derivation of Ga to impose the restriction that not only is "a" predicated by "F", but also "a" denotes an existent, or that there exists some "x", or "3x", such that "x" is "a", or "x=a", and that the object "x" called "a", has the desirable property "F". All this is written as 3x (x=a A FX). In case we ban non referring names, then we have Fa e 3x (x-a A FX), and the two systems of logic become one. So the two systems use the same existential 22 assumptions. Classical logic does this informally, covertly and implicitly, while free logic des it formally, openly and explicitly. There is a claim which, though not strictly false, is in actual practice highly misleading, and for this reason it ought to be examined in some detail. And this is that free logic is a rival system to classical logic. In examining this claim it is important to clarify the term "rival", and to make explicit, first, in how many different ways two logical systems can be said to be rivals, and, second, having done this, to decide in exactly what sense we use "rivals" when contrasting free logic with classical logic. First of all, a system of logic is said to be an extended system to classical logic, just in case the former, while keeping all the symbols of the latter, but also all of its axioms, introduces new symbols, each of which formalizes some new ideas that are foreign to classical logic, makes these new ideas explicit by means of new axioms, and, consequently, within this enriched system, in addition to all the already existing theorems of classical logic, certain new theorems are derivable. The paradigm case here is modal logic which, in addition to occupying a very central position in philosophical logic, has exhibited amazing fecundity by bringing forth some mighty offspring, such as tense logic, epistemic logic, deontic logic, and provability theory, to mention but a few. The hallmark of modal logic is the introduction of two new operators which act adverbially, that of necessity and that of possibility, the former represented by a square shaped symbol and the latter by a diamond shaped symbol. The one can be reduced to the other since the negation of the necessity of some state of affairs is equivalent to the possibility of the negation of that state. Neither, however, can be reduced to any combination of symbols of classical logic. These two new symbols, and the ideas they embody, (the latter to be set forth by appropriate axioms), must be introduced ab initio. It should come as no surprise then that there are theorems, derivable in modal logic, which have no counterpart in classical logic. On the other hand, every theorem of classical logic is derivable in modal logic. Thus modal logic is not a true rival of classical logic but only an extension thereof. 23
By what we have said, free logic is not an extension of classical logic. First, free logic does not introduce any new symbols save the soon to be explained E!, and, in the case the free logic we are using has the identity symbol, "E!" is reduced to the symbols of classical logic. In the case not, "El" introduces no really new ideas, and even when we use it as a primitive symbol, we can not derive with its aid any new theorems that can not be derived by using our old friend, "3" and identity. And second, what we have just said is really the heart of the matter. When we ask whether a proposed new system of logic is an extension of classical logic, the proof of the pudding is whether all the theorems of classical logic can be derived in the new system with the addition of some new non trivial theorems, which not only can not be derived in classical logic, but can not even be expressed in the language of classical logic, due to the latter's deficiency in symbols. And since free logic does not prove any new theorems, it is not a true extension of classical logic. The case of true rival systems of logic can be disposed of fairly quickly. A rival system to classical logic is one which admits of axioms that negate existing axioms of classical logic. As a consequence, there are theorems derivable in the rival system that flatly contradict theorems of classical logic. Intuitionism is one such. Intuitionistic mathematics, as developed by L.E.J. Brouwer, and the attending intuitionistic logic, as formalized by A. Hey ting, violate the principle of excluded middle, or |- P v ~P, and this is anathema to both traditional mathematics and to classical logic. Indeed, it is a theorem of intutitionistic mathematics that every real valued function is continuous and, whafs more, uniformly continuous. I don't want to say more because, unfortunately, my knowledge of intuitionism is next to nil. It should be fairly obvious, however, that free logic, by not negating any of the axioms of classical logic, is not a true rival system. Then in what way is free logic different from classical logic? If I am forgiven a neologism, I would say that free logic is an enlarged version of classical logic, in the sense that free logic's range of applicability is larger than that of classical logic, as the former admits of empty names, and allows them some very limited role, and the latter does not. Whafs more, free logic by delimiting the role of empty names to unquantified atomic sentences, seems to make a real effort, one is tempted to say a self conscious effort, to stay as close as possible in the domain of classical logic. And this is how it ought to be, since free logic was not developed in order to correct any defects, real or imaginary, of classical logic, but only to allow limited use of empty names. There is a glaring exception, however, and that is Fregean free logic. Both positive and negative free logics adhere faithfully to the principle of bivalence of classical logic, as they both stipulate that all unquantified atomic sentences be either true or false. Not so Fregean free logic. The latter stipulates that such sentences are neither true nor yet false, but simply meaningless. This stipulation introduces to the picture a logical gap which, it seems to me, renders Fregean free logic a branch of many valued logic. As such, Fregean free logic is a true rival to classical logic. Since my intention is to emend free logic by removing certain highly paradoxical results, that are part and parcel of it, and not to create a rival system, I will henceforth ignore Fregean free logic and concentrate entirely on positive and negative free logics, with emphasis on the former.
D. THE PROOF THEORY OF FREE LOGIC
By what we have already said in the last section, free logic is practically identical to classical logic save for the former's enlarged range of applicability. Hence its proof theory, and consequently also its model theory, is, for the most part, the same as its counterpart in classical logic. What differentiates the proof theory of free logic from the proof theory of classical logic, are the explicitly stated existential conditions in the former in order to accommodate the problems posed by the admission of empty names which is the hallmark of free logic. Two absolutely basic laws in the proof theory of classical logic are: the law of universal instantiation (U.I.), Vx Fx |- Fa, and the law of existential generalization (E.G.), Fa |- 3x Fx. These very same laws, suitably amended in order to exclude nonexistents, are also used in free logic: {VxFx,3x(x = a)} |- Fa (R.U.I.) { Fa, 3x (x = a)} |- 3x Fx (R.E.G.) where R.U.I, stands for restricted universal instantiation, and R.E.G. stands for restricted existential generalization. The new term 3x (x = a) is an existential condition whose only role is to exclude the use of empty names in free logic derivations. The above two can be written in greater generality as axiom schema as follows: { Vx m, 3x (x=a)} |- m, (R.U.I.)
{ 9% 3x (x=a)} |- 3x3*x (R.E.G.) The rest of the axioms of classical logic, such as substitutivity of identicals, the various distributive laws, the laws of quantifier negation, the laws of quantificational confinement, the transitive laws of conditionals, etc., carry over as they are from classical to free logic.16 In classical logic it is a trivial derivation, one that substitutes ~F for F and uses the law of contraposition, that (E.G.) and (U.I.) are equivalent. Availing oneself of these two devices, but also the Deduction Theorem (D.T.) and the Inverse Deduction Theorem (I.D.T.), since the presence of the term 3x (x = a) complicates a little bit the derivation, it is an entirely routine matter to ascertain that (R.U.I.) and (R.E.G.) are also logically equivalent. The derivation in one direction follows. The derivation in the other direction is entirely similar.
DERIVATION JUSTIFICATION (1) { Vx Fx, 3x (x - a)} |- Fa R.U.I. (2) Vx Fx |- 3x (x = a) D Fa (D,D.T. (3) Vx ~Fx |- 3x (x = a) D ~ Fa (2),~F/F (4) Vx ~Fx |- Fa D ~ 3x (x = a) (3) CONTRAPOSITION (5) { Vx ~Fx, Fa } |- ~ 3x (x = a) (4.),I.D.T. (6) Fa |- Vx ~Fx D ~ 3x (x = a) (5), D.T. (7) Fa |- 3x(x = a)D~Vx~Fx (6), CONTRAPOSITION (8) {Fa, 3x (x = a)} |- ~Vx ~Fx (7), I.D.T. (9) { Fa, 3x (x = a)} |- 3xFx (8), QUANTIFIER NEGATION
Thus we have (R.U.I.) |- (R.E.G.). In the above derivation, the term 3x (x = a), which serves to confine the derivation to existents only, plays a crucial role. This term explicitly uses the identity sign. Thus this derivation is a sound one only for a free logic with identity. The question naturally arises as to what happens when we are dealing with a free logic without identity. This problem can be tackled in two stages.17 In the first stage we introduce a new primitive term, "E!". How does the new term differ from the traditional existential quantifier "3"? 3xFx reads, "there exists an V predicated by '¥'". E!x Fx reads "Some V predicated by 'F' exists". The, admittedly subtle, distinction is that "E!..." is the existential condition that assures us that whatever follows E! is an existent. In free logic, the term E! applies mainly to names. "E!a" says in effect that "a" refers to some existent and is not an empty name. Thus in free logic df with identity we have "E!..." = "3x (x = ...)". So in free logic with identity "El" is a defined rather than a primitive term. Given the above, the laws of restricted universal instantiation and of restricted existential generalization take the form:
{VxFx,E!a}|- Fa (R.U.I.)
{Fa,E!a} |- 3xFx (R.E.G.) 27
Once again, using our old standby, the Deduction Theorem, the above can be written in a slightly different form which, sometimes, is more convenient for derivations.
E!a |- (VxFx D Fa) (R.U.I.)'
E!a |- (FaD3xFx) (R.E.G.)'
What the first tells us is that if "a" exists, then from the assertion that everything is predicated by "F" we can derive that also "a" is predicated by "F". The meaning of the second is entirely analogous. It is quite possible to start working in a system of logic without identity, one that uses "E!..." to express existential import, and then, for whatever reason, we may decide to enrich the system by introducing at some point the notion, and the symbol, of identity. In such a case we would like to be able to derive the logical equivalence, "E!..." = "3x (x =...)". This can be done. To show that |- E!a D 3x (x = a), and hence that the laws of restricted universal instantiation and restricted existential generalization using E!a, can be replaced by the same laws using 3x (x = a) we work as follows: We define the predicate "F ..." as "... = a". This gives Fx = (x = a), and, substituting "a" for "x" Fa becomes the reflexive law of identity, a = a.
Then we have the following derivation:
DERIVATION JUSTIFICATION
(1) E!a |- (Fa D 3xFx) (R.E.G.)' (2) E!a |- ((a=a) D 3x (x=a)) (1), DEFINITION of F (3) { E!a, a = a } |- 3x (x = a) (2), I.D.T. (4) (a = a) |- (E!a D 3x (x = a)) (3), D.T. 28
(5) |- (a-a)D(E!aD3x(x = a)) (4), D.T. (6) |- (a-a) REFLEXIVITYOF IDENTITY (7) E!a D 3x (x = a) (6), MODUS PONENS
So, indeed, E!a |- 3x (x = a) as desired. The derivation of the reverse follows The same strategy using the law of the indiscernibility of identicals (identicals share the same properties), or (a = b) |- (Fa D Fb) and letting the predicate "F ..." stand for "~E!...", or that"... does not exist", or, should the reader harbor any Kantian aversions about using the existence or non-existence of something as a predicate, "~E!..." can be read as "... is a fictitious entity". I will omit the derivation as I do not see what possible reason anyone can have to begin with a free logic with identity and then go to a free logic without identity, where of course we would want to have 3x (x=a) |- E!a. There is something profoundly disturbing about the term "E!...". In a free logic without identity this is a primitive term, which makes it look that free logic without identity is a rival system to classical logic which does not use this term, while in a free logic with identity this is a derived term, making free logic with identity a non rival system. It would be, therefore, highly desirable to be able to develop a free logic without identity, one that dispenses altogether with the worrisome term "E!...", which leads us naturally to the second stage. Such a system was developed by Lambert, which not only makes use of neither "E!...", nor yet "... = ...", but, also, as a most welcome bonus, does away with names in favor of variables. Consider the law of restricted universal instantiation. It says that every value of the universally bound variable has all the properties shared by all such values. Thus the law of restricted universal instantiation can be written as |- Vy (VxFx D Fy). Similarly, the law of restricted existential generalization can be written as |- Vy (Fy D 3xFx). And this brings to a close the outline for a proof theory of free logic. E. THE MODEL THEORY OF FREE LOGIC
As already stated, the founders of free logic made a real effort to develop their subject so that it would be as close as possible to classical logic. Their program was a conservative one in that in free logic we can not derive any theorems that are not already in classical logic, and certainly there are no results in free logic contradicting anything that there is in classical logic. Now first order classical logic is semantically complete. That is to say that those sentences and only those which are valid can be proved. It should come then as no surprise that free logic too is a semantically complete system, whose metalogic mimics closely that of classical logic, although we won't try to give the proof of this result. In order for a logical system to be semantically complete it is essential that the results which are derivable within its proof theory be also valid within its model theory, and vice versa. Given that the proof theory of free logic was developed chronologically well before its model theory was, it was inevitable that the proof theory of the system dictated the particular form that its model theory assumed. Thus every sentence that can be derived from true sentences using the proof theory of free logic must be valid, otherwise the system would not be sound. In such a case we would be able to prove too much. On the other hand, every valid sentence of free logic ought to be derivable within its proof theory, otherwise the system would not be complete. In such a case we would not be able to prove enough. In conclusion, in both free as well as in classical logic we have T |- A if and only if T |= A, or that the derivable sentences in either system and the valid sentences determine the same class. Given the above, the rather idiosyncratic way that the model theory of free logic was developed is in reality quite predictable, as no other way would make the model theory of free logic compatible with the proof theory of the same. Compatible in the sense that, when combined, the proof theory and the model theory constitute a semantically complete system. 30
The models of free logic contain two domains instead of the one of classical logic, the inner and the outer. A typical model is an ordered triple
OBJECTUAL SEMANTICS FOR FREE LOGIC
Not every fictional character has a name. Thus while we know the names of all three Furies we know the name of only one of the three Gorgons, the other two being nameless. If we have any hope whatever of being able to study literature, mythology and other related subjects whose stock in trade are non existents, then the need for the development of objectual semantics is pressing. Whafs more, given that we already have a very satisfactory substitutional semantics, the transition to objectual semantics ought to be fairly routine. One possible objection ought to be put to rest. In substitutional semantics, one may object, we use names, some of which are referring, in which case they play a role in deciding the truth value of quantified sentences, and some of which are empty, and hence play no role whatever. We can easily have names for non existents as well as existents, seeing that names are a matter of stipulation. In objectual semantics, on the other hand, we do not deal in names but in objects. So how can an object be a non existent? I will be not just pedantic but pedantic to a tiresome degree. In symbolic logic, as the name implies, we deal neither with objects, nor yet with names, but simply with symbols. We deal with three kinds of symbols to designate predicable entities, bound variables such as x, y, z, free variables such as u, v, w, and finally constants such as a, b, c. All these are just symbols, pure and simple; all else is interpretation. Thus we use bound 31 variables to lie within the scope of quantifiers in quantified sentences. These are the least interpreted of the lot; the most we can say about them is that they represent nouns of some kind or another. Free variables and constants, on the other hand, are specific kinds of nouns. Free variables stand for objects and are to be found in objectual semantics, while constants stand for names, and are to be found in substitutional semantics. Both are used to instantiate quantified sentences. But, after all has been said and done and even instantiated, all three are just symbols. And symbols are only stand-ins for other things, which is to say symbols are names. So whether we deal with objectual semantics or substitutional semantics we never deal with objects per se but only with names. In the first instance we deal with names which are names of objects, and in the second we deal with names which are names of names! Thus "Medusa" is the name of some entity, but when we use the lower case letter "a" to instantiate some quantified sentence for "Medusa", then "a" is the logical name for the being whose mythological name is "Medusa". Or, " 'a' " is the name of a name! Given this, the means we will adopt to develop our objectual semantics for free logic should be fairly obvious. We shall use two kinds of subscripted letters to denote objects. We shall use ulr u2,... etc. to denote existing objects, and wu w2,... etc. to denote non existing objects. And in order to keep our subject metaphysically neutral, I must hasten to add that I do not employ the term "non existing objects" in any Meinongian sense. The free variables wx, w2,... etc. are at most dummy symbols, and seeing that non existents can have no properties but at most names, wa, Wj,... etc. differ not at all from the names of non existents of substitutional semantics. The only difference between w1 and al7 say, is that the former is the symbol of a non existent, while the latter is the symbol of an empty name. And even if there were some confusion between these two it would not in the least impair our ability to construct models, as empty names (which are names attached to non existents) play no role whatever in the validation or invalidation of quantified sentences. 32
Once again a typical model looks like **• = < Dh D0, f > , but this time the inner and outer domains are no longer inhabited by referring and empty names respectively, but are the referents of variables that range over existing and non-existing objects, respectively. Once again the substitution of a free variable "u" for the bound variable "x" is denoted "u/x", the same way as the substitution of the name "a" for the bound variable "x" is denoted by "a/x". In classical model theory we use vectors to denote arrays of the elements of the domain,
a = < av a^... > where the a's are names, i.e. constants, when we are
in substitutional semantics, and u = < ulr \xv ... > where the u's are free variables when we are in objectual semantics. In free logic model theory we will use vectors once again to denote arrays of elements in the inner and outer
domains. For the inner domain a = < alr a^ ... > and u = < ux, u2,... > same as before. Arrays for names and objects in the outer domain will be
denoted by \y =
The vectors b and w of the outer domain are really only a formality done to give the system an aesthetically pleasing symmetry, as the members of the outer domain play no role in assigning truth values to quantified sentences. We are now in position to give the formal laws for the model theory for free logic using objectual semantics.
We define *^ |« Fx as follows:
(1) **• |= x = y iff x is the same element as y.
n (2) 21 |= Rn (Xi/ S2,..., xn) iff < njxv u2/x2,..., un/xn > € R 33
N.B. Here Rn is an n-ary relation and not the n-dimensional space whose coordinates are real numbers.
(3) *•*• \m ~Fx iff we do not have "•*• |=» F
(4) 21 |B F A G iff 21 |„ F and 21 |- G
(5) 2t |= p v G iff 21. |» F or 21 |„ G
(6) 2t |= FZ)G iff either 2t |„ „F or 21 |= G
(7) 21. |= FsG iff 21. j0 pDG and 21. |„ GDF
(8) *-*• J=» VxFx iff for every element u; in Dj there is some preimage Xj of
Uj under^such that **~ |« F x (Uj/xx, u2X2,...)
(9) ^- |=» 3xFx iff there is at least one element u; in Dj with preimage x;
7 under J such that *^~ \- F x (xlr x2,... \xjx{,...) Since F and G are any predicates we care to mention, the various axioms are in reality axiom schemas. All the axiom schemas are as before, but we replace F and G by the fancy upper case letters 3^and Q . And this takes care of the model theory of free logic.
F. FREE LOGIC AND THE THEORY OF DEFINITE DESCRIPTIONS
In an article that is definitive for free logic titled "Existential import, 'E!', and 'The'" Lambert makes two thought provoking assertions which on no account must be ignored. The first one is that Russell's theory of definite descriptions engenders two highly paradoxical results, the one being that it turns non existents into existents and the other that it leads to the 34 identification of all existents, or that two are really one. The Eleatic philosophers and all their ilk would have loved this but scarcely anyone else. The second assertion is that, because of the above, free logic and the theory of definite descriptions are incompatible.19 I totally disagree with both these assertions. Seeing that my overall intent is to emend the currently existing system of free logic to one that avoids certain infirmities that plague the standard system of free logic, but also to make sure that the new system is both sufficiently strong and flexible to handle non existents in a way that respects the context in which they occur, and at the same time to differentiate between existents and non existents, it is imperative that this thorny issue raised by Lambert be resolved. The reason is that in the proposed system the theory of definite descriptions will play an indispensable role in translating, in symbolic language, sentences encountered in literature in which non existents occur. If the theory of definite descriptions is incompatible with free logic then it can not be grafted onto the latter, and if it engenders false sentences then it can not be part of any system of logic. I shall begin by presenting an outline of Russell's theory.
1. RUSSELL'S THEORY OF DEFINITE DESCRIPTIONS
The starting point in Russell's theory is the observation that the superficial grammatical form of sentences can be highly misleading as it may serve to obscure the real logical substratum of the sentence which may lie underneath. Consider the following sentence: "The current king of France is a criminal." Superficially this seems to be a typical subject-predicate sentence, "criminal" being the predicate predicating the subject which here is "the current king of France". In symbolic form this translates as Gx with "G" being the predicate and "x" being the subject. While this looks reasonable, it creates problems. Every assertoric sentence must be true or false, and the ascertainment of this sentence's truth value must be a fairly routine matter.20 We work as follows: We request the French government to send us a list of all its residents, those having a criminal record, their names accompanied by their vocation. On scanning this list we fail to identify any of 35 them as being king. So we must assume that the sentence is false, and, consequently, that the king of France is not a criminal. We then request that a second list be sent to us, one similar to the first, only this time it must contain the names and the vocations of all residents with no criminal record. We scan this list too, fully expecting to find here the king of France, but to our consternation we fail to do so. The only conclusion is that the sentence "The king of France does not have a criminal record" is false and, hence, that the sentence "The king of France is a criminal," which we found to be false, is true after all. The disquieting conclusion is that we have on empirical grounds both V (Gx) = ^and also V (~Gx) = f, contrary to all expectations. The error we committed consists in that we treated such statements as the above as simple assertoric subject-predicate sentences, which is far from being the case; these statements rather than being subject-predicate sentences are instead definite descriptions, as is every statement beginning with the definite article "the". The difference between a subject-predicate sentence and a definite description rests on the fact that the former asserts one thing while the latter asserts three distinct things: that a certain entity exists, that it does so uniquely, and, finally, that it has some given property. Symbolically we represent a definite description as follows: 3x (Fx A Vy (Fy Dy = x) A Gx) In case "F ..." stands for "... is the current king of France," and "G ..." stands for "... is a criminal" then the above reads "there is a current king of France, there is only one king of France, and he is a criminal." The definite description can be abbreviated as 3x Vy (Fy Dy = x A GX). It can be further abbreviated by writing instead G (ixFx) which reads, "the 'x' predicated by "¥ has property 'G'." In Principia Mathematica Russell and Whitehead used the inverted Greek lower case iota to denote "the"; the best we can do is denote it by "i". As we have seen, a definite description is a tripartite conjunction, the first conjunct of which is an existential clause (3xFx); the second conjunct is a uniqueness clause Vy (Fy D y = x); and the third conjunct is a predication. Now it is quite easy to see what went wrong with our two statements concerning the king of France. Such a person does not exist, France no longer 36 being a monarchy. So the statement "The current king of France is a criminal" is false, not because the current king of France does not lie within the extension of the predicate "is a criminal" but rather because such a person does not exist, and so does not lie within the extension of any predicate, "criminal" or "not criminal". It is instructive to examine the correct way of negating a definite description. And that is not to say "the current king of France is not a criminal" for this is not the correct negation of "the current king of France is a criminal" as we already saw. The correct way, even though it sounds somewhat strange, is to say "it is not the case that the current king of France is a criminal." In symbolic language the former is translated as 3x (Fx A Vy (Fy D y = x) A ~ Gx ) which is patently false as it asserts that there is a unique king of France who is not a criminal. The only correct way to translate the latter is ~ 3x ( Fx A Vy (Fy D y = x) A Gx ). This, upon instantiation by u/x, and subsequently using De Morgan's law, becomes: ~ Fu v ~ Vy (Fy D y = u) v ~ Gu. So when we negate a definite description, what we finally end with is a triple disjunction, any of whose disjuncts may be false, and this is what we would expect. In the incorrect way of negating a definite description we placed the tilde sign somewhere within the definite description. In such a case we say that the negation has "narrow scope". In the correct way of doing the same we placed the tilde sign in the front of the entire definite description. In this case we say that the negation has "wide scope". And this is the correct way because for a conjunction to be true all its conjuncts have to be true, and consequently the negation of any one of these conjuncts falsifies the entire definite description. Given this it is easy to see what happens when we negate the uniqueness clause. In such a case we have more than one entity sharing in the predicate in the existence clause, and therefore we must use the indefinite article "a" instead of the definite article "the". As is well known, Bertrand Russell went to jail towards the end of WWI for his pacifist activities. Let "F .. ." stand for "... has authored Principia Maihematica" and "G ..." stand for ".. 37
. went to jail". Seeing that it was Bertrand Russell in cooperation with Alfred North Whitehead who coauthored Principia Mathematica it would be false to assert that "the author of Principia Mathematica went to jail" as this is a definite description and it implies uniqueness of authorship. The correct was is to say "An author (or one of the authors) of Principia Mathematica went to jail." This can be translated somewhat inaccurately as 3x (Fx A GX), which fails to tell us that there were exactly two authors. Far better, even if more laborious, is to write: 3x 3y (Fx A Fy A X * y A VZ ( Fz D (z-x v z=y)) A Gx). Finally, and this is a rather amusing demonstration of logician's pedantry, even to say that "the king of France does not exist" is false as it asserts the existence of someone who does not exist. The correct way is to say that "it is not the case that there exists a king of France."
2. THE DERIVATION OF THE ALLEGED PARADOXES
In symbolic logic all expressions such as "Fx" are subject-predicate sentences, with "x" being the subject and "F" being the predicate irrespective of what is the content of "F". In symbolic logic it is the form rather than the content that is the deciding factor. Thus "F" can take a truly vast range of interpretations. Thus if we interpret "F ..." to mean "3x (x = ...)" then "Fa" or "3x (x=a)" is a predicate notwithstanding all Kantian protests to the effect that existence is not a predicate. And this because in this case "F" when followed by "a" tells us something about "a", that it is an existent. So let us consider cases of predication which are bona fide cases but still a little unusual. Consider the definite description "G (ixFx)". What it says is that the uniquely existing "x" having property "F", and here "F" is some property possessed by a single existent, also has property "G". A typical case might be "the man who is talking to our host is an accountant". But now let "G (ixFx)" stand for "the uniquely existing x having property "F" does in fact have property "F". 38 df We can write this as Gx = (x = iy Fy) or even F (ix Fx). In ordinary language we encounter this meaning of "G" when we want to stress the fact that some "x" has indeed property "F", for emphasis. In this case the above when written out in full takes the form 3x (Fx A Vy (Fy D y = x) A Fx), which when instantiated for u/x becomes Fu A Vy (Fy D y = u) A FU, and by the commutative law of conjunctions and also P A P |- P, becomes Fu A Vy (Fy D y = u). By existential generalization we have Fu |- 3x Fx. So finally we have derived the troublesome: F (ix Fx) |- 3x Fx and F (ix Fx) |- 3xVy (Fy D y = x). These two read: From the sentence "the V predicated by '¥' is predicated by '¥'", we derive both the existence of such an "x" as well as its uniqueness. But what happens when "F ..." stands for "... is a Gorgon"? Then we derive, at least according to Lambert, that if "the Gorgon is a Gorgon", which is quite a reasonable thing to say, then Gorgons exist, at least one of them. The second is very much more troublesome as can be seen when we let "G" be as before, but this time we let "F ..." stand for "... = ...". Here "F" is the relation of self-identity which everything has by the reflexive law for identity. This implies VxFx, or that everything is self identical. Then the definite description "the self -identical thing" becomes 3x (x = x A Vy (y=y D y=x)), and the claim that it is self-identical is then the claim that there is some entity to which everything self identical is identical. Now Parmenides, Zeno, Spinoza and Bradley can all rest in peace! This is Lambert's claim and from this follows that any sound system of logic, not just free logic, can have no truck with a theory that engenders such absurdities. Our work is now cut out for us. There are two things we have to do. The first is to investigate whether Russell's theory of definite descriptions does in fact engender these two paradoxes. The second is to enquire whether free logic as has been developed so far is incompatible with the theory of definite descriptions, in which case these two can not be used together. 3. THE ALLEGED PARADOXES OF RUSSELL'S THEORY
There are two alleged paradoxes: that the theory of definite descriptions has as consequences, first, the existence of non existents and, second, the identity of non identicals. We shall treat them in this order. The first one is very easy to dispose of. The complaint is that from G (ixFx) we get 3xFx regardless of what "F" stands for. This is clearly not the case. A definitive description written out in detail is 3x (Fx A Vy (Fy D y = x) A Gx ), which by first instantiating and subsequently generalizing gives 3xFx. In order to be able to do these, however, there must be something predicated by 'F', so from the theory of definite descriptions we can derive 3xFx only in case the extension of F is not empty, which is exactly what Russell maintained and declared in the most uncompromising terms to begin with. As he said, "the V which is so and so has properties such and such," means, among other things, that such an V exists, and failing this we do not have a true description, as the example of the king of France should have made abundantly clear. The case, however, of the identity of everything requires closer analysis. The theory of definite descriptions allows the derivation of Vy (Fy Dy = x), but, once again, it does so because a definite description presupposes uniqueness which is embedded in G (ixFx). This is the very reason we call such a thing a definite description. Because it begins with the definite article 'the', which implies singularity, rather than with the indefinite article 'a', which implies plurality. If then the object predicated by 'F' is unique it should come as no surprise if any other object predicated by the same 'F' should turn out to be the original object. So far so good, but still a problem remains. And that is when 'F' is the predicate that whatever follows it must obey the reflexive law of identity. That is to say the case, 3x (x=x A Vy (y = y D y = x)). Here we have two distinct subcases. The first such subcase is the one according to which there is just one object in the universe. In such a case we can legitimately derive from the 40 above definite description the sentence Vy (y = y Dy = x) which here is entirely acceptable. After all, if, counterfactually speaking, there is just one object in all the plenum, then any other putative entity that we may bring to mind must be identical to that one object. The derivation therefore is legitimate, but, as we shall presently see, this is the only case in which it is legitimate. The second subcase occurs when there are at least two distinct, i.e. non-identical, objects in the universe. There may well be more than two which is in fact the case, but we may restrict the argument without loss of generality when we have just two objects. If we accept the reflexive law of identity - and we will - then any two objects one cares to consider are self identical. Observe now that since we consider two objects we no longer have a definite description and therefore Lamberf s objection is not valid. In particular, we do not have 3x (x=x A VZ (Z=Z D Z=X) A ...) for the very simple reason that now there is a second object, call it 'y', which though self- identical is distinct from 'x', and therefore we no longer have something as Vy (y=y D y=x). What we do have instead is the sentence 3x By (x=x A y=y A ~ (x=y) A Vz (z=z D (z-x v z=y))...) From the above we can legitimately derive Vz (z=z D (z=x v z=y) which is not the same thing as a uniqueness clause. And this because 'z' can not be identical to both 'x' and 'y', the clause ~ (x=y) blocking this derivation. In the absence of this clause there is nothing to prevent the two object universe to collapse in a one object universe. The same argument holds good for three or more objects. Thus we see that the theory of definite descriptions does not engender fallacious consequences, and Lambert's objection is groundless.
4. THE ALLEGED INCOMPATIBILITY BETWEEN FREE LOGIC AND THE THEORY OF DEFINITE DESCRIPTIONS
Once again we are confronted with two distinct cases depending on whether we deal with existents or with non existents. In the case when we are dealing with existents, free logic treats them in exactly the same way that 41 classical logic does. The theory of definite descriptions has been developed totally within classical logic and since, at least for existents, classical logic is compatible with free logic, then so is the theory of definite descriptions. Here I must add that, for this reason, if the theory of definite descriptions had been found to be wanting, as Lambert believed, then so would have been free logic. So let us look at the case of non existents which have created all the thorny issues that led scholars to develop free logic. Here we have two distinct systems of free logic to contend with, negative and positive free logic. Let us take them up in this order. Negative free logic's most conspicuous characteristic is that, in the case of non existents, every atomic sentence containing one of them is false on principle. Thus if "F ..." stands for "... is a Gorgon" and if "a" stands for "Medusa" then the statement, "Medusa is a Gorgon" is false, or V (Fa) =f. And this holds axiomatically for all predicates without exception so far as negative free logic is concerned. But then, in the case of non existents, if V (Fx) =$ then also V (VxFx) =yand also V (3xFx) =f Seeing that a definite description is a conjunction consisting of existentially and universally quantified sentences, in the case of non existents all definite descriptions are false. So the theory of definite descriptions does not conflict negative free logic. Coming finally to the subcase of positive free logic, this latter is more tolerant of non existents than negative free logic is. It admits that such atomic sentences as "Medusa is a Gorgon" can be true in certain contexts. But this tolerance does not extend to quantified sentences and hence this subcase reduces to that of negative free logic.
G. INFIRMITIES OF FREE LOGIC
Free logic, positive as well as negative, as it has been developed so far, is a deeply flawed system, one in need of radical surgery if it is to be salvaged. While we may applaud the efforts of free logicians to make explicit hidden existential assumptions of classical logic, and rightly so, still, after all 42 has been said and done, the final product is profoundly unsatisfactory for two reasons. First of all, free logic, by restricting so severely the role to be played by empty names as she does, fails quite grievously to respect the various contexts within which these are encountered. This surely delimits the applicability of free logic to situations where only referring names occur. If so, one may as well stay within classical logic with its blanket prohibition of all empty names, for indeed free logic as it stands can do no more than classical logic does. But this is only a secondary complaint, the main one being that free logic is the source of results which are highly controversial, totally counterintuitive, and even out and out fallacious. Let us begin with negative free logic which is the primary object of my wrath, as in addition to all the flaws of positive free logic, it contains some exclusively her own. Negative free logic begins by postulating that all atomic sentences, with no exception, that contain empty names are false. This is understandable. After all, only things that exist can have real properties. Thus to say that Medusa is dangerous to us is false since there is no such creature. Unfortunately, this policy creates some very thorny problems. First of all this principle, far from respecting the context of mythology, actually deprives it of all sense. Imagine somebody who is just beginning to study mythology and encounters the statement, "Medusa was a Gorgon". On hearing a negative free logician claim that this statement is false, he may not know how to interpret it. Could it mean that Medusa does not exist, or does it mean that Medusa was something else? A Cyclops, perhaps, or even a Centaur? The above statement is really a convert definite description which can be read as "there is a unique being called 'Medusa' which is a Gorgon". Here there seems to be some ambiguity as all three clauses in the description - the existence clause, the uniqueness clause, and the predication - are false. Here, however, all we have to identify the subject is that she is a Gorgon, and if this is false, how can we say that "Medusa" does not exist? The second strike against negative free logic is that the unrestricted stipulation that all atomic sentences containing empty names are invariably false, is in the final analysis self defeating. "Medusa" being an empty name, then every atomic sentence containing this name is deemed to be false. If "a" 43 stands for "Medusa" then for all predicates "F" we have V (Fa) = f. And this holds for "F" standing for "can turn humans into stone," or "has snakes growing on her head," or "is a malevolent creature," etc. But how about when we want to say that it is not the case that Medusa exists? We can write this using the simple subject-predicate form as "Fa" where "F ..." stands for "... is a fictitious entity". Then V (Fa) =^rwhich is exactly what all free logicians would want to avoid. When we write this out as a definite description it becomes positively absurd. It reads "there exists something unique called "Medusa" which is a fictitious entity". Then if the predication is false, there does exist something called "Medusa". Again let "H ..." stand for "... is an existent". Then V (Ha) = f but Ha = ~ Fa which implies that V (~Fa) = f. Combined with the above this gives V (Fa) = t which contradicts V (Fa) = f. So negative free logic is, on top of everything else, inconsistent. It is worth the time to examine this claim a little closer. First of all, negative free logicians assign a truth value of false to atomic sentences. Presumably, more complex sentences receive a truth value by using the ordinary laws of logic in such a way that the truth value of the atomic sentence within is not violated. So if V (Fa) =j then V (~ Fa) = t. Second, the predicate that assigns a name to something tells us precious little about that something. All alone it does not tell us if it exists or not. Thus if I say, "something is a Gorgon" then we know that that something does not exist as there are no Gorgons. If, on the other hand, I say "something is named 'Medusa'" then by itself this tells us next to nothing, names being arbitrary assignments. Thus if I say, "there exists something called Medusa, and this is not a fictitious entity," then this is a true definite description, even when what I have in mind is the mythological creature with the same name. Now observe that in the above two paragraphs we entered the domain of second order logic. We talked about all predicates predicating empty names. This could be written as V (( VF : F is a predicate) ( 3x : x is a non existent) (Fx)) =f. But this too is a predicate. It tells us, in the metalanguage, that all predicates when predicating something non existent give a false 44
sentence. The above could be written as G (Fx), where "G ..." stands for "... has the truth value 'false'.." Of course "G" does not predicate "x" itself but "F" and it is not a sentence in the object language but in the metalanguage. Still, according to negative free logicians, V (G (Fx)) = t whenever "x" is a non existent. This is something that negative free logicians ought to examine carefully. It may turn to be something of little account, but stall it ought to be examined. Finally, what I find totally unacceptable in negative free logic is that for the case some "x" turns out to be a non existent, then the sentence x=x is false. And what does this do to mathematics? If numbers do not exist the way doors and chairs exist then something like 3 = 3 is a false sentence. But do numbers exist? Only Platonists will subscribe to this view. Intuitionists believe that numbers have some sort of shadowy existence but certainly not an independent existence. For Intuitionists numbers arise from the consideration of the temporal sequence of our thoughts. Thus when I say "now", this "now", being unique, implies "one". But this "now" immediately recedes in the past, and I am presently faced with a new "now", and thus I have "two", and so on.21 Logicists, on the other hand, believe that numbers are but extensions of concepts. Thus "zero" is the extension of the concept of all contradictions, there being nothing which is self contradictory. But if we apply "zero" to all contradictions then this "zero" is unique. And thus we have the number of all "zeroes', and this number is "one", and so on.22 Finally, formalists believe that all mathematics is nothing more than the manipulation of formal rules which are a priori meaningless.23 Thus if we ask four philosophers whether numbers exist we will get four different answers. The Platonist will say "of course they do"; the intuitionist will say "they exist in some peculiar way and they depend for their existence on the passage of time"; the logicist will say that "numbers are only the extension of concepts"; and, finally, the formalist will say that our question does not make any sense at all. But enough with negative free logic. Time to have a look at positive free logic. There are only two objections here, but they are absolutely devastating. The first is that when we make the transition from simple quantified sentences to more complex such, patently false sentences are made to be true. The second is that true sentences, at least true in a certain context, are made to be false. Let us consider the sentence Vx (Fx D Gx). Let us assume that "F ..." stands for "... is a Gorgon" and also that "G ..." stands for "... is a chartered accountant". This sentence, "all Gorgons are chartered accountants" is false both in the context of mythology but also in real life. The only saving grace of such a statement might be that chartered accountants turn people into stone by their boring conversation. Gorgons, however, turn people into stone by their gaze, not at all the same thing. But how does this sentence fare in positive free logic? In free logic, positive as well as negative, all quantifications are supposed to have existential import. So, at first sight it seems that this sentence is ill formed. But wait. True enough, we can quantify over the simple atomic sentence "Fx" only in the case "x" has existential import. But what lies within the scope of Vx (Fx D Gx), namely "Fx D Gx" is not an atomic sentence, but instead a material conditional. Thus if "Fx D Gx" happens to be true, and this holds good for every "x", then Vx (Fx D Gx) will also turn out to be true, even if in both fact and mythology it is false. To do this let us use some set theory. Consider the extension of "F". This is the set of all Gorgons. How many such exist? The answer is none. Therefore the extension of "F", (all Gorgons) is the empty set. Now does there exist some member of the set of all Gorgons, which is not a member of the set of all chartered accountants? The answer is no. Therefore every member of the set of all Gorgons is also a member of the set of all chartered accountants. Hence all Gorgons are chartered accountants. This is a trivial application of the fact that the empty set is vacuously a subset of every set. So in the case of universally quantified material conditionals positive free logic turns false such into true ones. The same goes for negative free logic and more easily as V (Fx) -j. The reverse happens in the case of existentially quantified conjunctions. Let "F ..." stand for "... is the city of London in England" and "G ..." stand for "... one of its residents is Sherlock Holmes". Then 3x (Fx A GX) reads, "there is an entity which is the city of London in England and one of its residence is Sherlock Holmes". By using an argument analogous to the previous one we see that the above is a false sentence, which while true in fact fails to do justice to literature. All this shows that free logic as it stands fails to perform its assigned task unless it is emended. The emendation I propose to perform by developing a new version of positive free logic, one that uses insights and techniques from a branch of logic called "many-sorted logic". I will call this new logic "many-sorted free logic" and the free logic seen so far I will call "standard free logic" to distinguish between the two. First, though, I will present a very short outline of "many-sorted logic".
IV. MANY SORTED CLASSICAL LOGIC
A. INTRODUCTION
Let us now put free logic on the back burner for the time being, and let us have a closer look at classical logic from a rather novel and most flexible point of view, that of "sorts". Having done this we will return to our old friend, free logic, which we will develop ab initio using insights and tools gleaned from the "sortal" study of classical logic. To distinguish between these two ways of doing free logic, the already existing one and the new one developed by me, I will call "standard free logic" the old one, and the new one I will call "many-sorted free logic". Let us look at a typical universally quantified atomic sentence, Vx Fx. It claims that everything has a certain property, and this is such a sweeping assertion that very few predicates will validate this sentence. In fact the only such predicates that come to my mind are the logical tautologies, such as "Fx" stands for (x=x). For the existentially quantified sentence 3xFx the opposite holds good, and what invalidates this sentence is any logical contradiction such as "Fx" standing for ~ (x=x). From this it is quite clear that quantified atomic sentences are only a very small part of the predicate calculus, the largest part of which is concerned with the study of quantified compound sentences, such as Vx (Fx D Gx) and 3x (Fx A GX). The reason that such sentences are more useful is because the presence of two, instead of one, predicates restricts the range of discourse to more manageable proportions. This way the chances of arriving at a true universally quantified sentence, or a false existentially quantified sentence increases appreciably. Thus while we can not say that everything is yellow, we can certainly say that all lemons are yellow, and also that there is something yellow which is or isn't a lemon.
B. UNI-SORTED LOGIC
In quantified compound sentences what one of the two predicates accomplishes (the antecedent in universally, and one of the conjuncts in existentially quantified sentences) is to restrict the allowable range of the variable to some specific class of entities. Intuitively speaking we may say that this predicate assures us that the range of the variable will respect the context in which the overall sentence falls under. Thus the statement "all equilateral triangles are equiangular" is one which we would naturally expect to encounter in geometry. Therefore we can safely assume that the variable ranges over planar figures rather over lemons and yellow objects. This is the starting point in the development of uni-sorted logic, the key idea. Unisorted logic is just classical logic but with a slight modification in symbolism, one which makes this context respecting role of one of the two predicates stand out in relief. In the case of the universally quantified sentence Vx (Fx D Gx), where the entire conditional lies within the parentheses, we rewrite this by detaching the antecedent from the conditional, attaching it to the quantifier, and enclosing the lot in square brackets like so [Vx Fx]; the consequent follows enclosed in ordinary parentheses. The entire Vx (Fx D Gx) is rewritten as [Vx Fx] (Gx). Similarly, 3x (Fx A GX) is rewritten as [3x Fx] (Gx). 48
The significance of the square bracket enclosed expressions is that they choose the contexts within which the entire sentence will operate, that being lemons, triangles or what not. It ought to be obvious that this is still classical logic, with a slightly adjusted symbolism, and therefore every theorem derivable in the one system is derivable in the second. Since the square bracket enclosed expression sorts out in one specific class, one which is the extension of the predicate within it, all the objects that will be used in the instantiation of the quantifier, we call this way of writing out logical sentences "sortal logic". This, as we shall see shortly, allows for further development of classical logic, which endows it with great flexibility. In particular, when we deal with many predicates it is possible to sort out the various objects in several distinct classes, each of which is the extension of a different predicate. Quite appropriately, then, we refer to these extensions as "sorts". Since up till now we have sorted out objects in just one sort per sentence, we say that we are in the domain of "uni-sorted logic". When we sort out objects into two or even more sorts, then we say that we are in the domains of 'bi-sorted" and "many-sorted" logics respectively.24 Are there any advantages in using uni-sorted logic? So far as theorem proving goes, none whatever. But there is an advantage when we translate statements from ordinary human language to that of logic. And that is that in uni-sorted logic we define right from the start what exactly the context of discourse is going to be, something that we can not do in "un-sorted logic". Consider these two statements: "Some Buddhists are vegetarian" and "some vegetarians are Buddhists". In un-sorted logic they both translate as 3x (Fx A Gx), by virtue of the fact that conjunctions obey the commutative law. And yet these two statements do not say quite the same thing. In each case the context of discourse is different. The first is about adherents to a certain world view, while the second is about followers of a dietary practice - not at all the same thing. In unisorted logic the two statements translate as [3x Fx] (Gx) and [3x Gx] (Fx), which is more faithful to what we want to say. This benefit, however, real as it undoubtedly is, is rather thin and does not justify the recasting of the symbolism of classical logic in a new form. The real 49 benefit of introducing sorts in logic is that they allow one to develop many- sorted logic which has great flexibility and expressive power, and which incidentally will be of enormous service in developing an emended system of free logic.
C. MANY-SORTED LOGIC - INFORMAL DISCUSSION
Uni-sorted logic can be considered to be a degenerate case of many- sorted logic. As such it can functions a s a bridge linking unsorted logic and many-sorted logic. This latter is a logic in which entities are sorted out in more than one sorts. Before we go any further, let us consider three distinct examples, and let us discuss them. This will show that many-sorted logic is used on a practical level more often than one may think. The first is the telephone directory which comes in two volumes, the white pages and the yellow pages. The white pages makes a good example of an un-sorted logic, as every entry within appears in strict alphabetical order. The yellow pages is a good example of many-sorted logic, and this because the various entries within are first sorted out in different sorts. Thus churches are grouped together in one sort, law offices in a second, medical practitioners in a third, plumbers in yet another, and so on. Which of the two is more useful? This depends entirely on the situation. If we want to contact someone whose name we know but nothing else, we will consult the white pages. If, however, we need quite urgently to contact some professional, say a dentist, then we will consult the yellow pages. Absence of either of the two may complicate our life to an unreasonable degree. The second example is from classical logic which comes into two different forms, first order logic and higher order logic. When we first begin to study the subject, it is invariably first order logic. Here the quantifiers range over individuals. In higher order logic we have quantifiers ranging over sets of individuals, which sets are extensions of predicates, but also sets of sets of individuals, in which case we have predicates acting on other predicates, and so on. Bertrand Russell coined the name "types" to explain 50 this state of affairs. Type 0 consists of individuals; type 1 consists of sets of individuals; type 2 consists of sets of sets of individuals; and type n + 1 consists of sets whose members are of type n. In order to avoid the various paradoxes he stipulated that the expression x G y is admissible if and only if "x" is a member of a type one lower than the type of y. Thus in the theory of types, which has grown into a vast subject, we can legitimately write x€{xj€((x}).,. but never x Q {x} C {{x}} ... which differentiates type theory from set theory. It is quite clear that while first order logic is an example of un-sorted logic, higher order logic is an example of many sorted logic, one sort for each type.25 Our final example is from abstract algebra. Let us consider two typical structures we encounter in this field, that of "abelian groups", and that of "modules". All the elements of the abelian group are sorted together in one sort, and we can perform on any two of them the characteristic binary group operation. Thus the logic underlying the theory of abelian groups is un-sorted logic, or perhaps we had better say uni-sorted logic. By contrasts, the elements of a module are sorted out into two distinct sorts. The first sort is, with the appropriate binary operation and the necessary axioms, an abelian group. The members of the second sort form a "ring" and they can be multiplied with the members of the abelian group. Now that we have some significant examples let us tarry a bit and isolate some significant relations that obtain within sorts and between sorts, all in an informal way, the task of formalizing to be undertaken in the next few sections. Let us start with the theory of types which is the simplest of all due to the fact that this theory postulates an ascending hierarchy of all set theoretic objects. Each one of the latter must belong to a type, and, as we have already seen, it is impossible for an entity to belong to two distinct types. If "x" is a member of type n then it cannot be a member of type n + 1. The reason is that if "x" is a member of type n then "x" is a member of a type one lower than n + 1, and this is a necessary and a sufficient condition, which precludes it from being a member of any other type. In set theoretic language type theory is a 51 partition of all set theoretic entities. And if every type is a sort, then what we have here is a case where all sorts are mutually disjoint. The example of the phone directory is more interesting. First, not every name, of some individual person let us say, which is found in the white pages can also be located in the yellow pages. Also note that sorts in the yellow pages may occasionally overlap. Thus in some directories, the Chinese art of T'ai Chi Ch'uan and the schools that teach it may be located both in the yoga section and in the martial arts section. So the yellow pages do not constitute a partition. The most important thing, however, is that the yellow pages afford us our first significant case of several subsorts within a sort. If you look under the heading physicians, which is one particular sort among many, you will notice many subheadings, such as dermatology, cardiology, hepatology and the like. These are but subsorts in the more inclusive sort "physicians". Finally, let us consider modules. They are usually defined in such a way that the abelian group and the ring are two distinct sorts. But not always. Rings are themselves abelian groups. So when we use module theory to study rings we may take the same set, the ring to play a double role, as both an abelian group and also a ring. So it is quite possible that two sorts may be identical and yet each to play a somewhat different role.
D. THE LANGUAGE AND CONVENTIONS OF MANY-SORTED LOGIC
From now on we will be examining sentences involving many predicates, and so we will adopt a convention for distinguishing the one from the other. We will employ the upper case letter "F" indexed by natural numbers, such as ¥lr F2, ... etc. Similarly, the extensions of these predicates will be denoted by the upper case letter "X" indexed by the same subscript that is the index of the predicate whose extension it is. Thus x£Xj and Fxa will mean the same thing, and the one we choose to use will be the one we find most convenient. From what we said before, the extensions of two predicates may intersect or may be disjoint. This indexing convention, however, will not extend to variables. The subscript indexing a variable will 52 be used only to distinguish it from other variables, and by itself will not be an indication to what extension it belongs. Thus x1 £ X1 and x2 E X1, but also x2 £ X2 are all totally legitimate. Unlike predicates and their extensions, there will be no a-priori relation between the subscript of a predicate (or its extension) and that of a variable. Up till now, there is nothing in our conventions to distinguish standard classical logic from many-sorted classical logic. The next convention, however, is the one that will signify the transition, as from now on we will begin to index our quantifiers. We will be using the following abbreviations:
31 x for 3x FjX v\ x for Vx FjX
[3i x ] (F2 x) for 3x (Fj x A F2 X) 26 [VJ x ] (F2 x) for Vx (Fa x D F2 x)
Here, as can be seen, there is an identity of the subscripts we use for quantifiers, predicates and their extensions (the latter also called sorts). Note that when there is no risk of confusion we may omit the square brackets. There is one issue we will have to settle. Will the extensions of predicates be sets or classes, and, if sets, what kind? It is well known that not every set is the extension of a non trivial predicate, and also that not every predicate has a set for its extension. Let us consider the following: X = { carnation, "carnation", peacock, invoice, the square root of minus 1, Aristide Briand, the four aces card trick, your favorite joke}. This is a bona fide set, but it is hardly the extension of some predicate seeing that it is an entirely random collection. To say that "X" is the extension of predicate "F", and "F" is the predicate whose extension is just "X" will clearly not do, as such a definition is entirely circular. Let us now look at the reverse. Some predicates are predicable of themselves while others are not. Thus "... is an even number" is not predicable of itself, but"... is an abstract concept" is. Let us now define a 53 new predicate "I" meaning "it is not predicable of itself, or is impredicable". So for the first example we have I (F) while for the second we have ~ I (F). In symbols this is written as as I (F) = ~ F (F). But is the property of being impredicable itself predicable or impredicable? We have I (I) = ~ I (I), or if it is it isn't and if it isn't then it is. The extension of I is the set of all sets which are not members of themselves. In symbols, xEX s~(x€Ex). Then by substituting "X" for "x" we have X e X - ~ (X6X). This of course is the well known Russell's Paradox whose announcement caused so much consternation. Various axiomatic approaches were proposed to deal with the problem, two of those being Russell's theory of types and axiomatic set theory. Another is the theory of classes. According to this "X" is not a "set" but a "class". A set is a class that belongs to another class. Every set is a class but not the other way around, as sets have more properties than classes have. For example, subclasses, unions and intersections of classes can all be defined in the language of set theory by substituting the word "class" for "set". Where classes differ from sets are in defining the power class, and the quotient class. The power class of a class "X" is the class of all sets (and not the class of all classes) which are subclasses of "X". This because some subclasses of "X" may be proper classes and thus cannot belong to any class. If not so, then they would be sets rather than proper classes.27 In the work to follow we will impose on ourselves two restrictions, neither of which is onerous. First of all, we will not bother with sets which are not extensions of non trivial predicates. After all, we are primarily interested in predicates, our subject being logic, and we will need set theory only to facilitate the study of predicates. Thus pathological cases such as purely random collections of objects will be excluded. Second, seeing that many times sets are very convenient in the study of predicates, we will "use" sets exclusively, and classes we will only "mention". One such case of the latter is the example we availed ourselves of abelian groups. Abelian groups themselves are sets, but the collection of all such is a class, there being too many of them to constitute a set. E. RELATIONS WITHIN AND BETWEEN SORTS
Relations will be denoted by the upper case letter R, which will be indexed just as quantifiers are. A decision has to be made regarding the number of subscripts and also their numerical values for each relation symbol. A binary relation will have two subscripts, and, more generally, an n- ary relation will have n subscripts. Now, each variable occupies a certain position in the sequence of variables following the relation symbol. The corresponding subscript of the relation symbol will have the same numerical value as the extension that contains that variable. Thus R5 4 xy means that x E X5, y £ X4 and "x" and "y" stand to each other in the relation R. Let us consider a concrete example: "Every woman has a mother." We have
"F1... "-" ... is a woman"
"F2... "=" ... is a mother"
"F3... "=" ... is a daughter"
"R2 3 " denotes the mother-daughter relation.
Then the above statement is translated as [ Vax ] [ 3xy ] (R2 3 y x). Finally, the identity relation may hold for entities belonging to one or more sorts. A necessary condition for the latter is that the two sorts intersect. Thus a mother belongs to both the sort of all mothers and the sort of all daughters, as any mother is also someone's daughter. Let us now look at relations that hold not between members of sorts but within sorts themselves. Other than the trivial case when two sorts are identical (e.g. the sort of all equiangular triangles and the sort of all equilateral triangles), there are three cases. The first is that which arises when two sorts are disjoint. This is the case when the conjunction of the two predicates whose extensions are the sorts in question is a contradiction. The second is the case when a given sort is a subsort of another sort. Here we have that Xx C X2 is equivalent to [ Vxx ] [ 32 y ] (x = y). Finally we have the 55
case when two sorts intersect. In this case we have Xr D X2 * 0 which is equivalent to [ 3l x ] [ 32 y ] (x = y). And this covers all the basic facts of the many sorted logic we will be using to develop a much improved system of free logic. I do not intend to enter into the proof theory and the model theory of many-sorted classical logic because I intend to develop these ab initio for many-sorted free logic. By what we have already said, however, both in this section and in the previous one, it should be clear that standard free logic is a kind of many-sorted logic. This because every model of standard free logic has two domains, the inner and the outer, Dx and DQ. Quantification is allowed over the inner domain. So here we have the following situation with regards to quantifiers. These can be written in our notation as \/l x and 3t x . Our task with regards to the model theory of many-sorted free logic, and, by implication for the entire system, ought to be obvious. What we must do is to define quantification over the outer domain, or we have to define V0 x and 30 x, which will be the task of the following section.28
V. MANY-SORTED FREE LOGIC
To be is to be the value of a variable. -W.V.O.Quine-
A. INTRODUCTION
Having negotiated two rather technical excursions in the fields of standard free logic and also many-sorted classical logic we can finally bring together these two strands and develop the new system of many sorted free logic. Its advantage over classical logic is that it does not banish empty names but offers them equal though distinct status with referring names. Empty names, although they are attached to fictitious entities, are treated according to logical principles similar to the ones that hold for referring names. What's more, the way they are treated completely respects the context within which 56 they occur. Finally, it firmly differentiates between existents and non existents. In particular, the system will be ontologically neutral and will have sufficient flexibility to be of use to philosophers of both a Russellian and a Meinongian persuasion. The advantages over standard free logic are that the new system will avoid the stipulational approach regarding the truth value of sentences containing empty names as is the case of both positive and negative standard free logic. Instead it offers a new approach, one which is intuitively very natural. Once the truth value of atomic sentences has been established, the truth value of quantified atomic sentences will be decided in a completely non arbitrary way. Following this procedure one step further, the truth value of compound quantified sentences will be arrived at using exclusively well established logical principles. Thus the new system will not only respect context, but will avoid the many absurdities plaguing standard free logic, which absurdities we have already encountered. There were two distinct sources of inspiration for this new system both of which will play a crucial role in its development and, also, both are based on model theoretic considerations. The first we have already talked about. Standard free logic is a truncated bi-sorted logic. It is bi-sorted because its models contain two distinct domains, the inner and the outer. It is truncated because only members of the former play any role in its quantification theory. The new system will eliminate this asymmetry by introducing two distinct sets of quantifiers appropriately indexed, one corresponding to existents and the other to non existents. The interpretation function will map all existents to the inner domain and all non existents to the outer, and since the two will be distinguished by the subscript of the quantifier, quantification over non existents will be entirely legitimate and following accepted logical laws. The second source of inspiration is an insight I acquired from my very rudimentary study of the model theory of classical first order logic. In model theory, an atomic sentence "Fx" is true just in case "x" is a member of the extension of "F". Thus if "F ..." stands for "... is a triangle" then quite trivially "Fx" is true if "x" is a member of the set of all triangles. Going to quantified sentences let "G ..." stand for "... is a planar figure bounded by 57 three line segments" and "H ..." stand for "... has one 90° angle". Then we have V ( Vx (Fx DGx)) = t and V (3x (Fx A HX) ) = t. These can be translated in uni-sorted logic as V ([Vx Fx] (Gx)) = t and V ([3x Fx] (Hx)) = t. Going one step further, these are translated in set theory as V ([Vx :x G { y : Fy } ] (x : x E { y : Gy }) = t) and V ([3x :x e { y : Fy } ] (x : x e { y : Hy})) -1. What the above tells us is that the truth value of such sentences is decided by whether the variable in question belongs or not to some set or sets. Now the second sentence is an existential assertion. What it says is that there exists something of a given kind just in case some set has a member. This humble and at first sight insignificant observation is the beginning of the vast edifice of model theory. Truth of a sentence is a matter of set membership. And in case the sentence makes a claim about the existence of something, then existence also is a matter of membership. This is the starting point of bi-sorted free logic. It consists of replacing the concept of existence with the concept of location, location in some particular sort. And there can be one sort containing actually existing objects, and one sort containing fictitious objects. An example is in order. Sherlock Holmes was a consulting detective. He also never existed; he was the product of the pen of Sir Arthur Conan Doyle. Therefore we can not write 3x (x = a A
F2x) where "a" stands for "Sherlock Holmes" and "F2..." for "is a consulting detective". But we can introduce a new predicate as follows. "Fj..." stands for "... is a fictitious entity". And we can index the existential quantifier by indexing it the same way as we indexed "F/'. Thus "3^ ..." means "there is an Y in the sort of all fictitious entities". Sherlock Holmes was the fictitious hero who was depicted to be a consulting detective. We can write this as
3tx (x = a A F2x) without any fear of contradiction as "3" and "3/' do not mean quite the same thing. The former bespeaks of general undifferentiated existence. The latter tells us that the sort of fictitious beings is not an empty set, it has members and Sherlock Holmes is one of them. B. THE LANGUAGE OF MANY-SORTED FREE LOGIC
The language of many-sorted free logic can be treated fairly quickly as it is just the language of many-sorted classical logic with but one difference, and this difference does not affect the purely formal structure of the system, such as the derivation of proofs, in the least. And that is that the range of the objects under consideration is considerably larger. In classical logic, every object under consideration must belong to the following: Y = {all actually existing entities}, (counting numbers etc as actually existing entities). By contrast, the range of free-logic is considerably expanded; it includes all the members of Y and in addition quite a few more. These are objects encountered in mythology, literature, and also in science in the guise of putative objects which have been proposed in some theory but whose existence or non existence have yet to be empirically established. Last but not least, it must include all long departed historical personages who once existed but no longer do so. The new range of applicability is X = {all intelligible entities about which we can make meaningful assertions in a logically non self- contradictory way}. Obviously Y C X. the range X is truly enormous. There is, however, that clause at the end which places some restrictions. And that is that X can not include self-contradictory entities. Thus such Meinongian objects such as the non-circular circle are excluded. The restriction affects only logically self-contradictory entities. It does not extend to physically impossible mythological beings such as Gorgons, Centaurs and the like. Having done this, we subdivide X into all kinds of sorts. Here we have well nigh unlimited freedom. The only restriction being that a sort containing some kind of entities must be disjoint from all those sorts that logically
exclude these entities. Thus if Xt is the sort of all circles, and X2 is the sort of
all non circles, then we must have Xx n X2 = 0. If so, we can never have 32x
F2x which explains the exclusion form X of all logically self-contradictory Meinongian objects. And it also explains why we said "subdivide X" rather than "partition X". Had we partitioned X into mutually non intersecting sorts, then we would have ~ Vxx F2x. Not much of a logic! Now several of these sorts contain things (mythological entities, literary characters) that classical logic can not possibly admit. But the present system can. In particular it can quantify over any legitimate sort, and the truth value of the quantified sentence depends on the context, unlike standard free logic. Finally, the sort of actually existing objects being disjoint from that of fictitious ones, there is no danger of treating non existents as existents.
C. AN ASCENDING HIERARCHY AND THE ROLE OF CONTEXT
Up till now, all was plain sailing and yet there is still some ambiguity regarding what criteria one is to employ to differentiate between what exists and what is fictitious. In order to illustrate the point, I will have recourse to examples from literature. This is done for the sole purpose of clarifying a point, and is not meant to be a systematic analysis of how complex cases of existence can be translated within many-sorted free logic. This will be the topic of part VI. The fact is that the concept of existence as used in ordinary human speech (rather than how it is used in formal languages) is rather fluid; it can be given, in different contexts, a different sense. Imagine a nested sequence of sets. The innermost set will contain everything that presently exists. Let us, without any loss of generality, restrict our attention to persons. Then Ed Stelmach, the current premier of Alberta, is a member of this set which provisionally we will call Xx. Fx is the predicate whose extension is Xx. The superset immediately containing Xj will consist of all persons who either exist or have existed at some time in the past; we will call it X2 and the corresponding predicate, F2. This contains many more persons than Xv but is a weaker kind of existence. Thus it contains Ed Stelmach but also all the many personages of the Thirty Years War, such as Wallenstein. The superset just above X2 will contain literary characters such as Sherlock Holmes. This is 60
going to be X3 and it contains, among others, persons who have never existed but could possibly have existed. Finally, X4 will contain beings which, given the laws of nature, could not possibly exist, but the assumption of their existence does not engender any logical contradiction. Here we have Gorgons, Centaurs, vampires and the like. But before we go any further, please note that this ascending hierarchy is not unique. Scientists can create other such hierarchies. Astronomers, for example, may postulate a hierarchy of heavenly bodies whose existence has been confirmed, and immediately after that a super-set containing all hypothetical heavenly bodies. As I said, here we are in an area of great fluidity. Now consider the case of Wallenstein. He undoubtedly existed as, back in the 17th century, Europe bore the scars of his military genius. So for him we can write 32x (x=a) but not 3xx (x=a), as he was assassinated a very long time ago. To distinguish these fine points, let us employ the symbol for set theoretic difference to the predicates whose extensions these sets are. Thus for Wallenstein we can write (F2 \ FJ a or 32 \ jX (x=a). The same holds for all the other predicates that denote some kind of existence. When we enter the field of literature, the problem of what exists and what does not becomes considerably harder to settle, and in some cases it may even be impossible to find a truly satisfactory solution. To avoid undue repetition we will make the common sense assumption that most literary characters never existed, but even here the genre of historical fiction (e.g. novels by Alexander Dumas and Thomas Costain) is a notable exception. We will also assume that within the context of the stories the various protagonists are treated as fully existent. What about vampires? This depends on the story. In all Sherlock Holmes stories they do not exist at all and the chief protagonist is openly dismissive of supernatural solutions proposed to solve various cases. In the novel Dracula, however, vampires do exist. There is nothing unexpected about this, of course, but still we do draw conclusions that seem to be counterintuitive. The novel Dracula and also the Sherlock Holmes stories set for us contexts that restrict our horizons geographically, chronologically, and thematically. Every story is a "closed world," if I may be allowed a neologism, and only those persons exist in them 61 who are either explicitly mentioned or who are at least implied by the narrative. Therefore we may assume that persons who lie outside the story boundaries can be treated as non existent. Consider the Sherlock Holmes stories. Within their context does Ed Stelmach exist? This is something of a problem. On the one hand, since Ed Stelmach has actual existence, if nothing else, and since the sort X3 consisting of all possible beings, has the sort Xa consisting of all actual beings as a proper subsort, then Ed Stelmach ought to exist also in the context of the stories. Except that he doesn't. First of all, Ed Stelmach plays no part whatever in the Sherlock Holmes stories, either explicit or implicit. He is not mentioned once and little surprise seeing that Ed Stelmach was born long after Conan Doyle died. But, even if we were to credit the author with the gift of prognosis, Ed Stelmach could not exist in the stories, because the stories themselves depict a time period that passed before the present one was ushered in. Even within the stories, Ed Stelmach can be considered as someone physically impossible. Again there is a significant exception, that of science fiction stories that involve time travel. So in the interest of keeping the stories coherent, and within their context, we are justified in treating Ed Stelmach as a member of Xa but not X3. A truly counterintuitive solution. The justification is that the context of each story carves out a subset within X3 but does not include most members of Xa.
Again, the exception is that of historical novels, where some members of X2 may also be members of Xv
D. PROMISES KEPT
When we proposed our new system we made some requirements. We demanded of our system to fully differentiate between existents and non existents; to completely respect the context within which each individual entity is to be found; to allow for quantification over non existents; to formulate rules of quantification for non existents which will be entirely analogous to the ones that hold good for quantification over existents; and, 62 finally, to avoid the paradoxical results that are inherent in standard free logic. We will use an example from mythology to evaluate the degree of success of the new system.
Let "Vl..." stand for "... is a Gorgon", "F2..." stand for "... can turn humans into stone", "F3..." stand for "... is fictitious", "F4..." stand for
"... is mortal", "F5..." stand for "... is a superb pastry chef", and, finally, let "a" stand for "Medusa". Then we have the following statements and their translations: 1. Gorgons are mythological beings:
VjX F3x. 2. Gorgons can turn humans into stone:
Vax F2x. 3. There is just one mortal Gorgon: BjxtV^F^] (y=x)
The above statements and their translation are true in mythology and tell the truth about mythology, that truth being that the entities of mythology are imaginary. So let us now try something a bit more complicated, a definite description.
4. There is a fictitious being called Medusa who was a mortal Gorgon and she could turn humans into stone:
[3jx] (F2x A F3x A F4x A [Viy] (F4y D y = x) A (x = a))
There is one final issue to be settled, and that is whether many-sorted free logic succeeds in avoiding the paradoxical consequences of standard free logic. Consider the following: 5. All Gorgons are superb pastry chefs:
Vxx F5x . which in unsorted logic can be written as Vx (Fxx D F5x).
Now the reason that this absurd claim comes out true is because the extension of Fj is an empty set. And in unsorted logic it has to be empty, as the opposite 63 would assert the actual existence of Gorgons. But in many-sorted free logic this is not the case. The set of all Gorgons is a non empty subset of the non empty set of all fictitious beings. And since Gorgons were not superb, or even indifferent, pastry chefs, the universally quantified sentence is false in our system. Thus false sentences do not come out true which is the case in standard free logic. The opposite problem, that of existentially quantified true sentences coming out false, is treated in the very same way. Thus we were able to develop many-sorted positive free logic which does everything that standard free logic does and much more.
E. THE SEMANTIC PRINCIPLES OF MANY-SORTED FREE LOGIC
Up till now we developed the fundamental ideas of many-sorted free logic and tried to justify it both by argument and example. The last three sections are rather technical, and to prepare the ground we will offer four semantic principles, some of which we have already used without attracting attention to the fact that they constitute the heart of the entire subject. They are as follows:
1. V (Vxx : F2x) = t = Vx fox D F2x)
2. V (VlX : F2x) =/- 3x fox A ~F2x)
3. V fox : F2x) = t = 3x fox A F2x)
4. V fox : F2x) =/- Vx fox D ~F2x)
The significance of these four principles lies in the fact that they connect very effectively many-sorted logic with classical logic. These four are a real and true bridge between the object language of many-sorted logic and the metalanguage of classical logic. Thus any assertion we care to make in many-sorted logic about which we may harbour some doubts can be put immediately to the test by translating the assertion in the language of classical logic. On the level of metalogic this immediately tells us that, so far as 64 theorem proving goes, the two systems are completely equivalent; the only but very real advantage of using sorts is the vastly increased semantic expressibility that sorts confer. Again on the level of metalogic such results as consistency, completeness, compactness, etc. carry over to many-sorted logic. I have been using in this section the general term many-sorted logic rather than the more specific many-sorted free logic. The reason is obvious. The four semantic principles are sufficiently general to accommodate any two predicates "F/' and "F2" regardless of whether they mean that something is red or that it is fictitious. Thus these four principles and the proof theory that they will assist us in developing in the next section, apply to many-sorted logic in general and not just to free logic. When, however, we develop the model theory of many-sorted free logic, our results will be applicable to free logic exclusively. This because certain predicates in free logic will mean "actuality" or "fictitiousness"; here the meaning can not be ignored. Before we launch into the development of proof theory, a word is in order to indicate how the four semantic principles will be of assistance. Proof theory depends on uninterpreted axioms, which by definition can never be formally proved. Here, however, we are on terra incognita, one that presents us with its own specific problems. While axioms can not be proved, at least they ought to be given some justification for their adoption. The four semantic principles, by virtue of the fact that they connect the object language of many-sorted logic with the metalanguage of classical logic are ideally suited for that purpose. What I intend to do is to give the axioms of many- sorted logic and then to construct tree diagrams which, if nothing else, give ample evidence that these axioms do not lead to contradiction. The reason tree diagrams are so well suited is that by their very nature they introduce semantic considerations to the syntax of the system; they cut across both worlds. And now we are ready to discuss proof theory.
F. PROOF THEORY FOR MANY-SORTED FREE LOGIC
In developing a proof theory for many sorted free logic, there are two requirements. The first stems from the fact that this logic is many sorted, and 65 hence can be reduced to classical logic by means of the four semantic principles. This means that there must be nothing in the proof theory of the new system that in any way contradicts that of classical logic. And if this minimal requirement is adhered to then the proposed system of proof theory stands a fair chance of serving its purpose not only for many-sorted free logic but also for any other many sorted system whether free or not. The second requirement, however, is directly linked to the fact that the proposed new system is a system of free logic, and as such it must not do violence to standard free logic. In meeting this last requirement it is the issue of empty names that can cause some difficulty. So let us briefly recapitulate on how three distinct systems of logic treat empty names. The first is classical logic. Its method is effective but also unduly restrictive. It simply refuses to treat empty names. Its reason is that the admission of empty names leads to a contradiction. Universal instantiation, Vx Fx |- Fa, leads to the contradiction that if "a" is an empty name, then something that does not exist actually exists. And this because Fa |- 3x Fx. The second system is standard free logic. This is slightly more inclusive, but only slightly more. Hence universal instantiation is { Vx Fx, 3x (x=a)} |- Fa. This is almost the same as universal instantiation for classical logic, but with an existence clause added. The reason is that standard free logic admits empty names but excludes them from its quantification theory for the same reason that classical logic excludes them altogether, and that is that quantification over non existents leads to a contradiction. The third system is many-sorted free logic which admits empty names and collects all entities which are bearers of empty names in the sort of all fictitious beings. Since the human mind can imagine these and the human tongue can talk about these, the sort of all fictitious beings is non empty. For example, the set {Medusa, Alecto, Chiron} is not an empty set although its elements are empty names. And by specifying that its members can not belong to the sort of actual beings, the previously mentioned contradiction is avoided. If something is a figment of the imagination then it is not actual and hence does not exist the same way as chairs and pencils do. But they do exist 66 as fictitious entities. The immediate consequence is that many-sorted free logic does not have empty names. Every name refers to something, whether actual or imagined. The second consequence is that in universal instantiation the existence clause 3x (x=a) is both meaningless and redundant. Let us see why, as a clear understanding of this will help us formulate universal instantiation for many-sorted free logic. Many sorted free logic has two distinguishing characteristics. The first is that existence is replaced with location, and the second is that everything, whether an existent or a non existent, must be located in some sort which may be a set or a class. In many sorted free logic, to say that something exists does not make any sense at all. This claim must be replaced with one making the assertion that something is located in a specific sort, and this is the reason why we index our quantifiers. And if everything is located in some sort, then the fact that we have equated location with existence makes the existence clause 3x (x=a) redundant. And yet we can not dispense altogether with something pretty close to it. We need some kind of existential sentence, one appropriately indexed, which will tell us that the name we will use to instantiate a universally quantified sentence, does belong to the extension of the predicate that bears the same subscript with the universal quantifier. An example is in order. In our talk about mythological beings consider the statement "All Gorgons can turn humans into stone". Now it is entirely appropriate to instantiate using the name "Medusa" and derive " 'Medusa' can turn humans into stone," because Medusa was a Gorgon. But it would be terribly inappropriate to use "Chiron" to instantiate the above as Chiron was not a Gorgon but a Centaur and could not turn humans into stone. This is our big hint. In instantiating a universally quantified sentence, one whose universal quantifier has a certain numeral as its subscript, we must include an existential clause (or perhaps it is better to call it a location clause), one that will restrict the choice of names to be used from only those that are of the same sort as the one indicated by the universal quantifier. This is to say the universal quantifier and the existential quantifier must be 67 indexed with the same numeral. Given this, the form that universal instantiation ought to take is quite easy to arrive at. It is
{ VjX F2x, 3ax (x=a)} |- F2a.
What the above tells us is that if every member of some given sort has a certain property, and if something in particular belongs to that sort then it also has that certain property. From this it is easy to see that we could have easily used symbols, i.e. unbound variables, instead of names. We could have used objectual semantics instead of substitutional semantics. The second thing is that we can express the above more generally as an axiom schema, one using instead of F2.
{vlX m., 3,x (X=a)} i- m.
Before we go any further we ought to see how the above squares with standard free logic's treatment of quantification over existents. Let "¥1..." stand for "... is an actually existing heavenly body" and "F2..." stand for ".. . is observable". Then what we have is "if every actually existing heavenly body is observable, and if Neptune is an actually existing heavenly body, then Neptune is observable". Quantification over non existents also leads to no difficulties. Let "F3..." stand for "... is a fictitious being" and "F4..." stand for "... is non-observable". Then we have
{ V3xF4x,33x(x=a)} |-F4a. The above tells us that "if every fictitious being is nonobservable and if 'Medusa' is the name of a fictitious being then 'Medusa' stands for something nonobservable". We must now see if the above squares with how universal instantiation is handled in classical logic. To this end we will have recourse to the semantic principles. We must show that we have
{ VjX F2x, 3jX (x=a)} |- F2a. 68
The semantic principles give
VjX F2x = Vx (FjX D F2x), 3xx (x=a) = 3x (¥{* A x = a). So we must show that
{ Vx (FjX D F2x), 3x (Fax A x = a).} |- F2a. The tree diagram follows:
DERIVATION TUSTIFICATION
(1) Vx (FiX D F2x) ANTECEDENT
(2) 3x(Fax A x = a) ANTECEDENT
(3) ~F2a DENIAL OF THE CONSEQUENT
(4) Fjb A b = a (2), EXISTENTIAL INSTANTIATION, b/x
(5) Fib (4), CONJUNCTION ELIMINATION
(6) b = a (4), // //
(7) Fia (5) & (6)
(8) Faa D F2a (1), UNIVERSAL INSTANTIATION, a/x
(9) ~ Faa v F2a (6), CONDITIONAL ELIMINATION
(10) ~ F2a F2a (9), DISJUNCTION ELIMINATION
X X (3), (7), (9), LAW OF NON CONTRADICTION. So, { Vax F2x, 3jX (x=a)} |- F2a can be admitted as a valid axiom of universal instantiation. Having this at our disposal we must make a decision on what is the bare minimum of axioms we need. The answer depends on what method we are going to use for our theorem proving. If we choose natural deduction then we need four: Universal instantiation, existential generalization, existential instantiation, and universal generalization. Now the last two are very hard to come by and the most difficult is universal generalization. So we are better off if we opt for the method of tree diagrams which needs only two, universal instantiation and existential instantiation. The axiom of universal instantiation we already have, so let us see if we can come by with some reasonable candidate for existential instantiation. My proposed candidate is as follows:
{ 3ay (y=a), Vay (y = a D 32x (x=a) )} |- F2a Although it looks rather complicated, it is actually rather obvious once one breaks it down. Now the semantic principles give the following:
3ay (y=a) - 3y fa y A y = a)
Vay ( y = a D 32x (x = a)) -
Vy(FlY D(y = a D 32x(x = a))) =
Vy ((Piy A y = a) D 3x (F2x A x = a))
The tree diagram takes the form
DERIVATION TUSTIFICATION
(1) 3y (Fiy A y = a) ANTECEDENT
(2) Vy ((Fia A y - a) D 3x (F2x A (X = a)) ANTECEDENT
(3) ~ F2a DENIAL OF THE CONSEQUENT
(4) Fxd A d = a (1), EXISTENTIAL INSTANTIATION, d/y 70
(5) Fxd (4), CONJUNCTION ELIMINATION
(6) d = a // //
(7) Fia (5), (6), SUBSTITUTION OF IDENTICALS
(8) (Fja A a = a) D 3x (F2x A (x = a)) (2), UNIVERSAL INSTANTIATION, a/y
(9) ~ (Fxa A a = a) v 3x (F2x A (x = a)) (8), CONDITIONAL ELIMINATION
(10) ~ (Fxa A a = a) 3x (F2x A (x = a)) (9), DISJUNCTION ELIMINATION
(11) ~Fia v ~(a = a) (10), DE MORGAN'S LAW
(12) ~Fia ~(a = a) (11) DISJUNCTION ELIMINATION
(13) X X (5), (12), LAW OF NON CONTRADICTION
(continued next page) 71
(14) F2b A b = a (10), EXISTENTIAL INSTANTIATION, b/x
(15) F2b (14), CONJUNCTION ELIMINATION
(16) b = a (14), CONJUNCTION ELIMINATION
(17) F2a (15), (16), LAW OF SUBSTITUTION OF IDENTICALS
(18) X (3), (17), LAW OF NON CONTRADICTION.
So our proposed axiom for existential instantiation is valid, and together with the axiom of universal instantiation is sufficient to construct tree diagrams for any theorem. And this concludes our study of a proof theory form many-sorted free logic.
G. THE MODEL THEORY OF MANY-SORTED FREE-LOGIC
The way Lambert defines the semantics to be used in the model theory of free logic is a trifle unclear - page 30 of this thesis. He seems to equivocate between substitutional and objectual semantics. He claims that the interpretation function "f" maps names (substitutional semantics) into
Dl U D0 whose members are objects (objectual semantics) and then he asserts that every object in Dj U D0is assigned a name (substitutional semantics once more). Myself, I will employ mostly objectual semantics, but I will allow names whenever this is convenient and does not engender the kind of problems we have already encountered. Thus in speaking about the three 72 furies, all of whose names have come down to us, I see no problem in using substitutional semantics. But this is the exception and not the rule. Many-sorted free logic does not have empty names or non existing objects. Everything which is not self contradictory exists, in the sense of it being located in some sort. The sort in question is indexed by some appropriate subscript. This last point is important as it will help us define models in many-sorted free logic.
Let us suppose that we have two sorts X1 and X2 each of which is the extension of some predicate, V1 and F2. There is a system of models, one for each sort. We index them by using the same subscripts as the sorts that have them as models: 2la =
The interpretation function maps members of Xa into Dt €E
The situation of 2C-2 is analogous. When we want to consider the case when a given sort has more than one model, we differentiate between them by means of superscripts, or by using some other letter in the fractur script. So if we had two models for the first sort they would be ^l'i and ~*-2i. Given the above, one may get the impression that when we consider a problem where there are n sorts then we automatically have 2n domains, one inner and one outer for each sort. Such a conclusion is mistaken. We can not tell a priori how many distinct domains there are. This has to be decided by means of a careful analysis of the problem. It is quite possible that the inner domain of one sort is the outer domain of the other. This is always true when the two sorts are disjoint. Thus if X2 is the sort of all fictitious entities, and Xx,
(or X2 - Xx when we arrange our sorts in a hierarchy), is the sort of all
actually existing entities, then DT in ^ is identical to D0 in ^2 and vice versa. Given this, the assignment of truth values to quantified atomic sentences follows closely the rules laid down for standard free logic. 73
Thus V (VJXFJX) = t iff every member in the extension of Fx is also predicated by F2. And V (BiXFjx) = t iff there is a member in the extension of Fx which is predicated by F2. In the first case V (VaxF2x) = t iff every member in the extension Fx belongs to the inner domain of ^t2 =
Belongs to the inner domain of ^2 =
V (Vax F2x) = t just in case
{D.IDJ E &} C {DjrDje \ }, and
V (3:x F2x) = t just in case
{ Dj: Dx G \ } n { Dj: Dx E ^2} * 0 .
This looks a bit confusing since in both classical logic and standard free logic valuation is always relative to a single model. But in many-sorted free logic all the really important quantified sentences involve cases where the objects under consideration involve more than one predicate. Whaf s more, we have postulated one model, with its inner and outer domains for each sort. Thus in case the inner domain of ^ is not a subset of the inner domain of ^2 / then the sentence VjxF2x comes out false, as in such a case there are members of the inner domain of ^ which are not members of the inner domain of ^2 • VI. THE ROLE OF MANY-SORTED FREE LOGIC IN LITERATURE
For some time now it looked as if the theory of definite descriptions by Bertrand Russell had provided the metaphysics of Alexius Meinong with a decent funeral. Appearances can be deceiving. Meinongianism simply refused to repose in peace within the comfortable if somewhat narrow confines of its grave. Philosophers started asking pointed questions about the ontological status of non existing entities, difficult questions. The conceptual soil that proved to be especially fertile for the growth of such questions turned out to be literature. This is because stories and novels are replete with non existing persons who are predicated with properties, some of them real and some unreal, play significant roles in various events, interact with fellow human beings - and even sometimes with ghostly beings, and are depicted as existing. Obviously the boundary dividing existents from non existents is not a widthless line but a rather broad band. It is quite imperative to have a system of logic which is expressive enough and of sufficient flexibility to act as a tool for translating these matters in a formal language. Obviously this ought to be the first step to be taken if the many subtle questions that arise from the study of literature are to be settled in a convincing and unequivocal way. As always, precision is the first requirement, the lack of which will foster a counterproductive vagueness. And the best way to cultivate precision is to translate statements arising in literature from "humanese" into the formal language of logic. Provided of course that such a logical system powerful enough to perform the task is available. Once such translation is available, and not before, then the second step is to be taken by metaphysics as at this point logic will have already played its given role. I claim that many-sorted free logic is ideally suited for this purpose. In this, the final part of my thesis, I will consider a wide variety of problems having to do with existence or non existence that are forced on us when we read literature. No two of them are identical yet they all are variants of the same basic theme, that of empty names. I will offer translations to all of them thus demonstrating the wide range of applicability of the system. 75
The sources from which I will draw my examples are of two kinds: stories of the supernatural and detective fiction. The reason is that in these two genres we meet a wide variety of cases, each one of them illustrating a different aspect of the problem of empty names. Here is the list of the various problems I propose to consider. 1. Non existing persons whose existence is possible. 2. Non existing but possible persons interacting with impossible persons, such as vampires. 3. This is the reverse of 2. Non existing but possible persons interacting with historical personages. This is the case we encounter in historical novels by Dumas, Costain and others. 4. Issues of mistaken identity. Several varieties of the problem of mistaken identity are everyday occurrences in detective stories. Here are some: a. Two apparently distinct persons who in reality are one and the same. b. The reverse of the above is one person impersonating someone else. c. Mistaken predication. This is the case, frequently encountered in the novels of Agatha Christie, when suspicion initially falls on an innocent person. One fascinating variety on the theme of mistaken identity is the one of "mistaken cardinality". In nearly all detective stories the main issue is to identify the villain, the assumption being that there is only one. Very rarely, however, we have the case of confederacy between two villains or even more. The only such case I know of is provided by the novel Murder on the Orient Express by Agatha Christie, and is an extreme case. In a court of law a jury consists of twelve people. In this story twelve people, all of them suspects, acted as jury, judges and executioners when they killed a kidnapper and murderer of a little girl who had escaped justice. I believe the story was inspired by the Lindberg case. I will not even attempt to translate it as it involves twelve names and I wish this thesis to be in just one volume. In theory, however, it can be translated. This list is not exhaustive, but it gives ample opportunity to test the methods of many-sorted free logic. Each and every case considered involves 76 the identification of some person given by name with the role he or she plays in the story. Therefore all these cases involve a definite description whose quantifiers are indexed by numerals in accordance with the principles of many-sorted logic. Before we start, I will give a list of names, predicates and relation symbols which will be used in the translation of examples from literature in the language of many-sorted free logic. aj = "Sherlock Holmes", a2 = "Dr. Grimesby Roylott". a3 = "Mrs. Ferguson", a4 = "Dracula", a5 = "Lucy Westerna", a6 = "Sigmund Freud" a7 = "Vorobeichick" (modulo the spelling). a8 = "Ned", a9 = "Ted" , a10 = "Tim", au = "Jim". FjX = x is a character in a work of fiction.
F2x = x is an actual historical personage.
F3x = x is a non existing but possible human being
F4x = x is a non existing and also an impossible person who is treated in the story as existing.
F5x = x is a non existing and also impossible person who is treated in the story as a possible entity.
F6x = x is a consulting detective.
F7x = x is a villain.
F8x = x is a vampire
F9x = x is the originator of psychoanalysis.
R:xy: x brings about the death of y.
R2xy: x treats y.
R3xy: x causes event y to occur. R^yz: x believes y to be z.
R5xyz: x knows y to be z.
Here Ru R2 etc. do not stand for a relation in a particular sort, as used in part IV of the thesis. Here the index is merely used to differentiate between relations. Now that we have a complete list of symbols, let us use them to translate statements found explicitly or implicitly in fiction. We shall begin with the simplest case of all, one that can serve as a template for dealing with 77 the more difficult ones to follow. It is taken from the Conan Doyle story titled The Adventure of the Speckled Band. In this story Sherlock Holmes causes a venomous snake used by its owner, Dr. Grimesby Roylott, to commit his murders, to turn against its owner and kill him. The sentence we wish to translate is "Sherlock Holmes is a fictitious human being found in literature who is a consulting detective and who caused the death of Dr. Grimesby Roylott". This sentence is rather simple as it involves a fictitious human being performing an uncomplicated act. No ghostly entities here, and also no high placed historical figures. This sentence is obviously a definite description, one telling us who
Sherlock Holmes is by giving us some salient facts. And since we have set ax to be the name of Sherlock Holmes, the definite description must take the form aj = G (ixFx) where F is some predicate and G some other or rather a conjunction of predicates each one of them applicable to Sherlock Holmes. And here we are faced with a question. Of the many such predicates, which one is the best to use for "F"? Seeing that in this part we consider exclusively characters encountered in fiction, our first impulse is to use F1, but this would be a serious error of judgment. The reason is as follows: The compound sentence ixFx reads "the x which is predicated by F" or "there exists a unique x which is predicated by F". The word "unique" tells us why we can never use V1 for F. Sherlock Holmes is not the only character to appear in the story. The sentence ixFxx reads "there exists one and only one x which is fictitious," and clearly this is not the case. For the same reason, we can not use F3. What about F6? This is controversial. In this story there is only one consulting detective, namely Sherlock Holmes, so if we restrict ourselves to just this story F6 is acceptable. In detective fiction, however, there are many detectives (private, amateur, or even retired) who work outside the official police force. Therefore, if we wish to use F6 then we must introduce some new predicate, one that explicitly mentions the title of the story and restricts the choice of names, predicates and relations to just those described in the story. Such a step, however, in addition to being very cumbersome, will have the very unfortunate result of mixing in the definite description terms that belong to the object language such as Sherlock Holmes, with terms that belong to the 78 metalanguage such as the title of the story. After all, within the narrative of the story the events are depicted as real and not as part of a story. This last can happen only in the case where we have one of the characters in a story, herself tell another story. For these reasons, F6 also has to be rejected. There is one thing left. For the predicate F we must use a relation, the only relation we encounter in the statement, and that is that Sherlock Holmes caused the death
of Dr. Grimesby Roylott. Or Fx = R!xa2. More generally, we can use a binary relation as a predicated as follows, F = Rt . Here Rx and y are parts of F. Having found an acceptable candidate for F, let us now decide what G is going to be. This is easy. Sherlock Holmes is a fictional character who is depicted as a human being, which renders him possible, and he was a
consulting detective. Therefore, Gx = Fax A F3X A F6X . Finally, we have
ax = G (ix Rjxaj). Let us express this description more analytically. Every definite description starts with the existential quantifier 3. Here we can finally
use the predicate F2 to index the existential quantifier. We can safely do this since all entities, real as well as imaginary, encountered in literature are fictitious, in the sense that they are involved in events which never occurred.
The definite description is 3ax (Rxx a2 A Vay (Rx y a2 D y = x) A F3X A F6x). The above is an example of the interaction between fictitious but possible beings. Let us now look at an example of the interaction between fictitious beings some of which are quite possible and some are impossible. The novel Dracula will provide us with a case in point. The villain of the story was once a human being, but he is now one of the undead who, in order to sustain his ghostly existence, must drink human blood, and in doing so he kills Lucy Westerna. The sentence we want to translate is "Dracula is a vampire who although physically impossible is depicted in a story as both possible but also actually existing, who brought about the death of Lucy
Westerna". Or a4 = G (ixFx). The first order of business is to decide on a suitable candidate for F. As before, and always keeping in mind the uniqueness requirement, we designate Fx by "x brought about the death of
Lucy Westerna" or Ra x a5. And G = Fa A F4 A F8. The rest is routine. 79
From the case of the interaction of fictitious and possible beings we went to the case of fictitious and impossible beings interacting with fictitious and possible beings. Let us now go the opposite way and consider the case where fictitious and possible beings interact with actually existing beings in fictitious events. Sometimes detective stories that have become legendary have inspired latter day authors to write pastiches of these stories. In such pastiches the hero of the story becomes involved in new adventures but in every other respect is indistinguishable from the original. Now as every fan of the Conan Doyle stories knows, Sherlock Holmes, the abstemious Sherlock Holmes who is in every way a paragon of sobriety, has one fatal vice! He is addicted to cocaine. In one such pastiche titled The Seven Percent Solution by Nicholas Meyer, his friend, assistant and biographer, John Watson M.D. contrives to take Sherlock Holmes to Vienna so that he can be treated by the father of psychoanalysis in person, Sigmund Freud. The trip is a success; Sherlock Holmes is helped by Freud to kick this pernicious habit, and, as one good turn deserves another, he helps the latter solve a criminal case! This is very close to the second case we encountered, but instead of using RiXy we will use R2xy. Also, in the interests of illustrating the thorny issue of having existing persons play a role in a work of fiction, this time we will give a definite description of Freud, as the definite description of Sherlock Holmes follows the lines already developed. Here we are faced with a very serious difficulty. And that is that in such a case we are obliged to mix object language and metalanguage, otherwise it would seem that the events described in the story are real. To be precise we must now introduce a new symbol, a meta symbol, which is foreign to those already in our list of symbols as all those in the list are symbols of the object language. The new symbol is a sentential operator which acts as a "meta predicate" for which we will use the upper case D. The avoidance of using F with some appropriate index is deliberate. D is not by any stretch of the imagination a predicate of some person within the story. It is a predicate bringing to our attention the fact that whoever is predicated by D plays a role in a story, and everything that follows D is a fictional event. So let Dx stand for "x is depicted in a work of fiction as ..." 80
Before we go any further, we must first face a very serious objection. Can we call D a predicate at all, whether object or meta? The answer is yes. Imagine some gigantic book titled Trivia ofSigmund Freud. Here, among other things, may be listed some of the most noteworthy cases of Freud, where he treated actual patients suffering from hysteria, narcissism, etc. Each and every one provides us with a predicate of Freud. Now let us imagine that further on in the book there is a chapter titled "Sigmund Freud in the Imagination". Here are listed cases which never happened in actuality but were only imagined, such as the Sherlock Holmes case. The fact that a totally imaginary story was written about Sigmund Freud is a real fact. The story, while describing imaginary events, is real enough qua a story. So the fact that such a story was written about Sigmund Freud is something that predicates Freud and is one more property that distinguishes him from other analysts, such as, say, Alfred Adler. Thus for the real Sigmund Freud "D" acts as a predicate as it tells us something about him. But for Sigmund Freud, as he appears in the story, "D" is not a predicate. Here we must pause for one minute to ponder what possible objections there could be in calling D a predicate. In my mind there is only one such objection. Each and every predicate adds something to our store of information we have about the entity that is being predicated. And so far as the story goes, since every sentence beginning with "D" is true just in case its content is already in the story just as described by the sentence, to say that the content of the sentence is depicted in the story adds not one single iota to the story. In this case, however, the definite description is not meant to describe the story but only a character, real or imaginary, appearing in the story. And since the character is real and therefore had an existence independent of the story, with real life properties not described in the story, to say that this character appears also in a story adds to the stock of predicates that apply to
that person. So if we designate Dx (R2x a1) by G, we have a6 = G (i F9x). Up till now we have dealt with the ontology of fiction or how things are. Time to have a look at the epistemology of fiction or how people in a story believe or know how things are. A very good example that illustrates the above is provided by the Conan Doyle story The Adventure of the Sussex 81
Vampire. Here everyone, with the sole exception of Sherlock Holmes, believes the unfortunate Mrs. Ferguson to be a vampire. Sherlock Holmes knows better and proves that this is not the case. The symbolism used to translate the above gets to be a trifle unwieldy and so it is more convenient to break down the translation into stages. As in all definite descriptions we wish to derive a3 = G (ix Fx ), for some F and G. F must be a predicate that characterizes uniquely, within the bounds of the story, Mrs. Ferguson. Now Mrs. Ferguson is human, so we have F3x but there are many imaginary humans in the story so F3x alone will not do. What defines her uniquely is that everyone in the story save Sherlock Holmes believes her to be an vampire. This can be translated as
v\y ( ~ (y = ax) D (R4yxz A F8Z) ). The conjunction of this with F3x will furnish us with our F. G is going to be the fact that Sherlock Holmes knows, and finally proves to everyone's satisfaction, that Mrs. Ferguson is not a vampire, or,
aa xz A ~ F8z. And this gives us the desideratum. This is a case of mistaken predication as everyone believed Mrs. Ferguson to be a vampire, which she is not. The case where somebody is suspected mistakenly to be the villain is entirely analogous. I believe that in England a nickname for a detective story is a "who done it," for reasons too obvious to require any elaboration. The expression, ungrammatical though it may be, captures the essence of the genre. In most stories like these, the crux of the matter is to find out who the perpetrator of the crime is. In most cases the perpetrator has recourse to a number of ruses to cover his tracks. One of them is to provide the police with a perfect alibi. Another is to make it look as if someone else committed the crime. More decades ago than I care to count, when I was still a lad in Athens, Greece, I read a superb "who done it" by the expatriate Polish writer Sanislas Stiman (modulo the spelling). The hero of his stories, also an expatriate Pole living in Paris, answers to the unlikely name Vorobeichick. In this particular story whose title I can not remember, the villain has found a novel way to escape the clutches of the law. He has found a way to persuade the police to stop their investigation before it is yet over! 82
In a few words this is how the story goes. Paris is experiencing an unprecedented wave of brazen robberies and burglaries. Suspicion falls on someone whom we will call Ted, who is a thoroughly unsavory character. Ted has a brother, let us call him Ned, who is the polar opposite of his identical twin brother. At some point Ned makes the dark prediction that unless Ted mends his ways he will come to a bad end. And so it happens. During a burglary a fire breaks out and the building together with Ted is gutted. And the crime wave immediately comes to an end. After all, the villain has perished in the flames, hasn't he? Vorobeichick, however, is not satisfied. For one thing, while he has met face to face with both Ned and Ted, he has never met them together. The meetings were always separate. This and other tiny but most curious inconsistencies make him suspicious. Finally, the truth dawns on him. There never was a Ted. Ted is a phantasm, an illusion artfully created and foisted on the police by Ned who never had a twin brother and from the very start was the real villain. When finally Ned had made his pile, he contrived to have the nonexistent Ted perish in the flames, thus bringing the police investigation to a close, and then retired to enjoy his ill gotten gains. Here is our problem. While Ned is a work of fiction and does not actually exist, he still exists as a character in a story. But what about Ted? Does he exist as a character of fiction or not? On the one hand, Ted does not exist at all. Dracula, though an impossible being, still has a better claim to existence than Ted does seeing that there never was a person called Ted. While everyone thought that there were two distinct persons, Ned and Ted, there was in reality only one - Ned. And yet, seeing that Ted is none other than Ned and that the latter exists qua a character of fiction, it would be plainly absurd to claim that Ted does not exist. The solution to this problem is to give two definite descriptions in sequential order. The first is that of Ned, and the second is that of Ted, in which the name "Ned" will occur essentially. Now what do we know about Ned? Ned is a possible fictitious human who is a villain, but nobody,
including Vorobeichick, believes at first that he is one such. Now F7x, or x is a
villain, is a good candidate for a predicate which in conjunction with F3x, or x 83
is a non existing but possible being, define Ned uniquely. So Fx = F3x A F7X. As for G, we can use the fact that there is not one person in the story who believes Ned to be a villain, at least for the greater part of the story.
So Gx = ~3ay (R4yxz A ~F7Z). These give us a8 = G (ixFx). Having provided a definite description for Ned, we can use it to describe Ted. What we do know about Ted is that, in reality, the "reality" within the story of course, he is Ned and everyone, with the sole exception of Vorobeichick who finally realizes what is going on, supposes Ned and Ted to be two distinct persons. For F we chose the fact that Ted is Ned in conjunction with the fact that with the late exception of Vorobeichick, no one believed this to be so. Hence we have:
(x - a8) A v\y ( ~ (y = a7) D (R4 yxz A ~ (z = a8)). These give a9 = G (ixFx ). In the above story we have the case of one person pretending to be two. In the following we will examine the case of two distinct persons pretending to be one. In a story by Ellery Queen there are two identical twins, Tim and Jim let us call them; one of them, Jim, who is a scapegrace, has been out of sight for many years and when he finally comes back he confides only to Tim. Tim has a "brilliant" idea. Jim will be his double during a party in order to play on the guest a tasteless practical joke. This is what happens and then things go terribly wrong, when Jim commits a bad deed and everyone blames it on Tim, who, incidentally, underneath his staid, respectable exterior is quite as big a scoundrel as his brother is. Here, just as before, we will use in an essential way one brother to describe the other. Since Tim is entirely unproblematic, we will use him to describe Jim. What do we know about Jim? Jim is the person who committed a certain act (this gives us F), and the only one who did (now we have ixFx), who everyone believes mistakenly to be Tim (and this gives us G). So we have au = G (ixFx ). More analytically this is:
3jX (R3xu A Vjy (R3yu D y = x) A ( ~ (x = a10) A V\z R4zxa10)). It can be readily seen that many-sorted free logic affords us a system which makes a fairly routine matter the translation in formal language of statements describing fictional events within literature that would be formidably difficult or even impossible for other systems of logic. It may still have applicability in other areas. And this brings the present study to an end.
VII. ADDENDUM
THE EMPIRICAL DIMENSION OF SCIENCE IN CONFLICT WITH LOGIC
On page 16 of this thesis we discussed the problem of how abstract entities in science conflict with classical logic. These entities come into being in the theory construction aspect of science. Theory building involves a process of idealizing situations encountered in the physical world by the ruthless elimination of all properties that are temporarily considered to be irrelevant to the formation of a theory. This engenders "empty predicates" such as the perfect vacuum, the frictionless plane, the completely insulated system, etc. These predicates apply to objects which not only do not exist but can not possibly exist. But the theory building is but one dimension of the scientific endeavor. The other is the empirical dimension which collects the data necessary for both theory building and theory confirmation or refutation. This aspect involves subtle issues which are as difficult as they are fascinating and are best treated in an addendum rather than in the main body of the thesis. In the empirical aspect of science the problem that a logician must face is once again that of nonexistence. But it is not that of the nonexistence of objects characterized by idealized properties, i.e. "empty predicates," but rather it is the problem of "empty names". That is to say of objects which may well exist but whose existence or non existence has not yet been confirmed. A long time ago Pierre Duhem proposed the view that theories can never be disproved in isolation but in clusters with other theories which act as a conceptual backdrop for the theories under disputation.29 Thus the discovery of the lens made it imperative that astronomers, in addition to having knowledge of heavenly bodies, ought to have exact knowledge of optics. An astronomer, when faced with some unexpected result, must decide 85 before anything else if this result is a consequence of some hitherto unknown astronomical state of affairs or of some unknown property of light as it impinges her telescope. This is the first building block we shall need. The second building block was furnished by Imre Lakatos when he developed his "methodology of scientific research programs". According to this, mature scientific theories develop a sort of "conceptual intertia" (the neologism is mine) which, just like the physical inertia of material bodies, endows theories with a tendency to resist change. Scientific theories are meant to provide explanations for various phenomena and/or to give us the capacity to make predictions about such. The more successful a theory turns out to be, the less eager are scientists to abandon it when things begin to look bad as inevitably happens.30 To this end, in addition to the theory scientists wish to preserve, and in accord with the Duhemian postulate, they propose various auxiliary hypotheses whose main role is to form a protective belt around what Lakatos called "the hard core" of a theory. Should empirical evidence require some adjustment, the universal tendency is to modify one or more of these auxiliary hypotheses thus leaving the hard core intact. And now that we have all the necessary tools at our disposal, let us use them to examine one particularly fascinating example from astronomy. A revolution took place in the field of astronomy in the seventeenth century. Its empirical component was provided by the telescope which was invented by Galileo. Its theoretical component was provided by the development of celestial dynamics by Newton. In combination these two proved to be unbeatable as one success followed the other at a breathtaking speed. And then sometime in the nineteenth century a serpent found its way into the paradise fashioned by the great Newton himself, and this serpent turned out to be a veritable Lernea Hydra as the decapitation of its head engendered two such to grow in its place. A close study of the orbit of the planet Uranus revealed certain pronounced disagreements with the predictions of celestial dynamics. A clear conflict between a mature and immensely successful theory with the observed facts. Now what? Abandon celestial dynamics? Perish the thought! 86
And even if so, with what to replace it? Then Leverrier and Adams had a fine idea, one which if borne out would salvage the "hard core" of celestial dynamics. There had been an auxiliary hypothesis at work in the calculations regarding the orbit of Uranus. That was that there were just seven planets in the solar system with Uranus being the outermost. Well, suppose that there was still one more unobserved planet further out whose gravitational field caused the anomalies in the orbit of Uranus. This hypothetical planet was provisionally given the name Neptune, and on the basis of the disagreement between theory and fact, all the facts pertinent to Neptune were calculated using unmodified celestial dynamics. When this was done, it became quite clear just what the position of Neptune would be at any particular time. And when the telescopes were turned in that direction, Neptune revealed its presence. And even more, every particular about Neptune - mass, velocity, distance from the sun, etc. - turned out to be just as predicted by the theory, with one notable exception. The orbit of Neptune revealed anomalies quite similar to the ones exhibited by that of Uranus.31 But now Pandora's Box had been opened, and the astronomers knew what they had to do. Yet one more planet was postulated, this one called Pluto; similar calculations were performed with similar results. And now there were nine planets whose existence had been predicted by the theory and was empirically verified.32 A veritable triumph for astronomy. The next chapter, though, had contents of the most surprising nature. Turning their attention from the most distant planets to the one closest to the sun, Mercury, it was found out that its orbit, too, exhibited features not in conformity with celestial dynamics. Given the previous successes, no one was unduly worried. The astronomers, reasoning on the same lines that had brought to the surface the existence of Neptune and Pluto, assumed the existence of a tenth planet, Vulcan. All the necessary calculations were performed and when the attempt was made to locate Vulcan observationally, the results were totally negative. Vulcan does not exist. Time now to introduce a third philosopher of science, Thomas Kuhn. He had hypothesized that a theory, any theory, would last only for so long, until under the weight of increasing negative evidence the theory would 87 finally collapse thus necessitating a new theory. Kuhn called this process a "paradigm shift".33 Actually, Duhem had said the same thing and even better. A theory does not collapse so much under the weight of successive refutations but rather under the weight of the unending sequence of repairs needed to accommodate all those negative results.34 By now it was obvious that celestial dynamics as originated by Newton and further developed by Laplace and others could no longer hold its own. The time for a paradigm shift had come, and this was provided by Einstein's General Theory of Relativity. A glorious new dawn had arrived.35 And finally, after this unavoidably lengthy preamble we can say a few words about logic. Physically possible objects, be they planets or elementary particles or whatever, start their careers having some rather shadowy hypothetical existence. On this basis properties are predicated of them, properties which are consequences of some already existing or some proposed new theory. On the basis of these properties experiments or observations are undertaken to verify or to refute the existence of bearers of these properties. So from "hypothetical names" we finally arrive at either "empty names," such as Vulcan, or "denoting names" such as Neptune. And here is one more point that classical logic with its insistence that all that it examines must exist fails to do justice to science. Seeing that hypothetical names may turn out to be empty rather than denoting, classical logic can not possibly admit them. 88
ENDNOTES
1. The methods of natural deduction and of tree diagrams are thoroughly explained in [11] and [10] respectively. 2. See [22], chapters V, VI, VII, X, XI, XII. 3. See [8], pages 103,129-130,143-144, 249, and [22], pages 52-54, 56-60. 4. Oral communication from Dr. B. Linsky. 5. See [22], chapters V, X. 6. See [22], chapters VI, XI. 7. See [4], page 103. 8. See [22], chapters VII, XII, and [4] chapter 3. 9. See [8], pages 42-43. 10. See [7], chapter 6. 11. See [14], pages 16-18. 12. See [20], page 17. 13. See [3], page 375. 14. See [15] part III. 15. A comparison of the merits and demerits between positive free logic, negative free logic and Fregean free logic is offered by Sainsbury, see [21], pages 239-245. Some of the adherents to each of these systems are, according to Sainsbury, Lambert for positive free logic, Bostock and Burge for negative free logic, and finally Lehman for Fregean free logic. 16. For the proof theory of standard free logic see [14], pages 19-21. 17. See [3], pages 384-385. 18. See [14], page 22. 19. See [14], pages 26-27. 20. See [20], the entire article. 21. See [1], page 5. 22. See [23], pages 65-66. 23. See [23], pages 128-129. 24. See [17], the entire book is essentially a textbook on logic for linguistic analysis based entirely on uni-sorted logic. 89
25. See [5], for a very accessible introduction. 26. There are different conventions for representing sorts. The one I chose was developed by N.Rescher [19], pages 168-170, as being the most convenient of the lot and with least chance of engendering confusion. 27. See [18], pages 21-22. 28. For a brief but highly technical introduction to many-sorted logic, see [16]. 29. See [6], pages 187-194. 30. See [13], pages 195-211. 31. See [9], page 52. 32. See [9], page 72. 33. See [12], pages 316-325. 34. See [13], page 196. 35. See [9], page 54. 90
BIBLIOGRAPHY
1. Atten, Mark van. On Brouwer. Wadsworth Philosophers Series. A Division of International Thomson Publishing Inc. 1998
2. Bell, J. L., A. B. Slomson. Models and Ultraproducts, An Introduction. North-Holland/American Elsevier. 1971
3. Bencivenga, Ermano. "Free Logics". In D. Gabbay and F. Guenthne, Eds., Handbook of Philosophical Logic. Volume III. D. Reidel Publishing Company. 1986
4. Bridge, Jane. Beginning Model Theory. Oxford Logic Guides. Clarendon Press, Oxford. 1978
5. Copi, Irving, M. The Theory of Logical Types. Routledge & Kegan Paul. 1971
6. Duhem, Pierre. "Physical Theory and Experiment". In Janet A. Kourany, Ed., Scientific Knowledge: Basic Issues in the Philosophy of Sciences. Wadsworth Publishing Company. An International Thomson Publishing Company. 1998
7. Enderton, Herbert B. Elements of Set Theory. Academic Press. 1977
8. Haack, Susan. Philosophy of Logics. Cambridge University Press. 1978
9. Hempel, Carl. Philosophy of NaturalScience. Foundations of Philosophy Series. Prentice Hall. 1966
10. Jeffrey, Richard. Formal Logic, It's Scope and Limits. McGraw Hill, Inc. 1967 91
11. Kalish, Donald, Richard Montague, and Gary Mar. Logic: Techniques of Formal Reasoning. Harcourt, Brace Jovanovich College Publishers. 1980
12. Kuhn, Thomas. "The Nature and Necessity of Scientific Revolutions". In Janet A. Kourany, Ed., Scientific Knowledge: Basic Issues in the Philosophy of Sciences. Wadsworth Publishing Company. An International Thomson Publishing Company. 1998
13. Lakatos, Imre. "Falsification and the Methodology of Scientific Research Programs". In Janet A. Koruany, Ed., Scientific Knowledge: Basic Issues in the Philosophy of Sciences. Wadsworth Publishing Company. An International Thomson Publishing Company. 1998
14. Lambert, Karel. "Existential Import, E! and "The"'. In Karel Lambert, Ed., Free Logic: Selected Essays. Cambridge University Press. 2003
15. Lambert, Karel and Bas C. van Fraasen. Derivation and Counterexample: An Introduction to Philosophical Logic. Dickenson Publishing Company, Inc. 1972
16. Manazano, M. "Introduction to Many-sorted Logic" in K. Meinke and J. V. Tucker, Eds., Many-Sorted Logic and its Applications. Wiley Professional Computing. John Wiley and Sons Ltd. 1993
17. McCawley, James D. Everything that Linguists have always Wanted to Know about Logic. The University of Chicago Press. 1993
18. Murdeshwar, M. G. General Topology. Wiley Eastern Limited. 1990 92
19. Rescher, Nicholas. Topics in Philosophical Logic. D. Reidel Publishing Company. 1968
20. Russell, Bertrand. "Descriptions" in R. R. Ammerman, Ed., Classics of Analytic Philosophy. Hackett Publishing Company, Inc. 1994
21. Sainsbury, Mark. Logical Forms: An Introduction to Philosophical Logic. Blackwell. 2001
22. Thomason, Richmond H. Symbolic Logic, An Introduction. The McMillan Company. Conrad McMillan Canada Ltd. 1970
23. Weiner, Joan. Frege Explained: From Arithmetic to Analytic Philosophy. Ideas Explained Series. Open Court. 2004