Quantification by SidneyFelder

The truth-functional connectivesconstitute the expressive and deductive “engine” of propositional . These combine and relate complete atomic sentences whose internal structures are givenno distinctively logical role whatsoever. From one formal point of view, quantificational logic,inall of its infinite varieties, is marked by its penetration to a sentence’ssub-atomic level, a levelfrom which sub-sentential elements that are common to multiplicities of sentences can be discerned and exploited. It has been found, specifically,that just by isolating those aspects of that can be interpreted as making assertions about all or some individuals of one domain or another,a tremendously more powerful system of logic is created: In augmenting the propositional logic by the universal and existential quantifiers “for all x” (symbolized (x) or (∀x)) and “there exists an x” (symbolized (∃x)), and their associated apparatus, we both 1) enormously expand the sphere of abstract ideas and structures that can be givenprecise and distinctive expression in formal terms and 2) vastly increase the range of propositions that can be brought into non-trivial expressive and deduc- tive relationship with each other.

We are nowgoing to define the vocabulary and concepts of what is variously called the FirstOrder Predicate Calculus, Predicate Logic,the Lower Predicate Calculus (LPC), and FirstOrder Logic (FOL). (Another (nowseldom used) older name is the FirstOrder Functional Calculus). First Order Logic is simultaneously the most elementary and the most standard of the quantificational log- ics. (There is, for example, Second Order Logic, Third Order Logic, Quantified Modal Logic, Epis- temic Logic, etc.).

The First Order system whose axioms include only logical axioms (only axioms that are logically valid) singles out the class of logically valid formulae. However, there are manyother concepts and structures that can be expressed in a First Order language. Each augmentation of the logical axioms (which belong to all First Order systems) by a logically consistent of non-valid axioms (proper axioms)defines a class of more definite structures. While manydifferent choices of axioms will single out the same class of structures, and while manyconceptually natural structures are undefin- able in First Order systems, the expressive power of First Order languages completely dwarfs that of sentential calculi.

These notes are designed to provide a synoptic orientation to some of the simplest concepts, rela- tions, and structures that can be expressed in a First Order Language. In addition to the sentences and connectivesofpropositional logic, the basic elements of a First Order System include: 1) anon- empty domain or of individuals U; 2) aset of individual constants a, b, c,... each denoting some definite individual belonging to U; 3) aset of predicate constants A, B, C,... designating prop- erties and relations among the individuals of U (a monadic predicate corresponds to a of U and an n-adic predicate for n greater than 1 corresponds to a set of n-tuples of elements of U); 4) a set of variables x, y,z,... ranging overall the objects of U; 5) the quantifiers (∀x) and (∃x), repre- senting the expressions ‘for all x’ and ‘there exists an x’ respectively; and 6) the ‘≈’, invariably representing the relation of identity among individuals. While the designations of the predicate constants (predicates) and the individual constants (constants) vary from to interpretation, the quantifiers and the the identity relation are (likethe truth-functional connectives)

Course Notes Page 1 Quantification treated as logical constants, and their interpretations are fixed. (Although an infinite number of dis- tinct formulae can be composed from these linguistic elements, each well-formed formula of the lan- guage is assumed to be finite in length (i.e., to be composed of a finite number of symbolic tokens)).

Nowsuppose that U is the set of physical objects and the predicate G is interpreted as the property green.Ifthe constant a is interpreted as ‘the largest leaf on Earth’ and the constant b is interpreted as the sun, the sentence G(a)isinterpreted as the sentence “The largest leaf on Earth is green”, and the sentence G(b)isinterpreted as the statement “The sun is green”. It is a consequence of the interpretation chosen that G(a)istrue and G(b)isfalse. Or,suppose that U is the set of natural numbers {0,1,2,3,...}, that the > represents, as usual, the relation ‘numerically greater than’, and that the constants 2 and 3 represent, as usual, the numbers 2 and 3. Under this interpretation, the sentence >(0,5) (more commonly written 0>5) ,which represents the statement “0 is greater than 5” is , but the sentence 5>0, which represents the statement “5 is greater than 0”, is true. On the other hand, the simple predicate formula G(x) is neither true nor false categorically: G(x) is true for some assignments of objects to x and false for others. Adopting a slightly different point of view, wesay that the formula G(x) is satisfied by some assignments of values to x and is not satis- fied by others. Thus if we interpret the predicate G as the property green (or,equivalently,the set of all green objects), G(x) will be satisfied by all green objects and will fail to be satisfied by any non-green object. Analogously,x>y is true for some pairs of natural numbers and is false for oth- ers. Specifically x>y is satisfied by all ordered pairs of natural numbers whose first terms are numerically greater than their second terms, and fails to be satisfied by anyordered pair whose first is not greater than its second term.

We now describe howthe quantified formulae are to be understood. Giventhe non-empty U of all physical objects, and supposing for the present that the predicate letter G is interpreted as the color green, the formula (∀x)G(x) states that all objects of U possess the property green, and the formula (∃x)G(x) states that there exists at least one object in U that possesses the property green. In both of these formulae, all occurrences of x are bound,and we say that x appears in this formula only as a bound or apparent variable.The variables embedded in the quan- tifiers themselves, (∀x) and (∃x), are always classified as bound, and the other occurrences of the variable x in the above formulae are bound because (as indicated by the pattern of parentheses) these occurrences fall within the scope of the quantifiers. Because we normally desire the proposi- tion (∀x)G(x) to imply the (∃x)G(x) (i.e., because we want (∃x)G(x) to be true whenever (∀x)G(x) is true), it is standard to arrange things so that the formula (∀x)G(x) has existential import,meaning that the proposition that all objects are green is interpreted in such a way as to imply that there exists at least one green object. This is accomplished not by making anyparticular assumption about the meaning of the universal quantifier (∀x), but rather by imposing the constraint that the universe of discourse U can neverbeempty,that is, by imposing the demand that all admis- sible domains contain at least one object. Under the assumption that U has at least one object, the truth of the statement that “All objects are green” presupposes that at least one object in U is green, meaning that (∀x)G(x) is true in no domain in which (∃x)G(x) is false. If we admitted the possibil- ity of a domain U with no objects, (∀x)G(x) would be vacuously true, and we would be admitting a situation in which (∀x)G(x) was true and (∃x)G(x) was false. (The reader should be aware that there are variations of the standard formalism, called free ,inwhich empty domains are admit- ted).

What about a sentence of the form G(c), which states that the specific object designated by the

Course Notes Page 2 Quantification constant c is green? Such a statement is intermediate in logical strength between (∀x)G(x) and (∃x)G(x) (i.e., is logically implied by (∀x)G(x), logically implies (∃x)G(x), and is logically equiv- alent to neither). Forexample, the statement that all objects are green is logically implies the state- ment that Joe’scoat is green, and the statement that Joe’scoat is green implies the statement that there exists at least one green object. (In the peculiar but logically irreproachable universe whose only object is Joe’scoat, the following statements are all simultaneously true: All objects are green; Joe’scoat is green; Some object is green; Nothing other than Joe’scoat is green). Now, consider anydefinite domain U. If it is not the case that all objects in U are green, there must exist some object that is not green. Symbolically,the sentence ¬ (∀x)G(x) is equivalent to the statement (∃x)(¬ G(x) (¬ G is of course the predicate ‘is not green’). Analogously,itisclear that if there does not exist evenasingle green object in U, all objects in U (and there has to be at least one object in U) must be non-green. In symbols, ¬ (∃x)G(x) is equivalent to (∀x)(¬ G(x). This means that it is possible to define universal quantification in terms of existential quantification, and vice-versa.Ifwe takeexistential quantification as primitive,the universally quantified formula (∀x)G(x) corresponds to the formula ¬ (∃x)¬ G(x). In words (interpreting the predicate constant G as green), the assertion that all objects are green is logically equivalent to the assertion that there does not exist evenasin- gle non-green object. Conversely,ifwetakeuniversal quantification as primitive,(∃x)G(x) is the proposition ¬ (∀x)¬ G(x). In words, the statement that there exists at least one green object is logi- cally equivalent to the statement that not all objects are non-green. (Recall the the manner in which it is possible to define conjunction in terms of and disjunction, and disjunction in terms of negation and conjunction).

We now consider cases of universal generalization,cases possessing the form of the proposition “All ravens are black”. The first thing that needs to be said about this proposition is that its analysis should not proceed under the assumption that this proposition is an instance of the ascription of a predicate to a subject: Whateverthe superficial linguistic form of this sentence, ‘All ravens’ is not its subject. This proposition really concerns arbitrary individuals and the relationship, in a particular world, of the properties ‘raven’ and ‘black’. Letting R designate the set of ravens and B designate the class of black objects, this proposition is represented by the formula (∀x)(R(x)→B(x)), which is read “for all x, if x is a raven, then x is black”. Here, the coordinated occurrences of x servethe same function as do pronouns: This proposition states that for anyparticular object we choose, if that object is a raven, then that object is black—where in each possible case, “the particular object” and the twooccurrences of “that” represent the same object. From a slightly different point of view, this sentence states that R is a subset of B, i.e., that the set of ravens is a subset of the set of black objects. When we reduce the subset relation to the relation of membership from which the subset relation was originally defined, this means that each member of R is a member of B. Because we conceive the variable x as ranging overthe whole of U, what we really mean is: Givenany arbitrary x of U, if x belongs to the set of ravens, then x belongs to the set of black objects. Given that a universal generalization such as the one we are considering is in some sense a conjunction of conditionals (in asserting it, we are asserting the statement “If a is a raven, then a is black and if b is a raven, then b is black and if c is a raven, then c is black,...”), it is a generalization that does not possess existential import. (Although the domain overwhich the variable x ranges is necessarily non-empty,any particular subset of this domain, such as the set R of ravens, may be empty). This is to be contrasted with the standard formal rendering of the proposition “Some ravens are black”, which is “There exists at least one object that is both a ravenand black”, symbolized by the for- mula (∃x)(R(x)∧B(x)). This is a formula that obviously (and by design) does carry existential import.

Course Notes Page 3 Quantification

Those who find it odd that the universal generalization (∀x)(R(x)→B(x)) is not logically stronger than the existential generalization (∃x)(R(x)∧B(x)) will perhaps be mollified by the observation that the true universal correlate of (∃x)(R(x)∧B(x)) is not (∀x)(R(x)→B(x)) but rather the sentence (∀x)(R(x)∧B(x)), which says that all objects are black ravens. Confining ourselves to non-empty domains (as is our invariable practice here), it is indeed the case that (∀x)(R(x)∧B(x)) is nevertrue when (∃x)(R(x)∧B(x)) is false. (The closest thing to the universal generalization (∀x)(R(x)→B(x)) that implies (∃x)(R(x)→B(x)) —that is, the logically weakest statement that implies both (∀x)(R(x)→B(x)) and (∃x)(R(x)→B(x))—is the formula (∀x)(R(x)→B(x))∧(∃x)(R(x)). The logically weaker existential analog of (∀x)(R(x)→B(x)) is the very weak proposition (∃x)(R(x)→B(x)), which states that there exists at least one object x such that if x is a raven, then x is black, meaning “Among the class of ravens, if anyactually exist (which we don’tguarantee), at least one is black”. This is a statement whose truth does not imply that anyrav ens exist. When considered together, these formulae strongly suggest that the sentence (∀x)(R(x)→B(x)) is not about ravens, but rather about the relations among the elements and of the domain U. This is one reason that we are able to takethe logical equivalence between (∀x)(R(x)→B(x)) (“All ravens are black”) and its contrapositive (∀x)(¬ B(x)→¬R(x) (“All non-black things are non-ravens”) in stride.

The proposition (∀x)(R(x)→B(x)) must not be confused with the proposition (∀x)R(x)→(∀x)B(x)). In the former sentence, the whole conditional expression (R(x)→B(x)) falls within the scope of the quantifier (∀x), and hence all the occurrences of x in the expression are bound by the same quanti- fier.Inthe latter,the occurrence of the x in R(x) falls within the scope of the first quantifier,and the occurrence of the x in B(x) falls within the scope of the second quantifier but not the first. Consequently,the substitutions for x on the left of the disjunction sign are not coordinated with the substitutions for x on the right of the disjunction sign, and hence we can without anyalteration of meaning use different variables on the twosides of the disjunction. So, for example, the formula (∀x)R(x)→(∀y)B(y)) means the same thing as the formula (∀x)R(x)→(∀x)B(x)). Wetranslate these equivalent sentences as “If all objects are ravens, then all objects are black”. Although this is not a statement that anyone would be inclined to utter,itiscertainly true in anyworld reasonably similar to ours. This is because the antecedent is manifestly false in this world or in anyworld very much likeit, which implies (giventhe definition of the →)that the whole conditional is (vacuously) true whether or not all ravens are black.

An analogous contrast exists in the case of disjunction. The proposition (∀x)(A(x)∨ ¬Ax)), which states that for each object x, x has either the property A or the property ¬ A, is logically valid. The quite distinct expression (∀x)A(x)∨(∀x)¬ A(x), on the other hand, states “All objects have the prop- erty A or all objects have the property ¬ A”. This is a contingent statement, and a rather severe one. It implies 1) that if evenasingle object possesses property A, then all objects possess property Aand 2) that evenifasingle object does not possess property A, then none do. Looking nowata particular instance, suppose U is the set of natural numbers N {0,1,2,3,...}, O the property ‘odd’, and Ethe property ‘even’. Consider the propositions (∀x)(O(x)∨E(x)) and (∀x)O(x)∨(∀x)E(x). The first says “For anynatural number x, x is either odd or even”, a statement that is obviously and nec- essarily true. The second proposition makes the plainly absurd (and necessarily false) statement that all natural numbers are odd or all natural numbers are even. There are of course situations in which this kind of disjunction is appropriately asserted. If I believe that there is a good chance that a large commercial aircraft has crashed into the seas near the Arctic Circle and a good chance that it landed safely at its scheduled destination, it is extremely likely to be true that either all its passen- gers are alive or all its passengers are dead.

Course Notes Page 4 Quantification

The expressive capabilities of quantificational language can be seen yet more clearly in formulae in which n-place relations and a multiplicity of distinct variables appear.Phenomenae associated with alternations of universal and existential quantifiers are particularly revealing. Wemust here restrict ourselves to a consideration of the distinction between the meanings of the expressions (∀x)(∃y)(y>x) and (∃x)(∀y)(x>y). Suppose Uisagain the set of natural numbers N,and that the dyadic relation > is the irreflexive numerical relation ‘greater than’. Under this interpretation, (∀x)(∃y)(y>x) states that for anynatural number x, there exists a natural number y such that y is greater than x. This states, in effect, that there is no largest natural number,astatement that is obviously true. The sentence (∃x)(∀y)(x>y), on the other hand, states that there exists some natural number x, such that givenany natural number y,xisgreater than y.This says, in other words, that there is some particular natural number that is greater than anynatural number—an assertion that is more colloquially rendered by the statement that there is a single natural number that is simultane- ously greater than all natural numbers, a statement that is patently false.1

The distinction between these twostatements will perhaps be illuminated by a consideration of the following two“games” between twoopponents A and B: In the game corresponding to the sentence (∀x)(∃y)(y>x), A has the first move,and chooses anynatural number x he likes. If Bcannot name anumber that is greater than x, A wins the whole game; if B can name a number greater than x, B wins this round. Because A must select a definite natural number (though anynumber he pleases) on each round, and because A must commit himself first on each round, B clearly has a winning strategy because whateverAselects on anyround, there is always available to B, for example, the number that is one greater than the number A selects on that round. Although A’s choice of number on round n+1 may always be a larger number than anyofB’s choices on anyofthe rounds 1 to n, the winner of each round n is determined by the relative magnitudes of the choices on round n,and it is obvious that B has a winning strategy that determines the outcome of every round. Consider nowthe game corresponding to (∃x)(∀y)(x>y). A goes first, and selects a natural number.Howev er large this number is, it is nevertheless a finite number.Bwins this round by simply selecting its immediate successor.There is in fact only one round in this game, which reflects the fact that (∃x)(∀y)(x>y) asserts (falsely) that there is a single finite number that is greater than all finite num- bers simultaneously,asingle choice by A that exceeds all possible choices by B. In the case of (∀x)(∃y)(y>x), B merely has to demonstrate that each number has one successor or another,not necessarily the same for anytwo choices of A.

Sentences prefixed by the triple of alternating quantifiers (∀x)(∃y)(∀z) are critical in all mathematics involving the concept of limit.Thus consider a sequence such as 1/2,1/3,1/4,1/5,..., a sequence whose limit (in this case its greatest lower bound) is not itself contained as a term in the sequence. The numerical differences between successive terms of this sequence obviously become always smaller (i.e., smaller without exception) as we consider successively further terms in the sequence. The terms also obviously get closer to the number 0 as we consider successively further terms. Does the satisfaction of the twoconditions of the last twosentences justify the assertion that a sequence converges to 0 (i.e., that the limit of the sequence is 0) ?Consider the sequence obtained from the sequence 1/2,1/3,1/4,1/5... by adding 1 to each term. Although the numerical differences 1 What about the formulae (∀x)(∃y)(x>y) and (∃x)(∀y)(y≥x) (≥ means ‘greater than or equal to’)? The first for- mula says that for all natural numbers x, there exists a natural number y such that y is less than x. The second for- mula says that there exists a natural number x such that for all natural numbers y,xisless than or equal to y.The first formula states that there is no least natural number,astatement that is false. The second formula states that there is a least natural number,astatement that is true.

Course Notes Page 5 Quantification between successive terms of the sequence 1+1/2,1+1/3,1+1/4,1+1/5... become irreversibly smaller as we consider further terms in the sequence, and although the numerical values of these terms move successively closer to 0, the limit of this sequence is 1 and not zero. This shows that these two properties are not sufficient conditions for a sequence to have 0 aslimit. Interestingly,neither of these twoproperties is a necessary condition for a sequence to have 0 aslimit. Thus consider the sequence 1/2,1/3,1/4,2/2,2/3,2/4,1/5/1/6,1/7,2/5,2/6,2/7,1/8/1/9,1/10,..., which we will call s*. It is neither the case that the numerical differences of successive terms always diminish as we move fur- ther along the series (the difference between 2/4 and 1/5 is greater than the difference between 1/3 and 1/4) nor the case that successive terms always move closer to 0 (2/5 is further from 0 than is 1/5) . However, the limit of this sequence is 0. In the intuitively salient sense, it is plain that there is a progressive trend towards 0 as we consider evermore remote terms of the sequence, and it is plain (and can be demonstrated) that eventually the sequence comes arbitrarily near 0 and to no other number.

There is a precise expression of these intuitions in terms of alternating universal and existential quantifiers. Let δ represent a numerical separation greater than 0, N a particular ordinal position in asequence, n avariable ranging overthe positions of a sequence, and s(n) the numerical term occu- pying the position n in the sequence. (Remember,asequence is a function s from the natural num- bers to an arbitrary set. Thus the sequence 1/2.1/3,1/4,1/5,... is formally represented as the set of ordered pairs (0,1/2) ,(1,1/3) , (2,1/4) , (3,1/5) ,...). We say that the limit of a sequence is L ifffor anyarbitrarily chosen non-zero numerical separation δ,there exists an ordinal position N in the sequence s such that for all positions n of the sequence beyond N (i.e., for all n>N), the numerical difference between the numerical value of the term in the nth position (n>N) and L is less than δ. In symbolism, a magnitude L is the limit of a sequence s iff(∀δ>0) (∃N)(∀n>N)(s(n)-L<δ). The sequence s*above whose upward and downward oscillations progressively diminish in magnitude satisfies this condition, and hence the sequence s*converges to 0 as limit.

Among the infinitely manyother logical and mathematical notions that quantificational language per- mits us to define, I will only mention two.

The first is the general notion of an equivalence relation.Recall that an equivalence relation is a relation that possesses the properties of reflexivity,symmetry,and transitivity.(Numerical equiv- alence, logical equivalence, and the relation among natural numbers ‘leavesthe same remainder when divided by n’are notable equivalence relations). In symbolic terms, these properties are repre- sented as follows: (∀x)(xRx) states that for all x, x is equivalent to itself—Reflexivity. (∀x)(∀y)((xRy)→(yRx)) states that for anypair of elements x and y,ifxRy then yRx—Symmetry. (∀x)(∀y)(∀z)((xRy∧yRz)→(xRz)) states that for anythree elements x, y,and z, if xRy and yRz, then xRz—Transitivity.

With this same apparatus (the language of quantificational logic + Identity), we are also able to specify the precise cardinality of anyfinite domain U. (∀x)(∀y)(x=y) states that for anyassignment of objects from U to the variables x and y,the object assigned to x is identical to the object assigned to y.This sentence means that there exists at most one object (in U). When conjoined with the constraint that all domains are non-empty,the truth of this sentence implies that there exists exactly one object (i.e, that exactly one object belongs to U).

Course Notes Page 6 Quantification

(∃x)(∃y)(x≠y)∧((∀z)((z=x)∨(z=y))) states that the domain in question contains an object that we can assign to x and an object that we can assign to y such that 1) the objects assigned to x and y are distinct and 2) for anyassignment of an object in U to z, the object assigned to z is identical either to the object assigned to x or to the object assigned to y.This sentence means that there exist exactly twoobjects (i.e, that exactly twoobjects belong to U). This sentence may be regarded as the conjunction of the sentence (∃x)(∃y)(x≠y), which states that there are at least twoobjects, and the sentence ((∀z)((z=x)∨(z=y))), which states that there are at most twoobjects. (∃x)(∃y)(∃z)((x≠y)∧(y≠z)∧(x≠z))∧(∀v)((v=x)∨(v=y)∨(v=z))) states that U contains an object that we can assign to x and an object that we can assign to y and an object that we can assign to z such that 1) no twoofthe objects assigned to x, y,and z are identical and 2) for anyassignment of an object in U to v,the object assigned to v is identical either to the object assigned to x, the object assigned to y,orthe object assigned to z. This sentence means that there exist exactly three objects (i.e, that exactly three objects belong to U). This proposition may also be regarded as the conjunc- tion of twosentences, the sentence (∃x)(∃y)(∃z)((x≠y)∧(y≠z)∧(x≠z)), which states that there are at least three objects, and the sentence (∀v)((v=x)∨(v=y)∨(v=z)), which states that there are at most three objects.

Clearly,analogous constructions exist for anyfinite cardinality.

We now turn to more formal account of the semantics of First Order systems, in particular of the notions of logical validity, truth in an interpretation,and satisfiability.(This whole analysis is due to the great logician Alfred Tarski (1901-1983) ,and is expounded in his ground-breaking paper “The Concept of Truth in Formalized Languages (1935) ). Webegin by defining the general notion of interpretation of a first order language.

First, we specify the language of FirstOrder Logic With Identity (FOL≈). For purposes of exposi- tion, we makethe special assumption that the classes of variables, constants, predicates, and func- tions of the language each contain a denumerable set of elements, that is, a number of elements whose cardinality is equal to the infinite set of natural numbers N.

To the atomic sentences A, B, C,... AA, BB, CC,... AAA, BBB, CCC, ..., ... and truth-functional logical constants of propositional logic ∧, ∨,¬,->, ←→ ,the language of FOL≈ adds the following:

1) an infinite set of variables x, y,z,x1,y1,z1,... x2,y2,z2,..., ...;

2) an infinite set of individual constants a, b, c,... a1,b1,c1,... a2,b2,c2,..., ...; 3) an infinite set of predicate constants divided into a)monadic predicate constants P1,Q1,R1,... (P, 1 1 1 1 1 1 1 1 1 2 2 Q, R,... for short), P1,Q1,R1,... P2,Q2,R2,..., P3,Q3,R3,...; b)dyadic predicate constants P ,Q, 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 R ,..., P1,Q1,R1,... P2,Q2,R2,..., P3,Q3,R3,...; c)triadic predicate constants P ,Q,R,..., P1,Q1, 3 3 3 3 3 3 3 R1,... P2,Q2,R2,..., P3,Q3,R3,...; d)four-place predicate constants; e)five-place predicate constants, etc.; 4) an infinite set of function constants,divided into a)monadic functional constants f1,g1,h1,... (f, g, 1 1 1 1 1 1 1 1 1 2 2 2 2 h,... for short), f 1,g1,h1,... f 2,g2,h2,..., f 3,g3,h3,...; b)dyadic functional constants f ,g,h,..., f 1, 2 2 2 2 2 2 2 2 g1,h1,... f 2,g2,h2,..., f 3,g3,g3,..., c)triadic functional constants, etc.; 5) the universal quantifier ∀ (frequently symbolized ( )) and the existential quantifier ∃; 6) the logical constant =.

Course Notes Page 7 Quantification

An interpretation of the formulae of FOL= is a functional mapping I from linguistic entities to (most typically) non-linguistic entities that is defined by the following:

1) the assignment of a definite truth value to each propositional letter A, B, C,... and the assignment to the symbols ∧, ∨,¬, ⊃, ←→ of the previously defined truth functions conjunction, disjunction, negation, material conditional, and material biconditional respectively. 2) the specification of a non-empty universe of discourse U,which can possess anycardinality and whose elements can possess anycharacter,overwhich the variables x, y,z,... range. The elements of U are some number of individual objects oi. 3) the assignment to the symbolic complex(∀x) (sometimes simply (x)) of the meaning ‘for all x in U’and to the symbolic complex(∃x) of the meaning ‘there exists at least one x in U’. 4) the assignment to each individual constant a, b, c,... of a definite object belonging to the domain U (‘Socrates’ and ‘Joe’ are constants representing the definite individuals Socrates and Joe). Although the interpretation I may assign the same object in U to a multiplicity of constants, each constant can name only one individual in U. 5) the assignment to each monadic predicate constant P of a subset of U;the assignment to each dyadic predicate constant P2 of a subset of the Cartesian Product U×U; the assignment to each tri- adic predicate constant P3 of a subset of U×U×U, etc. Thus P is a set of elements of U, P2 is a set of ordered pairs of elements of U, P3 is a set of ordered triples of elements of U, etc. Asubset of U corresponds to a property (e.g., x is green); a set of ordered pairs corresponds to a two-place rela- tion (e.g., x is to the left of y); a set of ordered triples corresponds to a three-place relation (e.g., y is between x and z); etc. 6) the assignment to each monadic functional constant f of a one-variable function from objects of U to objects of U;the assignment to each dyadic function constant f 2 of a two-variable function from ordered pairs of objects of U to objects of U;the assignment to each triadic function constant f 3 of athree-variable function from ordered triples of objects of U to objects of U;etc. Within the domain of natural numbers, successor is a one-variable function (no twodistinct numbers have the same successor) and addition is a two-variable function (the operation of addition associates a single number (their sum) to a pair of numbers). In the domain of human beings, mother of is a multi- variable function from finite sets of human beings to human beings (more than one child can have the same mother). 7) the assignment to the logical constant = of the relation is identical to.(Note that FOL is often defined without a ‘built-in’ standard identity relation). The denotations of the logical constants ∧, ∨, ¬, ⊃, ←→ ,=, ∀, ∃ are the same in all interpretations. The denotations of propositional letters, indi- vidual constants, predicates, and functions in general vary from interpretation to interpretation; never- theless, propositional letters always denote atomic propositions, individual constants always denote individuals of U, predicates invariably denote sets (monadic predicates invariably denote subsets of U; dyadic predicates invariably denote subsets of U×U; triadic predicates invariably denote subsets × × n of U U U; function symbols f m invariably denote functions from n-tuples of elements of U to ele- ments of U. Unless there were some invariants in the interpretation of the elements of a formal sys- tem, the propositions corresponding to these expressions would have noform, and hence the basic idea that logical inferences and truths are valid in virtue of their form would collapse into nothing- ness.

We now discuss some of the most illuminating of the elementary aspects of the notions satisfiability, logical validity,and truth in an interpretation.Wefirst define the truth of formulae in which no

Course Notes Page 8 Quantification variables are present, and then go on to define the truth of formulae that contain variables.

The formula P(c)is true in interpretation I the object that the interpretation I assigns to the constant c is an element of the subset of U that I assigns to the monadic predicate P,and is false if the object designated by c does not belong to the set corresponding to P.(In this latter case, c belongs to the set corresponding to the complement of I(P) in U). Thus assume that we’ve chosen an interpretation in which I(Mortal) (the meaning the interpretation I assigns to the predicate con- stant Mortal) is the set of mortals, and that I(Socrates) (the meaning the interpretation I assigns to the individual constant Socrates) is the man Socrates. Under this interpretation, the sentence ‘Socrates is mortal’ is true because the object that the interpretation I assigns to the constant Socrates is an element of the subset of U that I assigns to the monadic predicate Mortal. The sen- tence ‘Jack is located directly to the left of Joan’ is true if the objects I assigns to the constants ‘Jack’ and ‘Joan’ form an ordered pair that belongs to the set of ordered pairs that I assigns to the dyadic predicate ‘left of’. If Jack is indeed directly to the left of Jill, the ordered pair (Jack,Jill) belongs to the set of ordered pairs representing the relation ‘directly to the left of’, and hence the sentence ‘Jack is directly to the left of Jill’ is true. On the other hand, because (assuming we are operating in the context of a space that is at least locally Euclidean) Jill cannot be directly to the left of Jack if Jack is directly to the left of Jill (the relation ‘directly to the left of’isplainly an asymmetrical relation), the ordered pair (Jill,Jack) does not belong to the set of ordered pairs repre- senting the relation ‘directly to the left of’. And, assuming that we have chosen an interpretation in which the constants 0, 1, 2, 3,... are the natural numbers 0, 1, 2, 3,... and are considering the rela- tion ‘x, y,and z are mutually distinct’, the statement that 2, 4, and 3 are mutually distinct is true because the ordered triple (2,4,3) will belong to the set of all triples of numbers whose components are distinct, and the statement that 2, 3, and 3 are mutually distinct will be false because the ordered triple (2,3,3) does not belong to the set of all triples whose components are distinct.

Note that while an interpretation I directly assigns a truth value to the elementary propositional sen- tences, I assigns an object of U to each individual constant and a subset of U to each predicate con- stant. However, just as the truth value of all propositions compounded from the elementary sen- tences by use of the truth-functional connectivesisuniquely determined by the truth values of the elementary sentences and the fixed denotations of the connectives, the truth value of every formula that associates n individual constants to an n-place predicate constant is a consequence—it should be clear—of the interpretations of the individual and predicate constants. Because a particular object either belongs to a particular set or doesn’t, formulae of the form P(a), P2(a,b), P3(a,b,c)are either true or false in a giveninterpretation, and hence theycan enter into truth-functional combinations with each other as well as with straight propositional formulae.

We now goontothe case of formulae that contain variables. The truth and falsity of these formu- lae are not in general determined by a specification of an interpretation. In order to define the truth of formulae with variables, we need to augment the giveninterpretation by a further semantical operation called an assignment or valuation,which associates objects of the domain to variables. We begin with the simple formula G(x). We assume that we are working within an interpretation I that assigns the property green to the predicate constant G. G(x) can be taken as meaning ‘x is green’. As we have emphasized before, such a sentence, likethe arithmetical statement ‘X is a prime number’, is neither true nor false. There are some numbers whose substitution for X will yield a true statement, and some numbers whose substitution for X will yield a false statement. From a somewhat different point of view, inwhich we keep our focus on the variable expression ‘X is a prime number’, we say that certain numbers satisfy the condition or property ‘is a prime

Course Notes Page 9 Quantification number’ and certain numbers do not. In the case ‘x is green’, the variable x is conceivedtorange overthe entirety of the domain U.The property green corresponds to a particular subset of U (the set of green objects): The individual objects that belong to the subset of green objects, such as blades of grass and leaves, satisfy the formula G(x); the objects that fall outside the subset of green objects, such as fire engines, roses, and ripe tomatoes, do not satisfy this formula. In relation to a giveninterpretation, G(x) is a ,afunction whose domain is U and whose co- domain is the set of definite propositions obtained when a definite object in U is substituted for (assigned to) the individual variable x.

Consider the dyadic predicate ‘less than’, assumed to be defined on the set Z of . The dyadic predicate ‘less than’, <, corresponds to the set containing all ordered pairs of integers whose first terms are less than their second. Thus this set contains an infinite number of elements (ordered pairs) such as (2,3) ,(-10,-3) , (0,27) , etc., and it excludes an infinite set of ordered pairs such as (5,5) , (3,2) , (-3,-10) , (27,0) . The formula x

Giventhe above interpretation of the monadic predicate constant G as the color green, the formula (∃x)G(x), obtained by attaching as prefix the quantifier (∃x) to G(x), is interpreted as the statement ‘There exists an object x, such that x is green’, or,more colloquially,‘There exists at least one green object’. Giventhe known character of our world, this statement (giventhe domain specified in our interpretation) is obviously true. Our present task is to relate the truth of this quantified statement to the satisfiability of the unquantified formula G(x), a formula that is not prefixed by the quantifier (∃x) or anyother quantifier.Most simply stated, the quantified sentence (∃x)G(x) is true if and only if there exists some object that satisfies the G(x). Note the manner in which the ascription of truth to one expression depends upon the satisfiability of another.Some objects satisfy the formula G(x) and some do not, but because at least one object answers to the description Green, and hence satisfies the formula G(x), a formally quite distinct but formally related expression (∃x)G(x) is categorically (i.e., unqualifiedly,unconditionally) true. Analogously,because not all objects are green, not all objects satisfy the formula G(x), and hence (∀x)G(x), the statement that all objects satisfy the formula G(x), is simply,unqualifiedly false.

This pattern reflects the essential distinction between an open formula on the one hand, and a closed formula or sentence on the other.When there is no expression of either the form (∃x) or (∀x) that is attached as prefix to the formula G(x), we say that the occurrence of x in G(x) is free or that x occurs in G(x) as a free variable.When G(x) is prefixed by either (∃x) or (∀x), the variable x in G(x) is said to fall within the scope of the quantifier (either (∃x) or (∀x), as the case may be), and we say that the occurrence of x in G(x) is bound or that x occurs in G(x) as a bound variable.(An occurrence of a variable in either the quantifier (∃x) or (∀x) is always classified as bound).

Course Notes Page 10 Quantification

Whether G(x) is satisfied depends upon whether or not the object in U that is associated with the variable x by a givenassignment belongs to the set corresponding to G. In other words, some assignments of objects to x produce true statements and some assignments of objects to x produce false statements. Indeed, G(x) may be conceivedasafunction from U to the set containing the two values true and false.Onthe other hand, because there exists at least one green object, (∃x)G(x) is simply true, and because there exists at least one object that is not green, (∀x)G(x) is simply false. So although the variable x appears in the expression (∃x)G(x), it does not behave likeatrue variable because the truth of these quantified statements does not depend upon what object is assigned to x. Because the attachment of the universal or existential quantifiers as prefix to G(x) has the effect of removing all variation from the quantified expression no less completely than would the substitution of a constant for the variable x, it has the effect of making the truth of these quantified statements something completely unconditional.For this reason, a variable x in a formula is sometimes referred to as a real variable when it does not fall within the scope of a quantifier,and is referred to as an apparent or dummy variable when all its occurrences fall within the scope of a quantifier. Thus x functions as a real variable in G(x), and functions as an apparent variable in (∀x)G(x) and (∃x)G(x).

There is a powerful and elegant device, invented by Tarski, that permits a unified treatment of all formulae possessing anynumber of free and bound variables. It works as follows: Wefixan arrangement of all individual variables of the language into a denumerably infinite sequence (x,y,z...). An assignment or valuation v of set of variables {x,y,z,...} associates with each variable in the set of variables {x,y,z,...} exactly one individual of the domain U. Although no variable is assigned more than one element of U (v is a function), anynumber of variables of the sequence can be assigned the same individual of U, meaning that among the class of possible valuations, there are an infinite number of valuations that assign the same individual to all variables. Each possible denumerable sequence s=(o0,o1,o2,...) of individuals from U represents a particular assignment v,a comprehensive assignment of objects of U to variables, associating o0 with x, o1 with y,o3 with z, etc. (The subscripts attached to the letter ‘o’ are present merely to distinguish among objects of U, and are not meant to indicate anyordering among these objects. Thus o1 and o4 may be the same object). There are a super-denumerably infinite number of possible assignments, as manyasthere are distinct ways to associate a fixed denumerable set of objects {o0,o1,o2,...} to the fixed infinite sequence of variables (x,y,z,...). (The cardinality of the set of possible assignments V is equal to the cardinality of R,the set of real numbers). In other words, there is a one-to-one correspondence between the class of possible valuations V and the class of infinite sequences S that can be formed from elements of U. Thus, for example, suppose that our underlying domain U is N,the set of nat- ural numbers {0,1,2,3,...}. The sequence of numbers (1,1,3,1,1,1,...) defines an assignment of 1 to x, 1toy,3toz,and 1 to every other variable; the distinct sequence (1,3,1,1,1...) defines an alternative assignment, one that associates 1 to x, 3 to y,and the number 1 to every other variable.

We say that the sequence s=(o0,o1,o2,...) satisfies a formula F if and only if the objects that are assigned to the free variables in F by the comprehensive assignment of objects to all variables (rep- resented by the sequence (o0,o1,o2,...)) satisfy the conditions defined by F.For example (a number of examples are indispensable here), suppose as givenaninterpretation I that assigns to the monadic predicate G the property green, and consider the sequence (the world’slargest blade of grass,Mars,the orange I just ate,the fire engine that just went by,the skyatnight,the skyatnight,...). The formula G(x) will be satisfied by this particular sequence, as well as by anyassignment v that assigns a green object (say a particular blade of grass) to the variable x, whateverobjects v assigns to the other variables y,z,etc. In other words, the formula G(x) is satisfied by all sequences, such

Course Notes Page 11 Quantification as the one above whose first six terms we listed, whose first element is a green object. What about the formula G(y)? Because y is the second variable in the canonical sequence of variables, G(y) is satisfied by all sequences whose second place is occupied by a green object, whatevercolor the objects are that occupythe first place, the third place, or anyplace beyond the third.

Consider nowaninterpretation J that possesses as domain the set of natural numbers {0,1,2,...} and that assigns to the two-place predicate G the relation ‘greater than’. The formula G(y,z) (y>z) is satisfied by all sequences whose second terms are greater than their third terms. Thus, for example, the sequences (5,7,6,1,6,1,6,1,6,...) and (9,8,4,9,9,9,...) satisfy the formula G(y,z), and the sequences (5,6,7,5,6,5,6,5,6,...) and (9,5,5,4,9,9,9,...) do not. Finally,consider the formula A(x,y,z), where A is interpreted as the predicate ‘x+y=z’. This formula is satisfied by the sequence (5,6,11,2,3,4,...) as well as by all other sequences possessing the property that the sum of the magnitudes of the first and second components is equal to the magnitude of the third.

Note the following general characteristic shared by all these cases: In determining whether a sequence s satisfies a formula F,weonly need to examine the terms of s that correspond to the positions (in the fixed ordering of variables) of the free variables of F.Thus supposing that G(x) is the simple formula above whose only free variable is x, we only need to look at the first element of asequence in order to tell whether that sequence satisfies G(x): If the first term of the sequence is a green object, the sequence satisfies G(x) whateverobjects occupythe other places of the sequence, and hence every sequence whose first term is a green object satisfies G(x). In other words, the pos- session by a sequence of a green object in its first position is a necessary and sufficient condition for its satisfaction of the formula G(x). Likewise, we need only look at he second element of a sequence to determine whether that sequence satisfies G(y). All sequences whose second position is occupied by a green object satisfy G(y), whateverthe color of the objects that occupyeither the first position or anyposition beyond the second. This pattern generalizes in the natural way to formulae containing more than one free variable. Determining whether a sequence of numbers satisfies the formula A(x,y,z) requires an examination of only three positions in the sequence—giventhe ordering of variables we have fixedabove,the three salient positions in the sequence are the first, the second, and the third. Every sequence whose first and second term adds up to its third term satisfies A(x,y,z), the other positions of the sequence playing no role whatsoever. Wesuppose (as is stan- dard) that every particular formula of our language possesses a finite number of symbols, and hence afinite number of free variables. Whythen do we bother with infinite sequences in these defini- tions? The reason that we employinfinite sequences is that there is no set bound either on the finite number of free variables a formula contains or on the position in the ordering of variables a free variable in a formula may be selected from.

Truth functional combinations of formulae with free variables behave aswould naturally be expected. If Gdenotes the set of green objects, ¬ Gdenotes the complement of G in U, the set of “non-green” objects, i.e., the set of all objects that are not green. Asequence s satisfies ¬ G(x) if and only if sdoes not satisfy G(x). In other words, anysequence that does not satisfy G(x) satisfies ¬G(y), and vice-versa. Extending and generalizing to possibly polyadic (n-place) formulae F, G, H, etc. with anynumber of free or bound formulae, we have the following: 1) Asequence s satisfies the formula ¬ F if and only if sdoes not satisfy F.2)A sequence s satisfies the formula F∧G,the con- junction of formulae F and G,ifand only if s satisfies Fand s satisfies G.3)A sequence s satisfies the formula F∨G,the disjunction of formulae F and G,ifand only if s satisfies Forssatisfies G (i.e., if and only if s satisfies at least one of the formulae F and G). 4) Asequence s satisfies the formula F->G,the conditional with formula F as antecedent and G as consequence, if and only if s

Course Notes Page 12 Quantification satisfies ¬ Forssatisfies G.5)A sequence s satisfies the formula F←→ G,the biconditional whose first formula is F and whose second formula is G,ifand only if s satisfies both F and Gorssatis- fies neither F nor G.

The non-obvious cases involvethe relationships between the satisfaction of formulae with free vari- ables x, y,z,etc. and the satisfaction of formulae in which these variables x, y ,z, etc. are bound by quantifiers. Because by this point the extensions to formulae with a multiplicity of free variables should be reasonably clear,werestrict ourselves to the simplest case of a formula with a single free variable, G(x). Consider again the interpretation I that assigns the property green to the predicate G. The basic idea is that the satisfaction of (∃x)G(x) by an arbitrarily chosen sequence (in a given interpretation I)iscontingent upon the existence of a sequence whose first term is a green object. In other words, an arbitrarily chosen sequence s satisfies the closed formula (∃x)G(x) with the vari- able x bound (quantified) because there exists a sequence that satisfies the formula obtained from (∃x)G(x) by removing the prefix that binds (quantifies) the variable x.

More precisely (nowhold on to your hats!), we say that a particular sequence s satisfies the closed formula (∃x)G(x) if and only if there exists a sequence differing from s in at most the first place that satisfies the open formula G(x). (Anysequence that differs from s in at most a single place w is called a w-variant of s). In other words, s satisfies (∃x)G(x) in the case that s itself satisfies G(x), as well as in the case that a distinct sequence—one that differs from s in its first position— satisfies G(x). Thus a sequence s whose first term is the fire engine that just went by satisfies (∃x)G(x) because there exists a sequence—in this case a sequence distinct from s—that satisfies G(x), for example, anychosen sequence whose first term is the largest leaf on Earth. In this case, the sequence s whose first term represents the fire engine that just went by satisfies (∃x)G(x) because there exists a sequence differing from s solely in its first place that satisfies G(x). In the case of a sequence s* whose first term is the world’slargest green leaf, s* satisfies (∃x)G(x) because s* is a sequence that doesn’tdiffer in anyplace at all from a sequence (s* itself) that satisfies G(x). It is because we want to accommodate both cases in a single clause that we employthe superficially puzzling phrase ‘at most’. Givenwhat was said before about the relevance of only those positions in a sequence corresponding to the free variables of a formula, the definition of satisfaction for an existentially quantified formula implies that if one sequence satisfies G(x), then all sequences satisfy (∃x)G(x).

The case of universally quantified formulae is exactly analogous. Asequence s satisfies (∀x)G(x) if and only if all sequences that differ from s in at most the first place satisfies G(x). Because not all objects are green, there exist sequences whose first positions are not occupied by green objects, which implies that there exist sequences that do not satisfy the formula G(x). Thus suppose that s is a sequence whose first term is a green object such as the world’slargest leaf. Because there exist sequences differing from s in at most the first place that do not satisfy G(x), s itself does not satisfy the formula (∀x)G(x). Consequently,ifthere exists evenasingle sequence that does not satisfy G(x), then no sequence satisfies (∀x)G(x). More generally,any formula that contains no free vari- ables is satisfied either by all sequences or by none.

We can nowdefine the notion of truth for an interpretation. Aformula F is true for an interpreta- tion I if and only if—in relation to the interpretation I—all sequences satisfy F. Aformula F is false for an interpretation I if and only if—in relation to the interpretation I—no sequence satisfies F.According to this definition, every formula F without anyfree variables—whether it is the case that F contains no variables at all or it is the case that all its variables are bound by quantifiers—is

Course Notes Page 13 Quantification either true or false for an arbitrary interpretation I.(Of course, F will be true for some interpreta- tions and false for others). Aformula F is true for an interpretation I if and only if ¬ F is false for I,and F is false for an interpretation I if and only if ¬ F is true for I.Aninterpretation relative to which each member of a set of formulae Γ is true is called a model of Γ.(According to our defini- tions, anytruth-functional is true for all interpretations).

Aformula is logically valid if and only if it is true in all possible interpretations, and is logically false if and only if it is false in all interpretations. Aformula F,whether it is open or closed, is satisfiable if and only if there exists a sequence that satisfies F in at least one interpretation. Sym- metrically, F is unsatisfiable or contradictory if no sequence satisfies F in anyinterpretation. A for- mula F is logically valid if and only if ¬ F is unsatisfiable, and F is satisfiable if and only if ¬ F is not logically valid.

Aset of formulae Γ is simultaneously satisfiable if and only if there exists at least one sequence s that satisfies all formulae in Γ.Aset of formula Γ logically implies aformula F,or(equivalently) F is a of Γ,ifand only if there is no sequence that simultaneously satisfies each formula of Γ butfails to satisfy F.This means that every model of Γ is a model of F.Two formulae F and G are logically equivalent if and only if Flogically implies G and Glogically implies F,which means that F and G are true in the same models.

Nowitmight seem that 1) if the monadic predicate G can be assigned the property ‘green’ in one interpretation and the property ‘red’ or even‘not green’ in another,and 2) if the the dyadic predicate G can be assigned the relation ‘greater than’ in one interpretation, ‘less than’ in another interpreta- tion, and ‘is equal to’ in a third, and 3) if the domain U may contain anynumber of elements of any character whatsoever, itwill be impossible to find anyformula that will be true in all interpretations or false in all interpretations. This impression is wrong. As we emphasized earlier,Uisalways a non-, G is always a subset of U, and all variables and constants are assigned elements of U. Thus, for example, whatevernon-empty set the domain U corresponds to, and whateversubset of U the predicate G stands for,itwill always be true that anyelement of U will either belong to G or to its complement in U, and hence the formula (∀x)(G(x) ∨ ¬G(x)) is logically valid. At the same time, because no element of U can both belong to a subset of U and not belong to that same subset of U, the formula (∀x)(G(x)∧¬G(x)) is logically contradictory.The formula (∀x)(G(x)∧H(x)→G(x)) is logically valid because G(x)∧H(x) is a subset of G(x) whateverthe identi- ties of G and H, which means that anyobject that belongs to G and Hnecessarily belongs to G. As a final example, we mention the formula ((∃x)(∀y)(xRy))->((∀y)(∃x)(xRy)), which means that if at least one particular object in a domain U possesses the relation R to all objects in U simultane- ously,then for each object y in U, there exists at least one object in U that has the relation R to y. (For example, the existence of evenone truly misanthropic person—the existence of a person who shuns everyone—is sufficient to ensure the truth of the statement that everyone is shunned by some- one or other.(On the other hand, the fact that each person is shunned by someone does not mean that there exists a person who shuns everyone)).

Aformula that is valid for all interpretations in a particular domain is said to be valid in that domain. For FOL=, it is easy to demonstrate that anyformula that is valid in a domain with a defi- nite finite number n of objects is also valid in any domain possessing n objects, and that anyfor- mula that is satisfiable in a domain possessing a definite finite number n of objects is satisfiable in any domain possessing n objects. Even more striking is that for FOL without Identity,itcan be demonstrated 1) that anyformula that is valid in a domain with a definite finite number n of objects

Course Notes Page 14 Quantification is also valid in every domain possessing norfewer objects,and 2) that anyformula that is satisfi- able in a domain possessing a definite finite number n of objects is satisfiable in every domain— finite or infinite—possessing normoreobjects.(Note that because anyformula that specifies the cardinality of a domain is false in all domains that possess a different cardinality,nosuch formula can be logically valid.)

Finally,something should be said about the relationship between provability and logical conse- quence. A well-formed formula (wff) F is provable or deducible from a set of formulae Γ of FOL if F can be derivedfrom Γ according to certain intuitively truth-preserving but formally well-defined rules (transformations) givenwith the definition of the system. Aformula F is a logical conse- quence of a set of formulae Γ if it is impossible for all of the formulae in Γ to be true while F is false. A logically valid formula F of FOL is a formula that no conditions can makefalse, and an unsatisfiable (or self-contradictory)formula F of FOL is a formula that no conditions can maketrue. (A central objective ofthe present exposition is to makethe notions of logical consequence, logical validity,and unsatisfiability precise). In 1930, Kurt G¨o del provedthat all logically valid formulae of FOL are formally derivable from a small set of transparently logically valid formulae of FOL, the (valid) logical axioms of FOL, a circumstance that could not be assumed to hold in advance. The proof that all logical consequences of the logically valid axioms of FOL are deducible from the axioms of FOL (i.e., are theorems of FOL), together with the far simpler proof that all formulae derivable from the axioms of FOL are logically valid, implies the critical metalogical result that the set of theorems of FOL and the set of logically valid formulae of FOL are co-extensive.2

2 This result must not be confused with G¨o del’sfar more well-known and surprising Incompleteness Theorem (usually denoted “G¨o del’sProof”), published in 1931. Very roughly stated, this latter remarkable metamathemati- cal result implies that anyconsistent finitary formal system rich enough to express all arithmetical propositions nec- essarily permits the construction of a formula that is demonstrably both true and unprovable. In other terms (still very rough), consider the class of systems yielded by the augmentation of the generic logical axioms by further axioms that 1) permit the definition of the three operations ‘successor of’, ‘addition’, and ‘multiplication’ and 2) permit us to formally distinguish in finite terms between the well-formed formulae that are axioms and the well- formed formulae that are not. G¨odel demonstrated that for anysuch system, there exists a definite and unambigu- ous proposition (closed formula) G—not necessarily the same for each system—such that neither G nor its negation can be derivedfrom the axioms of that system.

Course Notes Page 15