1 Introduction to Logic

Home , Double turnstile, Polish notation, Term (logic)

Reading: Metalogic Part I, 1-6; Part II,15-19

Contents 1.1 Preliminary Remarks ...... 3 1.1.1 Preliminary motivating remarks ...... 3 1.1.2 What is a Function? ...... 6 1.1.3 Use and Mention/Object Language and Meta Language ...... 9 1.1.4 Alternative Notations ...... 10 1.2 The System P: Informal Semantics ...... 11 1.2.1 Propositional Logic: Introductory Remarks . . . . . 11 1.2.2 Well-Formed Formulas ...... 13 1.2.3 Rules of Resolution ...... 15 1.3 Historical Remarks ...... 17

1.1 Preliminary Remarks

1.1.1 Preliminary motivating remarks

In characterizing a formal system, we state the Formation rules, i.e., the rules for setting down the sequences of expressions that will count as meaningful. Then we state the Transformation rules, i.e., the rules for generating the theorems of the system. And ﬁnally, we state the Translation rules, i.e., the rules identifying what the meaningful expressions speciﬁed by the Formation rules are supposed to mean, so that the theorems of the system will be interpreted as truths. The Formation and Transformation rules of the system are frequently called the Syntax of the system; the Translation rules are frequently called the Semantics of the system.1

1 The terms Formation Rules and Transformation Rules are Carnap’s (1937). (Translation Rules is my term); in the days when logicians were suspicious of 4 1 Introduction to Logic

One of the paradoxical results many of us are familiar with from elementary logic is that everything follows from a contradiction. An argument to this effect, which does not depend on the well-known peculiarities of the material conditional, goes as follows: 1. p ∧ ¬p Assumption 2. p Simplification, 1 3. p ∨ q Addition, 2 4. ¬p Simplification, 1 5. q Disjunctive Syllogism, 3, 4 So, assuming a contradiction, ‘p ∧ ¬p’, we have been able to derive an arbi- trarily chosen statement, ‘q’. This is an example of a derivation within a system of natural deduction, which we might call S. A derivation is a sequence of steps, each of which is a premise or an assumption or the result of applying one of the Deduction (i.e. Transformation) Rules in the system to earlier steps in the derivation. Since we have ‘q’ on the last line of the derivation and ‘p ∧ ¬p’ our only assumption, we say ‘q’ is derivable from ‘p ∧ ¬p’ in the system S. In mathematical notation, we express this as follows:

‘p ∧ ¬p’ `S ‘q’ (‘`’ is called the turnstile.) Three Deduction Rules have been used in the derivation:

A ∧ B Simpliﬁcation: A

A Addition: A ∨ B

A ∨ B, ¬A Disjunctive Syllogism : B To apply the rule of Simplification to line 1. in the derivation, we had to be able to identify the formula on line 1. as a conjunction. Our notation must be sufficiently precise so that the identification can be done purely on the basis of the shapes in the notation: a machine, or an individual lacking any understanding of the intended meaning of the shapes, should be able to make this identification. This is not a totally trivial problem. There are two connectives in the formula, and it must be clear that the conjunction- sign is the main connective. Similarly, to apply the rule of Addition, we had

anything semantical in logic, to specify a formal system was just to state the Formation Rules and the Transformation Rules. 1.1 Preliminary Remarks 5 to identify the formula on line 3. as a disjunction. So, we require a precise characterization of the notion of a meaningful sequence of expressions and a precise characterization of the notion of the main connective in a formula. Once we are satisﬁed that our notation is precisely characterized, then, to complete our description of the syntax of our language, we must specify the derivation rules and what is to count as a derivation. If we were to continue the list of rules in the spirit of those already given, we would produce a list of Introduction-and-Elimination Rules, or, as they are fondly known, Intelim Rules, for each of the logical symbols of our language. Addition is an introduction rule for ∨ because it enables us to introduce a formula using ∨ into the derivation; Disjunctive Syllogism is an elimination rule for ∨ because it enables us to break up a previously derived disjunction and write one of the disjuncts on a distinct line. Each of the derivation rules in the language is intended to capture some elementary valid inference. The rule of Simpliﬁcation only says that, given a line containing a conjunction, we can add on a new line to the derivation containing one of the conjuncts. (So it is an ∧-elimination rule.) Intuitively, this is supposed to capture the fact that if a conjunction is true, each conjunct is true. Addition holds because a disjunction is true if one of the disjuncts is true. We want the steps in our derivation to be such that if we start out with true premises, then each step the rules enable us to add on to the derivation will also be true. If we start out with truths and stay on the path of righteous- ness (in this context, that means obeying the laws of logic), then we will never stray into falsehood. We want the syntactical manipulations admissible in the setting up of a derivation to mimic the intuitive semantic notion of “follows from” or “is a logical consequence of.” We say that ‘q’ is a logical consequence of ‘p ∧ ¬p’ in the structure S. which is expressed in mathematical notation as:

‘p ∧ ¬p’ |=S ‘q’ (‘|=’ is called the double turnstile.) if any interpretation which makes ‘p ∧ ¬p’ true in a structure also makes ‘q’ true. We need, then, to set up the semantics of our language. We have to make precise the notion of an interpretation, which is supposed to set out what the elements of the notation mean. For the case of truth-functional logic, the intended structure will consist of the set of truth values T,F and the boolean functions on this set. An interpretation links up elements of the notation with elements of the structure. ‘∧’, for example, gets interpreted as the boolean function f∧ : {T,F } × {T,F } → {T,F } such that T if x = T and y = T f (x, y) = ∧ F otherwise 6 1 Introduction to Logic

That is, a conjunction of two truths is true; a conjunction of any other com- bination of truths and falsehoods is false. There are two key theorems we want to prove. The ﬁrst assures us that our rules are truth-preserving, i.e., that whenever we can derive a statement from a set of statements that the argument is intuitively a valid one. Let Γ be a set of statements and let γ be an individual statement. Then we can state this theorem as follows: Soundness If Γ ` γ, then Γ |= γ The second assures us that our rules capture all of those inferences we intuitively believe to be valid under the intended interpretation of the logical symbols: Completeness If Γ |= γ, then Γ ` γ

1.1.2 What is a Function?

We associate with each open sentence ‘Px’, that we intuitively take to express a property, a set P = {x | P x}.2 If we were to take ‘Px’ to be ‘x is a parent’, then P is the set of parents. If ‘Pa’ is true, we say a ∈ P, i.e., a is an element or member of P. Let C be the set of children. Since every parent is a child, we have the following relation holding between the two sets: If x ∈ P then x ∈ C. We abbreviate this P ⊆ C, i.e., P is a subset of C. Since there are children who are not parents, i.e., since there exists at least one x such that x ∈ C but x 6∈ P, P is said to be a proper subset of C, denoted P ⊂ C. And since there exists at least one x such that x ∈ C but x 6∈ P, the two sets are not identical, for sets are identical iﬀ their members are: P = C ↔ (V x)(P x ≡ Cx). We distinguish the element x from the singleton {x}: x ∈ {x} but x 6= {x}. We distinguish the set {x, y} from the ordered pair < x, y > which is frequently represented as {{x}, {x, y}}. The null or empty set, designated Φ is a set that has no elements so that for every set S, Φ ⊆ S. Here are some important operations on sets: • The intersection of P and C P ∩ C = {x | P x ∧ Cx}. P ∩ C is the set of all things that are in both sets, P and C. Following through with our example, it is the set of all things that are both parents and children.3 • The union of P and C P ∪ C = {x | P x ∨ Cx} P ∪ C is the set of all things that are in one or the other of the two sets P and C. It is the set of all things that are either parents or children.

2 Of course, such an unrestricted Comprehension Axiom of Set Theory would allow the paradoxes to be generated; precise statements guard against them. 3 Since in this particular case P ⊆ C, P ∩ C = P 1.1 Preliminary Remarks 7

• The Power Set of P 2P = {S | S ⊆ P}. 2P is a set, each of whose elements is a set. It consists of all possible groupings of parents. So, it includes P itself, the set of all parents, as well as the set of male parents, of female parents, of French parents, of parents of twins, and so on. We also have P + C = {x | x ∈ P or x ∈ C but not both }. And P − C = {x | x ∈ P ∧ x 6∈ C}. Satisfy yourself that each of the following is true: P ⊆ P ∪ C, P ∩ C ⊆ P, P ⊆ 2P

We associate with each open sentence Rx1x2 . . . , xn, that we intuitively take to express an n-ary relation, a set R = {< x1, x2, . . . , xn >| Rx1x2 . . . , xn} The most usual type of relation is a binary relation relating two things, e.g. η is a parent of ζ. In set theoretic terms, this relation associates elements of the set of parents with elements of the set of children, and we represent the relation as the set of ordered pairs < x, y > whose ﬁrst element x is a parent and whose second element y is a child of that parent. Here are some examples of such ordered pairs: Let us call the set of all such ordered pairs R. Note that we distinguish R from its inverse, designated R−1, where < y, x >∈ R−1 ↔< x, y >∈ R Where P is the set of parents and C is the set of children, P × C, called The Cartesian Product of P and C, is {< x, y >| x ∈ P ∧ y ∈ C}. R ⊆ P × C. Actually, R ⊂ P × C, because the ordered pair ∈ P × C but 6∈ R. We might restrict the ordered pairs in the relation R to those involving only children who are themselves parents. That restricted 2 relation is a subset of P × P, (denoted P ). X1 × X2 × · · · × Xn is the set of all n-tuples < x1, x2, . . . , xn > where xi ∈ Xi. X × X × · · · × X is the set of | {z } n all n-ary operations on X , and is designated X n.A function is a special kind of relation. The most usual kind of function, a singulary function, associates elements of one set, called the domain of the function, with elements of another (not necessarily distinct), called the range of the function. The domain of the function F, Dom(F) = {x |< x, y >∈ F}; the range of the function F, Ran(F) = {y |< x, y >∈ F}. A function, F is a relation that satisﬁes the following condition: If < x, y >∈ F and < x, z >∈ F, then y = z. 8 1 Introduction to Logic

If F is a function and < x, y >∈ F, then y is said to be the value or image of the function F for the argument x.4 The more usual notation for this is: F(x) = y. The relation R above is not a function, because it associates more than one element of the range with a given element of the domain: a parent can have more than one child. If, however, we were to restrict the relation so that a parent is associated with its first-born child, we would have a function. That function would contain the ordered pair but not the ordered pair . The function F from P into C has Dom(F)⊆ P and Ran(F)⊆ C. We say that a function G is one-to-one or 1 − 1 if If < x, y >∈ G and < z, y >∈ G, then x = z.5 F is not 1-1, because a child will have more than one parent. However, if we were to consider the relation between fathers and their first-born, we would have a 1-1 function. A function G is onto C if Ran(G) = C.6 Neither of the two functions just defined are onto C because not all children are first-born. X Y is the set of all functions from Y into X . This notation has already been used by us. We used 2X to specify the Power Set of X . We can explain this notation by means of an example. Let X = {a, b}, i.e., the set consisting of the two elements a and b. And we identify the number 2 with any two element set: the one we like in logic is {>, ⊥}. Now here are all the subsets {a, b}: Φ {a} {b} {a, b} These correspond, respectively, to the following functions from {a, b} into {>, ⊥}: {< a, ⊥ >, < b, ⊥ >} {< a, > >, < b, ⊥ >} {< a, ⊥ >, < b, > >} {< a, > >, < b, > >} Each of these is a characteristic function for the respective set, associating an element with > if it is in the set and associating it with ⊥ if it is not in the set.7 The notation X n for X × · · · × X is also to be understood as the set | {z } n 4 It is because a function associates a unique element with each argument that we can speak of the value for a given argument. 5 That is, in the more usual notation, G is 1-1 if x = y whenever G(x) = G(y). 6 Alternate terminology:an injection is 1-1, a surjection is onto, and a bijection is 1-1, onto. 7 These are Frege’s Werthverlau¨fe. 1.1 Preliminary Remarks 9 of all functions from the set consisting of n integers into X . Consider, as an example, X = {a, b}. Then X × X consists of the following elements: < a, a > < a, b > < b, b > < b, a > Then X 2 = {a, b}{1,2} has the following elements, respectively, {< 1, a >, < 2, a >} {< 1, a >, < 2, b >} {< 1, b >, < 2, b >} {< 1, b >, < 2, a >}

1.1.3 Use and Mention/Object Language and Meta Language

This is a course in metalogic, so it is about logical systems. It is important that we keep clear the diﬀerence between the language we use to talk about a logical system (the meta language) and the language of the logical system we are talking about (the object language). The standard philosophical convention for signaling that an expression is being talked about is to enclose that expression in single quotes. So, as Quine reminds us, it is not Boston that is disyllabic, but ‘Boston’. So, for example, when we said

‘p ∧ ¬p’ |=S ‘q’ we indicated that one expression was a logical consequence of another: so we placed each of the expressions in single quote marks because we were talking about them. Observing the distinction between use and mention is quite important; failure to observe it can lead to confusion and error.8 Frequently, however, the quote marks get in the way, and we drop them when there is little likelihood of confusion. So, we will (as Hunter does) drop quote marks in cases like the example above and write instead

p ∧ ¬p |=S q Hunter’s system of propositional logic, P, contains various symbols: the propositional symbols, connectives, and left and right parentheses. These be- long to the object language. But if we want to speak about various formulas in P, we must use the metalanguage. So, we will use calligraphic letters, A, B, C,... to speak generally about any formulas of the system. And the connectives ‘⊃ and ¬ will be included in the metalanguage as well as the object language. This enables us to speak of a conjunction A ⊃ B without specifying the actual propositional symbols occurring in it, or even specifying

8 The classic ﬁre and brimstone sermon on the topic is to be found in W.V.O. Quine, Mathematical Logic. 10 1 Introduction to Logic how complex A and B might be. Again, we trust that this will not cause any confusion.

1.1.4 Alternative Notations

English does not have parentheses to group phrases as our symbolic notation does. English has grouping words, however: ‘either’ goes with ‘or’, ‘both’ goes with ‘and’, and ‘if’ goes with ‘then’. A judicious placement of ‘either’ will render ‘A and B or C’ unambiguous. ‘Either A and B or C’ is unambiguous and symbolized as ‘(A ∧ B) ∨ C’, while ‘A and either B or C’ goes in as ‘A ∧ (B ∨ C)’. And ‘If if A then B then C’ goes in as ‘(A ⊃ B) ⊃ C)’, while ‘If A then if B then C’ goes in as ‘A ⊃ (B ⊃ C)’. It is possible to have a logical symbolism without parentheses. In Principia, Russell and Whitehead used a system of dots instead; Quine adopted this system in Methods of Logic. By placing a dot at the side of a connective, the scope of a connective occurring on that side is broadened. So, for example, (A ⊃ B) ⊃ C) would be represented as A ⊃ B. ⊃ C, while A ⊃ (B ⊃ C) would be represented as A ⊃ .B ⊃ C. It might turn out that more than one dot would be needed to disambiguate properly. For example, A ⊃: B ∨ C. ⊃ A is A ⊃ ((B ∨ C) ⊃ A). It is possible to have a logical symbolism with no punctuation needed to group subformulas. Such a symbolism was devised by the Polish logician Lukasiewicz, and has subsequently been called Polish Notation. (Prior (1955) favors this notation in his work.) Here is the notation:9 ¬ p Np p ∧ q Kpq p ∨ q Apq p ⊃ q Cpq p ≡ q Epq We leave it as an exercise to the reader to distinguish CCpqr from CpCqr, ApKqr from AKpqr, and to read CCpCqrCCpqCpr CCpAqrACpqCpr CCCpqpp In Polish notation, the logical symbols are prefixes. In traditional notation, they are infixes. They can also be postfixes. The HP calculators employ what is known as Reverse Polish Notation for symbolic entry. To calculate ‘3 + 5’, you first enter the ‘3’, then the ‘5’, and finally the ‘+’. How would you enter “5 × (4 + 2)’?

9 The quantiﬁer notation is as follows: V (W xi)B ΠxiB ( xi)B ΣxiB 1.2 The System P: Informal Semantics 11 1.2 The System P: Informal Semantics

1.2.1 Propositional Logic: Introductory Remarks

For the sentential connectives, we adopt the following truth table deﬁnitions:

A ¬A T F F T

AB A∧BA∨QA⊃BA≡B TT TTTT TF FTFF FT FTTF FF FFTT And we also adopt the following informal guidelines for translating the truth functional connectives:10 ¬A ... It is not the case that A A ∧ B ... A and B A ∨ B ... A or B, A unless B A ⊃ B ... If A then B, A only if B, B if A A ≡ B ... A if, and only if B As for the connectives, we note the following: conjunction The two constituent statements in a conjunction are called conjuncts. There is no temporal priority in the ordering to the conjuncts by contrast with our ordinary ‘and’ which frequently does carry such an im- plication. For example, compare ‘He got knocked on the head and fell down’ and ‘He fell down and got knocked on the head’. It is natural to regard the ﬁrst as implying that he fell down as a result of the blow to the head, and the second as implying that he received a blow to the head as a result of the fall. Logically, however, conjunction is commutative, i.e., ordering does not matter, so that A ∧ B is logically equivalent to B ∧ A. disjunction The two constituent statements in a disjunction are called disjuncts. (Quine favors the terminology of alternation and alternates.) We have deﬁned the inclusive ‘or’: one or the other or both. The exclusive ‘or’ (which computer scientists call ‘xor’) means: one or the other but not both. Latin has two words for ‘or’ corresponding to these two boolean functions,

10 There are a number of alternative symbolisms. For negation -, ¬; for conjunction &, ·; for disjunction +; for the conditional →; for the biconditional ↔. 12 1 Introduction to Logic

AB A vel BA aut Q TT TF TF TT FT TT FF FF Our ∨ is derived from the ‘v’ in ‘vel’. The mathematical ‘or’ is clearly inclusive. The situation in ordinary English is a bit more complicated. A high proportion of natural language informants who have not been exposed to logic will aver theirs is the ‘exclusive’ or. One fallacy in popular reasoning that leads to this belief is worth removing.11 One might think that ‘Sally is either at A&S or Macys’ exhibits the exclusive ‘or’. Sally cannot be at both places at the same time, so she is at one place or the other, but not both. But the question to bear in mind is, ‘What is doing the excluding here?’ Is it the ‘or’ that is forcing us to choose only one of the alternatives, or is it the fact that the alternatives are mutually exclusive? The alternatives are mutually exclusive, so they cannot both be true. Does this show that the ‘or’ is therefore inclusive? No. It cannot. It does, however, defang one of the most persistent reasons for thinking ‘or’ is exclusive. But why does it fail to show that ‘or’ is inclusive? Because, the only line on the truth table that distinguishes the inclusive from the exclusive ‘or’ is the top line when both disjuncts are true. To show we have an exclusive ‘or’, then, we need a situation in which both disjuncts are true and the disjunction is false. This is not an easy case to come by. (Test intuitions about ‘or’ by considering the negation of a disjunction, e.g., ‘Sally didn’t take either French or Spanish in high school’, and then calculate what this implies about the truth table for ‘or’.) conditional A ⊃ B symbolizes the material conditional, as it was dubbed by Russell (some call it the Philonian Conditional after the ancient Stoic philosopher who first clearly stated its truth conditions.12 The if clause is called the antecedent; the then clause is called the consequent. The material conditional corresponds to the mathematicians usage, on which A ⊃ B means Either A is false or else A is true and (so) B is true. There is considerable controversy about the connection between the material conditional and the ordinary conditional, and an even larger and ever-growing literature about different types of ordinary conditionals, e.g., indicative, subjunctive, contrary to fact. (Jeffrey (1990) uses Gricean ideas to defend the material conditional: his examples and discussion are ex- cellent. But the Gricean defense is deeply flawed.) It is, perhaps, worth remarking (and, on reflection, not so surprising) that the ‘if, then’ in computer languages is just the material conditional. Think of a command, e.g.,

11 The argument is from Quine (1950). 12 See Kneale and Kneale (1962) for discussion. 1.2 The System P: Informal Semantics 13

‘Shut the door,’ as telling someone to make a particular statement true, namely, ‘The door is shut.’ Then an ‘If, then’ command in BASIC, say, will be understood by the computer to make the ‘if, then’ statement true. Consider a typical BASIC command: (15) If x > 0 then goto step 25. How does the computer understand this? If the antecedent is true, i.e., if x > 0, then the computer does whatever the consequent tells it to do, in this case, go immediately to step 25. If the antecedent is false, the computer ignores the consequent and simply proceeds to the next command. That is, if the antecedent is true, the computer must make the consequent true for the conditional to be true; if the antecedent is false, the conditional is already true and the computer does not have to do anything to make it true. A last note, also from Quine. We must be careful about distinguishing If A then B from A implies Q. The former says that ‘A ⊃ B’ is true. The latter says that ‘A ⊃ B’ is logically true.

1.2.2 Well-Formed Formulas

Hunter’s system P has the following vocabulary: • propositional symbols: p, p0, p00, p000,... • connectives: ¬, ⊃ • punctuation marks: (, ) Of the infinite number of finite sequences of these elements, we single out the meaningful sequences, namely, the well-formed formulas.13 We define the set of wffs inductively: Definition 1.2.1 (Well-Formed Formula of PS) (1) A propositional symbol is a wff; (2) If A is a wff, then ¬A is a wff; (3) If A and B are wffs, then (A ⊃ B) is a wff; (4) The only wffs are those obtained by (1)-(3). None of the following are wffs of the system: (p00 ¬(p00) p00 ⊃ p0 Parentheses can only be introduced with the horseshoe. The first contains no horseshoe, so it cannot contain parentheses—and, of course, if parentheses are introduced, they would have to be introduced in pairs; the second contains no horseshoe, so it cannot contain parentheses; the third contains a horseshoe and no parentheses. On the other hand each of the following is a wff: p00 ¬(p0 ⊃ (¬p00 ⊃ p0)) (p000 ⊃ ¬(p0 ⊃ ¬p))

13 wﬀs, for short; the adjective well-formed is abbreviated wf. 14 1 Introduction to Logic

We can establish that the second, for example, is a wff of the system by means of the following construction sequence 1. p0 (1) 2. p00 (1) 3. ¬p00 (2), from 2. 4. (¬p00 ⊃ p0) (3), from 1. and 3. 5. (p0 ⊃ (¬p00 ⊃ p0)) (3), from 1. and 4. 6. ¬(p0 ⊃ (¬p00 ⊃ p0)) (2), from 5. Each line in this construction is either a propositional symbol (as per (1)), or the negation of an expression that has already been established to be a wff (as per (2)), or a conditional enclosed in parentheses, each of whose constituents is a previously established wff (as per (3)). Part (4) of the definition of a wff assures us that there is a construction sequence for every wff.14 Informally, it is clear from the definition of a wff that every wff will be a propositional symbol, or it will have either of the two forms ¬A or (A ⊃ B), where A, B are both wffs. Moreover, informally, it is clear that no complex wff can be viewed as having both forms, ¬A and (A ⊃ B). The notion of a construction sequence will come in handy later on, because we will prove some theorems about the system P by induction on the length of a construction proof for a wff. Definition 1.2.2 (Immediate Subformula) An immediate subformula of a wff A will be B if A is ¬B or either B or C if A is (B ⊃ C). Definition 1.2.3 (Subformula) A subformula of a wff A will be either an immediate subformula of A or an immediate subformula of a subformula of A.15 Having identified what we intend to be the meaningful sequences of expressions of P, we should now say something about what we intend them to mean. By an Interpretation for P, we mean an assignment of truth values to each of the propositional symbols, and an assignment of truth-functions to each of our propositional connectives in the intended way. That is, we assign ¬ the function f¬ : {T,F } → {T,F } such that T if x = F f (x) = ¬ F if x = T and we assign ⊃ the function

f⊃ : {T,F } × {T,F } → {T,F } such that 14 Question: Can there be more than one construction sequence for a wﬀ? 15 Question: Which are the subformulas of ¬(p0 ⊃ (¬p00 ⊃ p0))? 1.2 The System P: Informal Semantics 15

F if x = T and y = F f (x, y) = ⊃ T otherwise If all goes according to plan, then each interpretation provides us with a unique truth value for every complex wff of P. This is not a matter of chance. The idea is that the truth value of a complex wff is supposed to be a function (so unique) of the truth values of its constituent propositional symbols.16 It can be proved, though we will put off the proof until we have discussed proof by induction: we shall need to show that each complex wff decomposes uniquely. Note that Hunter takes only ¬ and ⊃ as primitive. The other connectives are easily defined in terms of these primitives. So,

• (A ∨ B) =def (¬A ⊃ B) • (A ∧ B) =def (¬(A ⊃ ¬B)) • (A ≡ B) =def ((A ⊃ B) ∧ (B ⊃ A))

17 ℵ0 For ℵ0 propositional symbols, there are 2 distinct interpretations. In any given wff, there will only be a finite number n of propositional symbols, and so only 2n distinct interpretations of that formula. When we set up a truth table for a wff, we consider all possible assignments of truth values to the propositional symbols occurring in the wff, and then calculate the truth value of that wff for each such assignment. Let’s work out a truth table for the wff ¬(p0 ⊃ (¬p00 ⊃ p0)). We consider all possible truth value assignments to the propositional symbols p0, p00, and then calculate the truth value of each of the subformulas of the wff until we can calculate the truth value of the wff itself. The truth table looks like this:

p0 p00 ¬p00 (¬p00 ⊃ p0)(p0 ⊃ (¬p00 ⊃ p0)) ¬(p0 ⊃ (¬p00 ⊃ p0)) TT FTTF TF TTTF FT FTTF FF TFTF

We ﬁnd out that our wﬀ is F for all possible assignments.

1.2.3 Rules of Resolution

Truth tables grow geometrically in size with each new propositional letter. A wff containing 1 variable will have 21 = 2 lines, a wff containing 2 variables will have 22 = 4 lines, a wff containing 3 variables will have 23 = 8 lines, a wff containing 4 variables will have 24 = 16 lines, and so on. So truth tables, although a mechanical way of computing the truth value of a complex for any assignments of truth values to its constituent propositional symbols, becomes

16 This property of the logical system is known as Compositionality. 17 This is the number (in the sense of cardinal number of the integers. It is the smallest transﬁnite number. 16 1 Introduction to Logic

unwieldy rather fast. In this section, we consider an alternative procedure described in Quine’s Methods of Logic. This is the method of Resolution. There are 8 Rules of Resolution to bear in mind. (i) Delete T as a component of conjunction. (ii) Delete F as a component of disjunction. (iii) Reduce a conjunction with F as a component to F. (iv) Reduce a disjunction with T as a component to T. (v) If a conditional has T as antecedent or consequent, drop the antecedent. (vi) If a conditional has F as antecedent or consequent, negate the antecedent and drop the consequent. (vii) Drop T as a component of a biconditional. (viii) Drop F as a component of a biconditional and negate the other side. In addition, we have the obvious rules governing negation: ¬T becomes F ; ¬F becomes T . Each of these rules is easily justified. Corresponding to each of the rules are the following facts: (i) (T ∧ B) ≡ B. (ii) (F ∨ B) ≡ B. (iii) (F ∧ B) ≡ F . (iv) (T ∨ B) ≡ T . (v) (T ⊃ B) ≡ B and (B ⊃ T ) ≡ T . (vi) (F ⊃ B) ≡ T and B ⊃ F ) ≡ ¬B. (vii) (T ≡ B) ≡ B. (viii) (F ≡ B) ≡ ¬B. We use the rules of resolution as follows. We pick one of the propositional symbols occurring in the wff and first assign it T and then assign it F . In each case, we resolve as far as we can. If we reach only T ’s and F ’s, we are done. But if we resolve to another wff, then we follow the same procedure, picking one of the propositional symbols in this wff and first assigning it T and then F . We continue in this way until we end up only with T ’s and F ’s. If we end up with all T ’s, our original wff is a tautology; if we end up with all F ’s, it is a contradiction. If we end up with at least one T , it is consistent. Here is an example of how we would resolve for the wff

(p0 ⊃ (p0 ⊃ p00))

p0 = T p0 = F T ⊃ (T ⊃ p00) F ⊃ (F ⊃ p00 T ⊃ p00 T p00 p00 = T p00 = F TF 1.3 Historical Remarks 17

We pick a propositional symbol, p0, and first assign it T and then assign it F . Working down the left side, we dropped T as antecedent twice, resolving to p00. We then had to let this propositional symbol vary among interpretations. In terms of truth tables, we found that our wff was T for the top line and F for the second. We now work out way down the right hand side to see what happens for the bottom two rows. And we note that since we have F in the antecedent, we can negate it and so it becomes T . Or wff, then, is consistent and equivalent to (p0 ⊃ p00) Here is how we would work out our wff from before:

¬(p0 ⊃ (¬p00 ⊃ p0))

p0 = T p0 = F ¬(T ⊃ (¬p00 ⊃ T )) ¬(F ⊃ (¬p00 ⊃ F )) ¬(T ⊃ T ) ¬T ¬TF F We choose p0, and set it ﬁrst T and then F . Working down the left branch, we drop the antecedent when the consequent is T —twice; and then, the negation of T is F . So, the top two rows of the table are F . Working down the right hand side, we know that F in the antecedent becomes T , and since this is negated, we resolve to F . The bottom two rows are also F . So, ours is a contradiction, just as we found out earlier.

1.3 Historical Remarks

The study of Logic goes back deep into ancient times. Aristotle is widely regarded as the one who put the study of Logic on a firm foundation. His Organon consisted of four books on logical themes. De Interpretatione set out principles governing logical form, truth, and grammatical form; the Prior Analytics included a study of deductive logic, primarily of the syllogism; the Posterior analytics included a study of scientific reasoning; the Sophistical Refutations included a study of informal arguments, debating tricks and fal- lacies. Other work in antiquity was carried through by the Stoic logicians— Diodorus, Cleanthes, and Philo—who laid out the rules for the propositional operators. Insofar as we focus on deductive logic, that was the major advance in the subject, and although there was considerable interest and work on logic for many centuries, there were no significant advances in the science until well into the 19th century. Boole and Schrödermade progress with traditional logic, the American Peirce too, but it was not as well known. Modern Logic fundamentally starts with Frege (1879), in which we find an axiomatization of propositional logic, first-order logic, higher-order logics and the definition of the ancestral. The mathematical advances subsequent were in set theory, with Cantor, Dedekind, 18 1 Introduction to Logic

Peano, Zermelo largely taking the lead. Whitehead and Russell (1910) was of primary importance, but that was pretty much the end of the line for philosophically-led developments in logic. All eyes had shifted toward mathe- matics, where Hilbert, Bernays, Ackermann, G¨odel,Heyting, Lewis, Gentzen, Herbrand, Church, Kleene made the most signiﬁcant advances. 1.3 Historical Remarks 19 Problems

1.1. Which of the following are wﬀs of the system P, following the strict deﬁnition given? (i) (¬p0) (ii) ¬(p0 ⊃ q0) (iii) ¬p00 ⊃ p00 (iv) (¬p0 ⊃ p00)

1.2. Put each of the following into Polish Notation. (i) (p ⊃ (q ⊃ r)) ⊃ ((p ⊃ q) ⊃ r) (ii) ((p ∨ q) ⊃ r) ≡ (¬r ⊃ ¬(p ∧ q))

1.3. Determine whether the formulas are logically valid, logically consistent, or logically inconsistent using either truth tables or rules of resolution.

(i) ¬(p ∨ q) ⊃ (¬p ∧ ¬q) (ii) ¬(p ≡ q) ⊃ (p ≡ ¬q) (iii) ¬p ⊃ (p ∧ q) (iv) (p ∧ (q ∨ r)) ⊃ ((p ∧ q) ∨ (p ∧ r)) (v) ((p ⊃ q) ⊃ p) ⊃ p

1.4. What further truth values can be deduced from those shown? (i) ¬p ∨ (p ⊃ q) given p ⊃ q is F. (ii) ¬(p ∧ q) ≡ (¬p ⊃ ¬q) given p ∧ q is F. (iii) (¬p ∨ q) ⊃ (p ⊃ ¬r) given the whole formula is F. (iv) (p ⊃ ¬q) ⊃ (r ⊃ q) given the whole formula is F.