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Mathematical Logic and Computability Contents Mathematical Logic and Computability J. Keisler, K. Kunen, T. Millar, A. Miller, J. Robbin This version is from Spring 1987 Contents 1 Propositional Logic 5 1.1 Syntax of Propositional Logic. 8 1.2 Semantics of Propositional Logic. 10 1.3 Truth Tables . 12 1.4 Induction on formulas and unique readability . 14 1.5 Tableaus. 16 1.6 Soundness and Confutation . 21 1.7 Completeness . 23 1.8 Computer Problem . 28 1.9 Exercises . 30 2 Pure Predicate Logic 36 2.1 Syntax of Predicate Logic. 38 2.2 Free and Bound Variables. 41 2.3 Semantics of Predicate Logic. 42 2.4 A simpler notation. 47 2.5 Tableaus. 49 2.6 Soundness . 53 2.7 Completeness . 53 2.8 Computer problem using PREDCALC . 58 2.9 Computer problem . 59 2.10 Exercises . 61 1 3 Full Predicate Logic 65 3.1 Semantics. 67 3.2 Tableaus. 69 3.3 Soundness . 72 3.4 Completeness . 73 3.5 Peano Arithmetic . 76 3.6 Computer problem . 77 3.7 Exercises . 79 4 Computable Functions 84 4.1 Numerical Functions. 84 4.2 Examples. 84 4.3 Extension. 86 4.4 Numerical Relations. 87 4.5 The Unlimited Register Machine. 87 4.6 Examples of URM-Computable Functions. 89 4.7 Universal URM machines . 106 4.8 Extract and Put . 111 4.9 UNIV.GN commented listing . 117 4.10 Primitive Recursion. 122 4.11 Recursive functions . 124 4.12 The URM-Computable Functions are recursive. 128 4.13 Decidable Relations. 133 4.14 Partial decidability . 134 4.15 The Diagonal Method . 134 4.16 The Halting Problem. 135 4.17 The G¨odelIncompleteness Theorem . 136 4.18 The Undecidability of Predicate Logic. 138 4.19 Computer problem . 138 4.20 Computer problem . 139 4.21 Exercises . 141 5 Mathematical Lingo 144 5.1 Sets. 144 5.2 Functions. 146 5.3 Inverses. 149 5.4 Cartesian Product. 152 5.5 Set theoretic operations. 154 2 5.6 Finite Sets . 157 5.7 Equivalence Relations. 158 5.8 Induction on the Natural Numbers . 161 6 Computer program documentation 164 6.1 TABLEAU program documentation . 164 6.2 COMPLETE program documentation . 172 6.3 PREDCALC program documentation . 174 6.4 BUILD program documentation . 180 6.5 MODEL program documentation . 184 6.6 GNUMBER program documentation . 185 A A Simple Proof. 194 B A lemma on valuations. 194 C Summary of Syntax Rules 199 D Summary of Tableaus. 201 E Finished Sets. 204 F Commented outline of the PARAM program 206 G Index 208 List of Figures 1 Propositional Extension Rules. 22 2 Proof of Extension Lemma . 25 3 Quantifier Rules for Pure Predicate Logic. 51 4 flowchart for add . 91 5 ADD.GN . 92 6 flowchart for multiplication . 93 7 assembly program for mult . 94 8 MULT.GN . 95 9 Psuedocode for Predescessor . 96 10 flowchart for predescessor . 97 3 11 PRED.GN . 98 12 flowchart for dotminus . 99 13 dotminus assembly code . 100 14 DOTMINUS.GN . 101 15 divide with remainder (divrem) . 102 16 expanding part of the flowchart for divrem . 103 17 divrem assembly code . 104 18 DIVREM.GN . 105 19 flowchart for univ . 108 20 flowchart for nextstate . 109 21 Jump: magnification of J box . 110 22 UNIV2.GN part1 . 119 23 UNIV.GN part 2 . 120 24 UNIV.GN initialization routine . 121 25 Hypothesis Mode . 166 26 Tableau Mode . 169 27 Map Mode . 173 28 Help box for the & key . 178 29 Help box for the R(...) key . 178 4 1 Propositional Logic In this book we shall study certain formal languages each of which abstracts from ordinary mathematical language (and to a lesser extent, everyday En- glish) some aspects of its logical structure. Formal languages differ from natural languages such as English in that the syntax of a formal language is precisely given. This is not the case with English: authorities often disagree as to whether a given English sentence is grammatically correct. Mathe- matical logic may be defined as that branch of mathematics which studies formal languages. In this chapter we study a formal language called propositional logic. This language abstracts from ordinary language the properties of the propo- sitional connectives (commonly called conjunctions by grammarians). These are “not”, “and”, “or”, “if”, and “if and only if”. These connectives are ex- tensional in the sense that the truth value of a compound sentence built up from these connectives depends only the truth values of the component sim- ple sentences. (The conjunction “because” is not extensional in this sense: one can easily give examples of English sentences p, q, and r such that p, q, r are true, but ‘p because q’ is true and ‘p because r’ is false.) Further- more, we are only concerned with the meanings that common mathematical usage accord these connectives; this is sometimes slightly different from their meanings in everyday English. We now explain these meanings. NEGATION. A sentence of form ‘not p’ is true exactly when p is false. The symbol used in mathematical logic for “not” is ¬ (but in older books the symbol ∼ was used). Thus of the two sentences ¬2 + 2 = 4 ¬2 + 2 = 5 the first is false while the second is true. The sentence ¬p is called the negation of p. CONJUNCTION. A sentence of form ‘p and q’ is true exactly when both p and q are true. The mathematical symbol for “and” is ∧ (or & in some older books). Thus of the four sentences 2 + 2 = 4 ∧ 2 + 3 = 5 5 2 + 2 = 4 ∧ 2 + 3 = 7 2 + 2 = 6 ∧ 2 + 3 = 5 2 + 2 = 6 ∧ 2 + 3 = 7 the first is true and the last three are false. The sentence p ∧ q is called the conjunction of p and q. For the mathematician, the words “and” and “but” have the same mean- ing. In everyday English these words cannot be used interchangeably, but the difference is psychological rather than logical. DISJUNCTION. A sentence of form ‘p or q’ is true exactly when either p is true or q is true (or both). The symbol use in mathematical logic for “or” is ∨. Thus of the four sentences 2 + 2 = 4 ∨ 2 + 3 = 5 2 + 2 = 4 ∨ 2 + 3 = 7 2 + 2 = 6 ∨ 2 + 3 = 5 2 + 2 = 6 ∨ 2 + 3 = 7 the first three are true while the last is false. The sentence p ∨ q is called the disjunction of p and q. Occasionally, the sentence p or q has a different meaning in everyday life from that just given. For example, the phrase “soup or salad included” in a restaurant menu means that the customer can have either soup or salad with his/her dinner at no extra cost but not both. This usage of the word “or” is called exclusive (because it excludes the case where both components are true). Mathematicians generally use the inclusive meaning explained above; when they intend the exclusive meaning they say so explicitly as in p or q but not both. IMPLICATION. The forms ‘if p, then q’, ‘q, if p’, ‘p implies q’, ‘p only if q’, and ‘q whenever p’ all having the same meaning for the mathematician: p ‘implies q’ is false exactly when p is true but q is false. The mathematical symbol for “implies” is ⇒ (or ⊃ in older books). Thus, of the four sentences 2 + 2 = 4 ⇒ 2 + 3 = 5 6 2 + 2 = 4 ⇒ 2 + 3 = 7 2 + 2 = 6 ⇒ 2 + 3 = 5 2 + 2 = 6 ⇒ 2 + 3 = 7 the second is false and the first, third and fourth are true. This usage is in sharp contrast to the usage in everyday language. In common discourse a sentence of form if p then q or p implies q suggests a kind of causality that is that q “follows” from p. Consider for example the sentence “If Columbus discovered America, then Aristotle was a Greek.” Since Aristotle was indeed a Greek this sentence either has form If true then true or If false then true and is thus true according to the meaning of “implies” we have adopted. However, common usage would judge this sentence either false or nonsensical because there is no causal relation between Columbus’s voyage and Aristotle’s nationality. To distinguish the meaning of “implies” which we have adopted (viz. that p implies q is false precisely when p is true and q is false) from other possible meanings logicians call it material implication. This is the only meaning used in mathematics. Note that material implication is extensional in the sense that the truth value of p materially implies q depends only on the truth values of p and q and not on subtler aspects of their meanings. EQUIVALENCE. The forms ‘p if and only if q’, ‘p is equivalent to q’; and ‘p exactly when q’ all have the same meaning for the mathematician: they are true when p and q have the same truth value and false in the contrary case. Some authors use “iff” is an abbreviation for “if and only if”. The mathematical symbol for if and only if is ⇔ ( ≡ in older books). Thus of the four sentences 2 + 2 = 4 ⇔ 2 + 3 = 5 2 + 2 = 4 ⇔ 2 + 3 = 7 2 + 2 = 6 ⇔ 2 + 3 = 5 2 + 2 = 6 ⇔ 2 + 3 = 7 the first and last are true while the other two are false. Evidently, p if and only if q has the same meaning as if p.
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