PHI 120 Introductory Logic Consistency of the Calculus

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1 PHI 120 Introductory Logic Consistency of the Calculus What we will show: 1. All derivable sequents are tautologous, i.e., valid. these results are metatheorems as they express results about the calculus, they are not results obtained in the propositional calculus itself 2. Every theorem provable from the rules is tautologous by a truth-table test. no contingent theorems exist no inconsistent theorems exist no contradiction Ö & ~Ö can be derived 3. The propositional calculus is consistent. 1. All derivable sequents are tautologous, i.e., valid. I. Preliminary definitions: A. By definition, a sequent is derivable if a proof can be found for it employing only the ten primitive rules of derivation B. Any sequent-expression has the following structure: A 1,..., A n B 1. where A 1,..., A n equals the total assumption set 2. B equals the conclusion to be derived II. III. IV. A derivable sequent has, by definition, a proof. A. any proof is finitely long, though its length is indeterminate B. any proof proceeds in stages C. every proof begins necessarily with an application of the rule A D. any later step (unless also the application of the rule A) is based on earlier lines of the proof in a definite way Note: the number of derivable sequents is indefinitely large since this includes all possible substitution instances of a derivable sequent A. hence we cannot inspect every individual sequent B. we need a more general method akin to the mathematical method of induction 1. If we wish to show that all (natural) numbers have a certain property, it suffices to show that 0 has the property, and that if a given number has the property then the next number in the sequence has the property. Given that 0 has the property, we can show that 1 has it; given that 1 has it, we can show that 2 has it, and so on. Tautologous (or Valid) Sequent Expressions: 1: 1. A A 1,..., A n B is tautologous (valid) if, for every assignment of truth-values to its variables for which all of A 1,..., A n take the value T, B takes the value of T also. ARE OUR RULES OF DERIVATION SAFE.WPD 7.Dec.05 (11:01) p. 1

2 for example: P Q P -> Q, ~ Q ~ P T T T F F T F F T F F T T F T F F T T T 2 A : A 1,..., A n B is tautologous (valid) if there is no assignment for which A 1,..., A is all true and B false. n B: For every sequent-expressions which has no assumptions, i.e., theorems, B takes the value T for all assignment of truth-values to its variables 1. B will be a tautologous sequent-expression insofar as B is a tautologous wff 2.. A sequent-expression is tautologous iff its corresponding conditional is tautologous A: Every sequent-expression can be associated with a single wff which is called the corresponding conditional: A 1 -> (A 2 -> (... (A n -> B)... )). e.g., (P -> Q) -> (~Q -> ~P) B: if there are no assumptions, the corresponding conditional is simply B itself 1. although B may not actually be a conditional at all. Outline of proof: V. Two general stages to our present proof: á. any application of A yields a tautologous sequent â. if at a given stage in a proof the earlier lines correspond to tautologous sequents, then an application of the other nine (9) rules to some of these lines yields a resulting line which also corresponds to a tautologous sequent 1. thus this stage fall into nine phases corresponding to the nine rules 2. in each phase we can establish a corresponding conditional proposition Proof of (á): 1. Any sequent derivable by the rule of assumption alone is tautologous. P P is obviously tautologous. P P P -> P T T T F F T ARE OUR RULES OF DERIVATION SAFE.WPD 7.Dec.05 (11:01) p. 2

3 Proof of (â) in nine stages: 1. &I: P, Q P&Q or P -> (Q -> (P & Q)) P Q P, Q P & Q P -> (Q -> (P & Q)) T T T T T T T F F T T F F T F T F F F F F T T F 2. &E: P&Q P or P&Q -> P P Q P & Q P P & Q -> P T T T T T T F F F T F T F F T F F F F T 3. vi: P P v Q or P -> P v Q P Q P P v Q P -> (P v Q) T T T T T T F T T T F T T T T F F F T F 4. ve: PvQ, ~P Q or PvQ -> (~Q -> P) P Q P v Q, ~ P Q P v Q -> (~ Q -> P) T T T F T T F T T F T F T T T T F T T T T T F T F F F T F T T F ARE OUR RULES OF DERIVATION SAFE.WPD 7.Dec.05 (11:01) p. 3

4 5. ->I: P Q->P or P -> (Q->P) P Q P Q -> P P -> (Q -> P) T T T T T T F T T T F T F T F F F T T T 6. ->E: P->Q, P Q or (P -> Q) -> (P -> Q) P Q P -> Q, P Q (P -> Q) -> (P -> Q) T T T T T T T F F F T F F T T T T T F F T T T T 7. <->I: P->Q, Q->P P<->Q or (P -> Q) -> ((Q -> P) -> (P <-> Q)) P Q P -> Q, Q -> P, P <-> Q (P -> Q) -> ((Q -> P) -> (P <-> Q)) T T T T T T T T T T T F F T F F T T F F F T T F F T T F T F F F T T T T T T T T 8. <->E: P<->Q P->Q or (P<->Q) -> (P->Q) P Q P <- > Q P -> Q (P <-> Q) -> (P -> Q) T T T T T T T T F F F F T F F T F T F T T F F T T T T T Proof continued on next page. ARE OUR RULES OF DERIVATION SAFE.WPD 7.Dec.05 (11:01) p. 4

5 9. RAA: (a more complicated procedure because the tautologous sequent has a contradiction as its conclusion) a. RAA has a peculiar form by virtue of the fact that we introduce an assumption that brings about a contradiction i. we begin an RAA with the following structure: (á) A 1,..., A n, A n+1 B & ~B ii. Given the inconsistency in (á), using RAA we conclude (â) A 1,..., A n ~A n+1 b. Suppose the sequent (á) is tautologous but sequent (â) is not i. then some assignment of truth-values to the variable in (â) gives A 1,..., A n all the value of T and ~A n+1 the value of F (1) hence A has a value of T ii. iii. n+1 A n+1 ~ An+1 T F F T this supposed assignment gives all assumptions in (á) a value of T if we suppose (á) is tautologous (or valid), then B&~B must have a value of T also (1) this is absurd (because every contradiction is necessarily false) B B & ~ B T F F F F T c. Hence, the former sequent (á) is not tautologous d. If the supposition (á) is false, then it turns out that (â) is a tautologous or valid sequent expression. Examples of RAA: P P (see also T1 on page 36 of The Logic Primer) 1 (1) P A Note: the sequent is proven on the first line; 2 (2) ~P A so lines 2 and 3 are technically unnecessary. 1,2 (4) P 1,2 RAA(2) See "incompatible premises" in Logic Primer, p.45. P P ~ P T T F F F T ARE OUR RULES OF DERIVATION SAFE.WPD 7.Dec.05 (11:01) p. 5

6 S18: ~P P->Q 1 (1) ~P A 2 (2) P A <Step #1 of ->I AND Step #2 of RAA> 3 (3) ~Q A 1,2 (4) Q 1,2 RAA(3) 1 (5) P->Q 4->I(2) P Q ~ P P -> Q T T F T T F F F F T T T F F T T If any application of the nine rules is made on tautologous sequents, then the result is a tautologous sequent. Since by A [cf. proof of (á)] we can only begin with tautologous sequents, any sequent we can derive by application of the ten primitive rules is tautologous. In other words, all derivable sequents are tautologous, i.e., valid. 2. Every theorem provable from the rules is tautologous by a truth-table test. Theorems are the special case of derivable sequents having no assumptions. Given that all theorems are derivable sequents, they are all tautologous. Hence every theorem provable from the rules is tautologous by a truth table test. 3. The propositional calculus is consistent. A logical system is consistent if the rules of the system do not enable us to generate as a theorem a contradiction. Since a contradiction is an inconsistency and not a tautology, the propositional calculus is consistent. Source: Lemmon, E. J. Beginning Logic. Indianapolis: Hackett Publishing Company, 1978, ARE OUR RULES OF DERIVATION SAFE.WPD 7.Dec.05 (11:01) p. 6