04-proptop

04-proptop - Topics in propositional logic Readings: None...

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Topics in propositional logic Readings: None from the textbook. In this module, we will look at alternate methods of demonstrating the truth or falsity of statements in the propositional calculus, and consider some implications of soundness and completeness. We will be quickly covering a number of related topics whose combined scope is considerable. 1
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Variations on natural deduction One of the advantages of natural deduction is that tautologies can be built up starting with nothing at all. Other related systems reduce the number of rules by introducing axioms , basic formulas assumed to be true (and therefore usable on the left-hand side of any sequent). It is possible to define a proof system with a single derivation rule (MP) but with thirteen axioms. An example of an axiom would be φ φ ψ . This is really an axiom schema , since substitution of any formulae φ and ψ produces an axiom. Such a system may have some technical advantages (it shifts work from the inductive step to base cases in some proofs) but it does not correspond as well to mathematical reasoning. 2
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The sequent calculus Instead of using a proof as a method of showing validity of a sequent, we could describe transformations on sequents that preserve validity. This is the basis of Gentzen’s sequent calculus, which he introduced at the same time as natural deduction. Here is a sample rule in the sequent calculus. Γ Δ L 1 Γ ψ Δ The intuitive meaning of the label is that a formula involving is introduced on the left side of the sequent. 3
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There are also rules for introducing connectives on the right side of the sequent. Here is one: Γ φ, Δ R 1 Γ φ ψ, Δ Notice that these sequents, unlike ours, permit a set to appear on the right-hand side. Other rules deal with managing the left-hand and right-hand sets. The sequent calculus is more complicated than our system of natural deduction, but it simplifies treating a proof as a formal object, for example to show that a given formula cannot be proved in intuitionistic logic. We will not explore this idea further. 4
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We previously defined two formulas φ and ψ as provably equivalent if φ a‘ ψ . We can use our notion of semantics to say that φ and ψ are semantically equivalent if φ | = ψ and ψ | = φ . In this case, we write φ ψ ( is pronounced “is equivalent to”). Because we have proved the soundness and completeness of propositional logic, we know that these two notions coincide. Thus if we wish to show whether a formula is provable or not, we can try all valuations, though this may be inefficient. Our method of assigning a truth value to a formula is very algebraic in nature, and this was the original source of the connectives we use (George Boole and what is now called Boolean algebra). This suggests yet another method of proof. 5
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This note was uploaded on 12/08/2011 for the course CS 246 taught by Professor Wormer during the Spring '08 term at Waterloo.

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04-proptop - Topics in propositional logic Readings: None...

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