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Logic-Argument.References

Logic-Argument.References - Non-Classical Logics...

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Deductions in Modal Logic Introduction Before Kripke came up with a decent semantics for modal logics we were forced to deal with them only using the tools of proof-theory. We'll be looking at how this is done in the next two lectures. We'll begin by looking at proofs using natural deduction. Natural Deduction Standard Rules We'll assume that we have most of the rules that were introduced in the introductory logic course. Replacement rules: ~~p : : p ( DN ) Double Negation p v q : : q v p ( Com ) v Commutation p & q : : q & p ( Com ) & Commutation p v (q v r) : : (p v q) v r ( Assoc ) v Association p & (q & r) : : (p & q) & r ( Assoc ) & Association p v (q & r) : : (p v q) & (p v r) ( Dist ) v Distribution p & (q v r) : : (p & q) v (p & r) ( Dist ) & Distribution ~(p & q) : : ~p v ~q ( DeM ) DeMorgans Law ~(p v q) : : ~p & ~q ( DeM ) DeMorgans Law p v p : : p ( Idem ) v Idempotence p & p : : p ( Idem ) & Idempotence p É q : : ~p v q ( IMP ) Material Implication p É q : : ~q É ~p ( Cont ) Contraposition (p & q) É r : : p É (q É r) ( Exp ) Exportation p É (q É r) : : q É (p É r) ( Perm ) Permutation p º q : : (p É q) & (q É p) ( Equiv ) Equivalence p º q : : (p & q) v (~p & ~q) ( Equiv ) Equivalence p : : p & (q v ~q) ( TConj ) Tautologous Conjunct p : : p v (q & ~q) ( TConj ) Contradictory Disjunct Inference rules: p & q ( Simp ) Simplification p p ( Conj ) Conjunction ... Non-Classical Logics http://www.uq.edu.au/~uqswats2/courses/phil2100/... 1 of 9 06/17/2008 12:00 AM
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q p & q p É q ( MP ) Modus Ponens, or p ( AA ) Affirming the Antecedent q p É q ( MT ) Modus Tollens, or ~q ( DC ) Denying the Consequent ~p p É q ( HS ) Hypothetical Syllogism, or q É r ( ChArg ) Chain Argument p É r p v q ( DS ) Disjunctive Syllogism, or ~p ( DC ) Denying a Disjunt q p ( Add ) Addition, or p v q ( DA ) Disjunctive Addition p v r ( CD ) Constructive Dilemma p É q r É s q v s p ( CP ) Conditional Proof ç ... ç q ç p É q ~p ( RAA ) Reductio ad Absurdum ç ... ç q & ~q ç p Modal Rules Replacement rules: ~<>~p : : []p ~[]~p : : <>p ~[]p : : <>~p ~<>p : : []~p Inference rules: []p ( NE ) Necessity Elimination Non-Classical Logics http://www.uq.edu.au/~uqswats2/courses/phil2100/... 2 of 9 06/17/2008 12:00 AM
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p p ( PI ) Possibility Introduction <>p []p ( MRT ) Modal Reiteration T p []p ( MRT ) Modal Reiteration S4 []p <>p ( MRT ) Modal Reiteration S5 <>p _______ ( NI ) Necessity Introduction ç ... ç p ç []p The NI rule is an application of a null assumption deduction, which is like a CP but with no initial assumption to be discharged at the end. Since there are no assumptions, and the result is proved, the idea is that the consequence must be a necessary truth - it doesn't depend on any contingencies. Here is an example of the use of these inference rules. We wish to prove that ([]p & []q) É [](p & q), a theorem in T: 1. []p & []q Assumption | 2. []p 1, Simp | 3. []q 1, Simp | ___ 4. Null Ass | | 5. p 2, MRT | | 6. q 3, MRT | | 7. p & q 5, 6, Conj | 8. [](p & q) 4-7, NI | 9. ([]p & []q) É [](p & q) 1-8, NI Prove the S4 theorem []p É [][]p: 1. []p Assumption | ___ 2. Null Ass | | 3. []p 1, MRS4 | 4. [][]p 2-3, NI 5. []p É [][]p 1-4, CP Prove the S5 theorem <>p É []<>p: 1. <>p Assumption Non-Classical Logics http://www.uq.edu.au/~uqswats2/courses/phil2100/...
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