Logic_and_Philosophy__A_Modern_Introduction

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1、ValidFormsforSententialLogicValidArgument1.ModusPonens(MP):5.Conjunction(Conj):FormsofInferencep
qpp/∴qq/∴p⋅q2.ModusTollens(MT):6.HypotheticalSyllogism(HS):p
qp
q~q/∴~pq
r/∴p
r3.DisjunctiveSyllogism(DS):7.Addition(Add):p∨qp/∴p∨q~p/∴q8.ConstructiveDilemma(CD):p∨qp∨q~q/∴pp
r4.Simplification(Simp):q
s/

2、∴r∨sp⋅q/∴pp⋅q/∴qValidEquivalence9.DoubleNegation(DN):14.Contraposition(Contra):Forms(Ruleofp::~~p(p
q)::(~q
~p)Replacement)10.DeMorgan’sTheorem(DeM):15.Implication(Impl):~(p⋅q)::(~p∨~q)(p
q)::(~p∨q)~(p∨q)::(~p⋅~q)16.Exportation(Exp):11.Commutation(Comm):[(p⋅q)
r]::[p
(q
r)](p∨q)::(q∨p)17.Tautology(T

3、aut):(p⋅q)::(q⋅p)p::(p⋅p)12.Assocation(Assoc):p::(p∨p)[p∨(q∨r)]::[(p∨q)∨r]18.Equivalence(Equiv):[p⋅(q⋅r)]::[(p⋅q)⋅r](p
q)::[(p
q)⋅(q
p)]13.Distribution(Dist):(p
q)::[(p⋅q)∨(~p⋅~q)][p⋅(q∨r)]::[(p⋅q)∨(p⋅r)][p∨(q⋅r)]::[(p∨q)⋅(p∨r)]ConditionalandConditionalProofIndirectProofIndirectProof→pAP/∴q→~pAP/∴p.

4、.....qq⋅~qp
qCPpIPRulesforPredicateLogicRuleUI:(u)(...u...)/∴(...w...)Provided:1.(...w...)resultsfromreplacingeachoccur-renceofufreein(...u...)withawthatiseitheraconstantoravariablefreein(...w...)(makingnootherchanges).RuleEI:(∃u)(...u...)/∴(...w...)Provided:1.wisnotaconstant.2.wdoesnotoccurfreeprev

5、iouslyintheproof.3.(...w...)resultsfromreplacingeachoccur-renceofufreein(...u...)withawthatisfreein(...w...)(makingnootherchanges).RuleUG:(...u...)/∴(w)(...w...)Provided:1.uisnotaconstant.2.udoesnotoccurfreepreviouslyinalineobtainedbyEI.3.udoesnotoccurfreepreviouslyinanassumedpremisethathasnotyetbee

6、ndischarged.4.(...w...)resultsfromreplacingeachoccur-renceofufreein(...u...)withawthatisfreein(...w...)(makingnootherchanges)andtherearenoadditionalfreeoccurrencesofwalreadycontainedin(...w...).RuleEG:(...u...)/∴(∃w)(...w...)Provided:1.(...w...)resultsfromreplacingatleastoneoccurrenceofu,whereuisaco

7、nstantoravari-ablefreein(...u...)withawthatisfreein(...w...)(makingnootherchanges)andtherearenoadditionalfreeoccurrencesofwalreadycontainedin(...w...).RuleQN:(u)(...u...)::~(∃u)~(...u...)(∃u)(...u...)