Logical Entailment - Stanford Logic Group

1 Logical EntailmentComputational LogicLecture 3 Michael Genesereth Autumn 20102 Logical ReasoningLogical Reasoning relates premises and conclusion does not say whether conclusion is true in general says conclusion true whenever premises are trueLeibnitz: The intellect is freed of all conception ofthe objects involved, and yet the computation yieldsthe correct : Math may be defined as the subject inwhich we never know what we are talking about norwhether what we are saying is EntailmentA set of premises logically entails a conclusion (written as |= ) if and only if every interpretationthat satisfies the premises also satisfies theconclusion.{p} |= (p q){p} |# (p q){p, q} |= (p q)4 Logical Entailment Logical Equivalence{p} |= (p q){p q)} |# pAnalogy in arithmetic: inequalities rather than equations35 Truth Table MethodWe can check for Logical Entailment by comparingtables of all possible the first table, eliminate all rows that do not the second table, eliminate all rows that do notsatisfy the the remaining rows in the first table are a subset ofthe remaining rows in the second table, then thepremi

7 13 Rules of Inference A rule of inference is a rule of reasoning consisting of one set of sentence patterns, called premises, and a second set of sentence patterns, called conclusions. ϕ⇒ψ ϕ ψ 14 Rule Instances An instance of a rule of inference is a rule in which all meta-variables have been consistently replaced by

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