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CSC 685 Logic Review

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Logic: Modeling Human Reasoning syllogistic logic Syllogistic Logic (Aristotle). all/some X are/not Y Propositional Logic (Boole). if X and Y then not Z or W First-order Logic (Frege). for all X there is a Y such that if not Px then Qy propositional logic first-order logic

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Frege (1879) AA A AA A B ABAB Px x x Px Px x Qx

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"Term t is free for variable x in formula " means: all variables in t remain free when t is substituted for x in . [t/x] = with all free occurrences of x replaced by t. x( y Rxy Pxy) [f(x)/y] = [[t/x]] = with all free occurrences of x replaced by t as long as t is free for x in x( y Rxy Pxy) [[f(x)/y]] = x( y Rxy Pxf(x)) x( y Rxy Pxy) (no change) Variables in plugged-in terms shouldn't be "captured"

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ie , , [ … ], [ … ] [ … ] , [ … ] , = t = t t = s, [ t/x ] [s/x] y[ … [y/x] ] x x [[ t/x ]] [[t/x]] x x , y[ [y/x] … ] where does not contain x copy First-Order Natural Deduction PBC [ … ] MT , LEM

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Why is e this: x [[ t/x ]], not this: x [ t/x ] ? x y Lxy y Lay x y Lxy y Lyy... ok.... not ok.

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An Axiomatic System Axiom schemas A1 ( ) A2 ( ) (( ) ( )) A3( ) ( ) A4 x [[t/x]] A5 x if x is not free in A6 x ( ) ( x x ) A7 x , where is an axiom Rule MP ,

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bool truth( Sentence , Model m ) { if ( = = ) return ! truth( ,m) ; else if ( = = ) return truth( ,m) && truth( ,m) else if ( = = ) return truth( ,m) || truth( ,m) else if ( = = ) return ! truth( ,m) || truth( ,m) else if ( = = x (x) ) for each a in m.A if ! truth( (a), m ) return false ; return true ; else if ( = = x (x) ) for each a in m.A if truth( (a), m ) return true ; return false ; else // = = Pf 1 (a 1 a 2 …)f 2 (b 1 b 2 …)… return P m ( f 1 m (a 1 m a 2 m …) f 2 m (b 1 m b 2 m …)… ; } Truth in a model m

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|= | syntax (proof theory) semantics (model theory) sound complete

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Compactness Theorem If all finite subsets of a set of formulas G are satisfiable, then G is satisfiable. Proof Assume all finite subsets of G are satisfiable. Assume G is NOT satisfiable. Then G |= Then G| (by completeness) Then there is a finite proof of from formulas in G. This proof only uses a finite set D of formulas in G. So D | where D is finite. So D |= (by soundness) So some finite subset D of G is unsatisfiable. Contra! So G is satisfiable.

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Expressing reachability u v v is reachable from u R In SECOND-ORDER logic: reachable( R,u,v):= P x y z( Pxx (Pxy Pyz Pxz) Puv (Rxy Pxy)) Can we express this in first order logic?

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NO! Reachability is not first-order u v R Suppose (u,v) were a first-order formula with predicate R expressing that v is reachable from u. Define n = x 1... x n-1 (Rcx 1 ... Rx n c') Let D = { (u,v), (c=c'), 1, 2,... } D is unsatisfiable, but every finite subset of D is satisfiable. This contradicts the Compactness Theorem. So such a first-order formula (u,v) cannot exist.

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