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# E 45 - ?x.p i.e for any M and ai’s M{ai/xi}G implies...

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Proof/Deduction Rules p !x.p ?x.p !x.p ?x.p {t/x}p {t/x}p p q q !-i !-e ?-i ?-e + Natural Deduction Rules

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The Notation We write p q iff q is deducible/provable given p using the proof rules. We write p1,…,pN q iff q is deducible/provable given p1 ,…, and pN using the proof rules.
Are the Rules Sound? (1/4) p !x.p !-i Can we put !x for any x? Zero(x) Zero(x)->!x.Zero(x) We can put !x only when x is not free in live/charged assumptions. Soundness: G !x.p implies G !x.p. I.H. G p i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}p. T.S. G !x.p i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}!x.p. !x.Zero(x)

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Are the Rules Sound? (2/4) !x.p {t/x}p !-e Can we put any term t? Soundness: G {t/x}p implies G {t/x}p. I.H. G !x.p i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}!x.p T.S. G {t/x}p i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}([t/x}p). Yes, we can put any term.
Are the Rules Sound? (3/4) ?x.p {t/x}p ?-i Can t be any term? Soundness: G ?x.p implies G ?x.p. I.H. G {t/x}p i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}({t/x}p) T.S. G

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Unformatted text preview: ?x.p i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}?x.p Yes, t can be any term. Are the Rules Sound? (4/4) ?x.p p q q ?-e Can q be anything? q’s free ivar should not be free in p. Soundness: G q implies G q. I.H. G {t/x}p and G,p q i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}({t/x}p) and M {ai/xi}(G,p) implies M {ai/xi}q T.S. G q i.e., for any M and ai’s M {ai/xi}G implies M {ai/xi}q Proof/Deduction/Inference Examples (1/2) !x.P(x)->Q(x) !x.P(x)->!x.Q(x) !x.(P(x)->Q(x)) !x.P(x) P(x)->Q(x) P(x) Q(x) !x.Q(x) !x.P(x)->!x.Q(x) Proof/Deduction/Inference Examples (2/2) P(x)->!x.Q(x) !y.(P(x)->Q(y)) !y.(P(x)->Q(y)) P(x)->!x.Q(x) P(x) Q(y) !x.Q(x) P(x)->Q(y) (?x.p) !x.( p) (?x.p) !x.( p) p p {x/x}p ?x.p Soundness/Completeness • the deduction rules for the first-order predicate logic are sound and complete • every provable formula is true/valid • every true/valid formula is provable p <=> p...
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E 45 - ?x.p i.e for any M and ai’s M{ai/xi}G implies...

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