Poet: Au Predicate Logic,III All that glitters is not gold. - PDF document

Mathematics for Computer Science Math vs. English MIT 6.042J/18.062J G Poet: � Au � Predicate Logic,III “All that glitters is not gold.” IMPLIES NOT ( ∀ ∃ in English ∀ x. [G(x) Au (x))] Two Meta-Theorems No : gold glitters like gold! Albert R Meyer, February 17, 2012 Albert R Meyer, February 17, 2012 lec 2F.1 lec 2F.2 Math vs. English Math vs. English Poet: “There is a season to every Poet: necessarily � � � � �� � purpose under heaven” “All that glitters is not gold.” ∃ s ∈ Season ∀ p ∈ Purpose. s is for p NOT ( x ∀ . [ ( G x) IMPLIES A u(x)]) Some season, say Summer, is good for all Purposes? (Poetic license) NO , Summer no good for snow shoveling Albert R Meyer, February 17, 2012 Albert R Meyer, February 17, 2012 lec 2F.3 lec 2F.4 Math vs. English Math vs. English Poet: “There is a season to every Poet: “There is a season to every purpose under heaven” purpose under heaven” ∃ s ∈ Season ∀ p ∈ Purpose. s is for p ∀ p ∈ Purpose ∃ s ∈ Season. s is for p � � � � �� � � � � � �� � Poet’s meaning flips the quantiers Poet’s meaning flips the quantiers Albert R Meyer, February 17, 2012 lec 2F.5 Albert R Meyer, February 17, 2012 lec 2F.6 1

Power & Limits of Logic Math vs. English Poet: “There is a season to every Two Profound purpose under heaven” Theorems about Meta- ∀ p ∈ Purpose ∃ s ∈ Season. s is for p for snow shoveling, Winter is good Mathematical Logic for planting, Spring is good for leaf watching, Fall is good Albert R Meyer, February 17, 2012 Albert R Meyer, February 17, 2012 lec 2F.7 lec 2F.8 Gödel's Completeness Theorem Axioms & Inference Rules Thm 1, good news: only need to Rules are just UG and modus know a few axioms & rules to ponens. Most of the valid prove all valid formulas. axioms shown already. (in theory; in practice need lots of rules) Albert R Meyer, February 17, 2012 Albert R Meyer, February 17, 2012 lec 2F.9 lec 2F.10 Profound Meta-Theorems Validity is undecidable Thm 2, Bad News: there is no We won't examine these Theorems procedure to determine whether further. Their proofs usually require half a term in an intro a quantified formula is valid logic course after 6.042. But they (in contrast to propositional are interesting to think about. formulas). Albert R Meyer, February 17, 2012 lec 2F.11 Albert R Meyer, February 17, 2012 lec 2F.13 2

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