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# Logic1209 - More propositional axjY iomatics 1 2 Deduction...

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Final exam: Monday (12/14) 7pm , Room 412 Covers Chs.13-21 (except for Ch.19) Monday: Ch.22 (briefly), Ch. 27 (mainly)
F (11 axioms) F1. ((X&Y)=>Z) => (X=>(Y=>Z)) (ex- portation) F2. ((X=>Y)?(Y=>Z)) => (X=>Z) (syl- logism) M (9 axioms) – instead of F2-F4, a single axiom

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K (10 axioms) – no exportation K1. X=>(Y=>X) K2. (X=>Y) => ((X=>(Y=>Z))=>(X=>Z)) K3. X=>(Y=>(XXY)) Without exportation, the proof of X=>X in K (disaster!) (1) X=>(X=>X) K1 (2) (X=>(X=>X) ) =>
Deduction Let S be a set of formulas. Then a deduc- tion of X from S in U is a finite sequence of formulas (the last being X) such that each term is An axiom of U A member of S Derivable from an earlier term by an infer- ence rule of U If there is such a deduction, X is said to be

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Deduction property In mathematics, in order to prove p=>q, you first assume p, and then derive q from it (together with other assumptions in the question). Similarly in logic, if a system U has the deduction property , if X is deducible from S:Y, then Y=>X is deducible from S. Ex. K has the deduction property. Obvi- ously, X is deducible from X, and there- fore, X=>X is deducible in K.
Deduction theorem U has the deduction property Y in addi- tion to MP, K1 and K2 hold in U:

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Logic1209 - More propositional axjY iomatics 1 2 Deduction...

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