By Peter B. Andrews
This advent to mathematical good judgment starts off with propositional calculus and first-order common sense. issues coated comprise syntax, semantics, soundness, completeness, independence, common varieties, vertical paths via negation common formulation, compactness, Smullyan's Unifying precept, ordinary deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The final 3 chapters of the ebook supply an creation to kind idea (higher-order logic). it really is proven how a variety of mathematical options might be formalized during this very expressive formal language. This expressive notation allows proofs of the classical incompleteness and undecidability theorems that are very dependent and effortless to appreciate. The dialogue of semantics makes transparent the very important contrast among typical and nonstandard versions that is so vital in figuring out difficult phenomena resembling the incompleteness theorems and Skolem's Paradox approximately countable versions of set idea. many of the a number of routines require giving formal proofs. A desktop application referred to as ETPS that is to be had from the net enables doing and checking such workouts. viewers: This quantity should be of curiosity to mathematicians, desktop scientists, and philosophers in universities, in addition to to computing device scientists in who desire to use higher-order common sense for and software program specification and verification.
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Additional resources for An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof
If 54 CHAPTER TWO x, Y E F and x =I- y then x i y or y i x. Assume the former; then (x ::::} y) =I- 1 in L. 15 let F be an prime filter on L not containing (x ::::} y); then in LjF, [X]F i [Y]F' hence [X]F =I- [Y]F and therefore i(x) =I- i(y). 17 Associate with each formula tp of BL a term tpe of the language of residuated lattices by replacing the connectives -+, &, /\, V, 0, I by function symbols and constants ::::}, *, n, U, 0,1 respectively and replacing each propositional variable Pi by a corresponding object variable Xi.
If Ij results by modus ponens from previous members Ii, hi -t ,j) then, by the induction hypothesis we assume T f- VJn -t Ii, T f- VJm -t hi -t ,j), thus by (7), T f- (VJn&VJm) -t hi&hi -t ,j)), thus T f- VJ n +m -t Ij (cf. (4)). This completes the proof. 10 valid for classical logic; we shall see below that only one of our logics (G6del logic) will have the classical deduction theorem. Also note that the formula VJn -t 'ljJ may be replaced by VJ -t (VJ -t ... -t (VJ -t 'ljJ) ... ) (n copies of VJ).
B) Now let us prove the second fonnula. e. e. ff- [(ep&X)&((ep&X) -+ 'ljJ&X)] -+ ((ep 1\ 'ljJ)&x),or ((ep&X) -+ ('ljJ&X)) -+ [(ep&X) -+ ((ep 1\ 'IjJ)&X)]· Denote the last fonnula by (*); we prove f- (ep -+ 'ljJ) -+ (*) and f- ('IjJ -+ ep) -+ (*) (this suffices by axiom (A6». 8 (6», thus BL f- (ep -+ 'IjJ) -+ (anything -+ ((ep&X) -+ ((ep 1\ 'ljJ)&X))); on the other hand, BL f- ('IjJ -+ ep) -+ (('ljJ&X) -+ ((ep 1\ 'IjJ)&X)) ,thus BL f- ('IjJ -+ ep) -+ ([('ljJ&X) -+ ('ljJ&X)] -+ [('ljJ&X) -+ ((ep 1\ 'IjJ)&X)]·) Using (A6) we get (*).
An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof by Peter B. Andrews