Advanced Logic for Applications by Richard E. Grandy (auth.) PDF

By Richard E. Grandy (auth.)

ISBN-10: 9027710341

ISBN-13: 9789027710345

ISBN-10: 9401011915

ISBN-13: 9789401011914

This publication is meant to be a survey of an important leads to mathematical good judgment for philosophers. it's a survey of effects that have philosophical importance and it's meant to be available to philosophers. i've got assumed the mathematical sophistication got· in an introductory common sense path or in interpreting a simple common sense textual content. as well as proving the main philosophically major leads to mathematical common sense, i've got tried to demonstrate a variety of equipment of facts. for instance, the completeness of quantification concept is proved either constructively and non-constructively and relative advert­ vantages of every kind of facts are mentioned. equally, optimistic and non-constructive models of Godel's first incompleteness theorem are given. i'm hoping that the reader· will enhance facility with the equipment of facts and in addition be as a result of contemplate their variations. i guess familiarity with quantification conception either in below­ status the notations and find item language proofs. Strictly talking the presentation is self-contained, however it will be very tough for somebody with out historical past within the topic to persist with the fabric from the start. this is often useful if the notes are to be obtainable to readers who've had various backgrounds at a extra straight forward point. even if, to cause them to obtainable to readers with out history will require writing another introductory common sense textual content. a variety of workouts were incorporated and plenty of of those are critical components of the proofs.

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By our definition a(jt l ... tn) = I(j)(a(tl) ... a(tn» so by our induction hypothesis a(jt l ... tn) = I(j)(IL(tI) ... 1L(tn», and the definition of I(jn) is {(1L(t1), ... 1L(tn), JL(jt l ... tn»: ft l ... tn E and so a(jt I ... tn) = lL(jt I . . tn)· n U sing this fact we can show that for all atomic A E T, a satisfies A in MiffAET. Proof. If A is Ft l ... tn then by definition a sat Ftl ... tn iff (a(tl)' . a (tn» E l(F), but we know that I(F n) = ((JL(tI), . . JL(tn»: Fnt l ... tn E r} and since a(tj) = JL(t;), a sat Ft l ...

Vn) in M then a also sat 3 vB for if we take {3 "'"v a such that (3(v) = a(f(vi ... vn »{3 sat B. Conversely, if all a satisfy 3 vB in M then we know that for each n-tuple (d" ... dn ) in the domain there is at least one d such that if a sequence {3 assigns d 1, ••• dn to VI'" • Vn respectively and (3(v) = d then {3 sat B. To define the interpretation of f we choose for each (d l , . . dn ) a unique d and assign (d l , . . dn, d) to f. ] Hence, for any A *, by induction we have shown that a formula Af exists such that A * is true in M iff At is true in an extension of M.

By strong completeness, if T is consistent then it has a model. A formula A is a theorem of T iff T f- A. One interesting property of theories is categoricity. Two models M\ and M2 are isomorphic iff there is a 1-1 function T such that T maps D\ onto D2 and (1) 12(c n ) = T(l\(Cn » (2) 12(pn) = {(T(d\), ... T(dn»: (dlo ... dn) E I\(pn)} (3) 12(fn) = {(T(d\), ... T(dn), T(d n+\»: (dlo ... d n+\) E ltU n)}. EXERCISE 1. Show that if a satisfies A in M\ and M\ and M2 are isomorphic then there is a ~ such that ~ satisfies A in M 2• A theory is categorical iff all of its models are isomorphic.

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Advanced Logic for Applications by Richard E. Grandy (auth.)

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