logic with quantifiers (informally)
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1 EDAA40 Discrete Structures in Computer Science 8: Quantificational logic Jörn W. Janneck, Dept. of Computer Science, Lund University logic with quantifiers (informally) Given a logical formula that depends on a variable x : represents forall x, alt. notations represents there exists an x, such that Forall is universal quantification, exists is existential. Examples: SLAM, p the language of quantificational logic terms also 1-place! formulae also other relation symbols SLAM, p
2 more examples... Zermelo-Fraenkel Set Theory w/choice (ZFC) extensionality regularity specification union replacement infinity power set choice 4 equivalences: quantifier interchange The following equivalences hold for any formula : Remember de Morgan's laws? Explain. 5 finite transforms Suppose we quantify all variables over a finite set D, and we have constant symbols for each of its elements. A finite transform of a universally/existentially quantified formula removes the quantifier, and instantiates the body for each element of D in a chained conjunction/disjunction. Do this for the following formula, until all quantifiers are gone. Assume a domain with two values, with constant names a and b. 6
3 equivalences: distribution The following equivalences hold for any formula : This should be easy to see if you think about what this would look like in a finite transform. 7 quantifier scopes variable uses, and the quantifier they are bound by Y x x z quantifier scopes, and the variables bound in/by them 8 free and bound variable occurrences A variable occurrence is bound iff it occurs inside the scope of a quantifier that binds that variable. It is free otherwise. A formula with no free variable occurrences is called closed. A closed formula is a sentence. bound occurrences free occurrences 9
4 equivalences: vacuity, relettering Vacuity: If x does not occur free in, then Can you think of a case where this is not true? Relettering: If x does not occur at all in, and is the result of replacing every bound occurrence of some variable y in with x, then 10 interpretations The value of a formula depends on how you read the symbols in it. Also, we need to determine what values the quantified variables can assume. A domain or universe D is the set of values that quantified variables range over. An interpretation v is a function assigning mathematical objects to the symbols occurring in a formula. Specifically... - to each constant a - to each variable x - to each n-place function letter f - to each n-place relation letter P - to the identity symbol the identity over D 11 evaluating terms and formulae Given a domain D and an interpretation v, the value of a term t is defined as follows (reusing the v): With this, we can determine the truth value (0 or 1) of a formula as follows: We are overloading the name of the interpretation, like we did for assignments and valuations in propositional logic. 12
5 method 1: substitution Given a formula, a variable x, and a term t, is the formula resulting from substituting every free occurrence of x in with t. Now we make two assumptions: (1) There are enough constant letters so each value in D can have its own constant letter. (2) The interpretation from the constant letters is onto D, i.e. every value in D is represented by (at least) one constant letter under v. 13 method 1: substitution With this we can evaluate quantified formulae as follows: It sounds a bit circular, but it does give an algorithm for computing the truth value of a quantified formula. Also, larger domains require more constant letters. 14 method 2: x-variant To avoid having to tinker with the constant symbols, we can instead tinker with the interpretation. Given an interpretation v and a variable x, an x-variant of v (with value c) is another interpretation v' that agrees with v on all constants, function and predicate letters, and variables other than x, and that assigns x the value c. With this we can evaluate quantified formulae as follows: 15
6 logical implication Given a set of formulae and a formula, we say that A logically implies iff there is no interpretation v such that all the formulae in A are true under v, but is false: Also: logical equivalence logical truth contradiction A formula that is neither logically true nor a contradiction is contingent. 16 some implications Why isn't this an equivalence? 17 clean substitution Given a formula, a variable x, and a term t, a substitution is clean iff no free occurrence of x is in the scope of a quantifier binding a free variable in t. y 18
7 instantiation & generalization The following are general rules regarding quantifiers: universal instantiation (UI) if substitution is clean universal generalization (UG) if x not free in existential generalization (EG) if substitution is clean not vacuity, as x could be free in! existential instantiation (EI) if x not free in Why does UG / EI work? EG involves reverse substitution 19 replacement Given a formula, and a term t, a occurrence of t in in a scope that binds a variable in t. is free iff it is not non-free occurrences free occurrences A replacement of a term t by t' in a formula free occurrences of t in by t'. is the result of replacing It is clean if the t' replaced for t are also free. 20 principle of replacement Principle of Replacement: If the replacement is clean. 21
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