EDAA40EDAA40Discrete Structures in Computer ScienceDiscrete Structures in Computer Science

8: Quantificational logic8: Quantificational logic

Jörn W. Janneck, Dept. of Computer Science, Lund University

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logic with quantifiers (informally)

Given a logical formula that depends on a variable x :

represents “forall x, “

represents “there exists an x, such that ”

alt. notations

Examples:

SLAM, p. 218

“Forall” is universal quantification, “exists” is existential.

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the language of quantificational logic

also 1-place!

terms

formulae

SLAM, p. 219

also other relationsymbols

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more examples...

Zermelo-Fraenkel Set Theory w/Choice (ZFC)

extensionality

regularity

specification

union

replacement

infinity

power set

choice

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equivalences: quantifier interchange

The following equivalences hold for any formula :

Remember de Morgan's laws?

Explain.

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finite transformsSuppose we quantify all variables over a finite set D, and we have constantsymbols for each of its elements.

A finite transform of a universally/existentially quantified formula removesthe quantifier, and instantiates the body for each element of D in a chainedconjunction/disjunction.

Example:

Do this for the following formula, until all quantifiers are gone.Assume a domain with two values, with constant names a and b.

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equivalences: distribution

The following equivalences hold for any formula :

This should be easy to see if you think aboutwhat this would look like in a finite transform.

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quantifier scopes

Y

x

x

z

quantifier scopes, and the variables bound in/by them

variable uses, and the quantifier they are bound by

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free and bound variable occurrences

A variable occurrence is bound iff it occurs inside the scope of a quantifierthat binds that variable.

It is free otherwise.

A formula with no free variable occurrences is called closed.A closed formula is a sentence.

free occurrences

bound occurrences

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equivalences: vacuity, relettering

Vacuity: If x does not occur free in , then

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

Example:

Can you think of a case where this is not true?

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interpretationsThe value of a formula depends on how you read the symbols in it.

A domain or universe D is the set of values that quantified variables rangeover.

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

Also, we need to determine what values the quantified variables can assume.

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evaluating terms and formulaeGiven a domain D and an interpretation v, the value of a term t is definedas 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 wedid for assignments and valuations in propositional logic.

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method 1: substitution

Given a formula , a variable x, and a term t,

is the formula resulting from substituting every free occurrence of x inwith 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.

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method 1: substitution

With this we can evaluate quantified formulae as follows:

It sounds a bit circular, but it does give an “algorithm” forcomputing the truth value of a quantified formula.

Also, larger domains require more constant letters.

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method 2: x-variant

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.

To avoid having to tinker with the constant symbols, we can instead tinkerwith the interpretation.

With this we can evaluate quantified formulae as follows:

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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 iscontingent.

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some implications

Why isn't this an equivalence?

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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 afree variable in t.

y

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instantiation & generalizationThe following are general rules regarding quantifiers:

universal instantiation (UI)

universal generalization (UG)

existential generalization (EG)

existential instantiation (EI)

if substitution is clean

if substitution is clean

if x not free in

if x not free in

Example:

not vacuity, as xcould be free in !

EG involves “reverse substitution”

Why does UG / EI work?

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replacementGiven a formula , and a term t, a occurrence of t in is free iff it is notin a scope that binds a variable in t.

free occurrences

non-free occurrences

A replacement of a term t by t' in a formula is the result of replacingfree occurrences of t in by t'.

It is clean if the t' replaced for t are also free.

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principle of replacement

Principle of Replacement:

If the replacement is clean.

Example: