Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 204

k 4.3 Natural Deduction for first-order logic I now extend propositional Natural Deduction to first-order logic by adding rules for the quantifiers.. C C *where c is a constant symbol, n

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4.3 Natural Deduction for first-order logic

I now extend propositional Natural Deduction to first-order logic by adding rules for the quantifiers

4.3.1 Natural Deduction rules for the quantifiers

(∀I)* A[c/x]

∀xA(x) (∀E)** ∀xA(x)

A[t/x]

(∃I)** A[t/x]

∃xA(x) (∃E)***

∃xA(x)

[A[c/x]]

C C

*where c is a constant symbol, not occurring in A(x) or in any open assumption used in the derivation

of A[c/x];

**for any term t free for x in A;

***where c is a constant symbol, not occurring in A(x), C, or in any open assumption in the derivation

of C, except for A[c/x].

Let us start with some discussion and explanation of the quantifier rules

(∀I) : If we are to prove a universally quantified sentence,∀xA(x), we reason as fol-lows We say “Letcbe any object from the domain (e.g., an arbitrary real num-ber).” However, the namecof that arbitrary object must be new – one that has not yet been used in the proof – to be sure that it is indeed arbitrary We then try to prove thatA(c) holds, without assuming any specific properties ofc If we

succeed, then our proof will apply to any objectcfrom the structure, so we will have a proof thatA(x) holds for every objectx, that is, a proof of∀xA(x)

(∃I) : If we are to prove an existentially quantified sentence ∃xA(x), then we try to

find an explicit witness, an object in the domain, satisfyingA Within the formal system we try to come up with a termtthat would name such a witness

(∀E) : If a premise is a universally quantified sentence∀xA(x), we can assumeA(a) for

any objectafrom the structure Within the formal system, instead of elements of

a structure, we must use syntactic objects which represent them, namely, terms3 (∃E) : If a premise is an existentially quantified sentence∃xA(x) then we introduce a name, sayc, for an object that satisfiesA That is, we say “if there is anxsuch

3 Note that this sounds a little weaker because in general not every element of the domain has a name, or is the value

of a term However, it is sufficient because ad hoc names can be added whenever necessary.