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

180 Logic as a Tool 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 Introduction rules (∀I )* (∃I )** Elimination rules A[c/x] ∀xA(x) A[t/x] ∃xA(x) (∀E )** (∃E )*** ∀xA(x) A[t/x] [A[c/x]] ∃xA(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 follows We say “Let c be any object from the domain (e.g., an arbitrary real number).” However, the name c of 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 that A(c) holds, without assuming any specific properties of c If we succeed, then our proof will apply to any object c from the structure, so we will have a proof that A(x) holds for every object x, 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, satisfying A Within the formal system we try to come up with a term t that would name such a witness (∀E ) : If a premise is a universally quantified sentence ∀xA(x), we can assume A(a) for any object a from 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, say c, for an object that satisfies A That is, we say “if there is an x such 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