Chapter 4 Deductive Reasoning in First-order Logic
4.3 Natural Deduction for first-order logic
4.3.2 Derivations in first-order Natural Deduction
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181 Deductive Reasoning in First-order Logic that A(x), then take one and call it c.” We then replace the premise by A[c/x] If we succeed in deriving the desired conclusion C from that new premise, then it can be derived from ∃xA(x), provided c does not occur in A or C , or in any other assumption used in the derivation of C from A[c/x] This means that the name c must be a new one that has not yet been used in the proof At that moment we can discard the assumption A[c/x], as it was only an auxiliary assumption used to justify the derivation of C from ∃xA(x) From this point onward I write A(t) instead of A[t/x] whenever it is clear from the context which variable is substituted by the term t Remark 135 A common mistake is to use this simpler rule for elimination of ∃: (∃E ) ∃xA(x) A(c) where c is a new constant symbol Although simple and natural looking, this rule is not logically sound! For instance, using it I can derive the invalid implication ∃xA(x) → A(c) which can be interpreted as, for example, “If anyone will fail the exam then YourName will fail the exam,” which you certainly not wish to be valid 4.3.2 Derivations in first-order Natural Deduction Derivations in first-order Natural Deduction are organized as for derivations in Propositional Natural Deduction: as trees growing upward (but usually constructed downward); with nodes decorated by formulae, where the leaves are decorated with given or introduced assumptions; every node branching upward according to some of the derivation rules; and the formula decorating that node is the conclusion of the rule, while those decorating the successor nodes are the premises The root of the derivation tree is decorated with the formula to be derived In particular, the Natural Deduction derivation tree for the logical consequence A1 , , An |= B has leaves decorated with formulae among A1 , , An or with subsequently canceled additional assumptions, while the root is decorated with B If such a derivation tree exists, then we say that A1 , , An |= B is derivable in Natural Deduction and write A1 , , An ND B Here are some examples of derivations in Natural Deduction Check that the rules for the quantifiers have been applied correctly ND ∀x∀yP (x, y ) → ∀y ∀xP (x, y ) : [∀x∀yP (x, y )]1 ∀yP (c1 , y ) (∀E ) P (c1 , c2 ) (∀I ) ∀xP (x, c2 ) (∀I ) ∀y ∀xP (x, y ) (→ I ) ∀x∀yP (x, y ) → ∀y ∀xP (x, y ) (∀E )