PHIL 250: Introduction to Symbolic Logic

tennant.9@osu.edu

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INTRODUCTION RULE FOR THE UNIVERSAL QUANTIFIER ( ∀ I)

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		:
		F(a)

__________

∀xF(x)

Explanation in words: In order to infer to a universally generalized conclusion ∀xF(x) , one must prove the `arbitrary instance' F(a). By taking the indicated step of universal introduction, one thereby rests the universally generalized conclusion ∀xF(x) on the assumptions on which the arbitrary instance F(a) depends.

The syntactic criteria by which one determines that F(a) is indeed an `arbitrary instance' legitimating the inference to ∀xF(x) are as follows:

1. in the sentence F(a), no occurrence of the parameter a lies within the scope of a quantifier binding the variable x;
2. F(x) is the result of replacing every occurrence of the parameter a in F(a) by an occurrence of the variable x (every one of these new occurrences of x then being free in the resulting formula F(x), because of condition (1)); and
3. the parameter a does not occur in any assumption on which F(a) depends.

Conditions (1) and (2) entail that the parameter a cannot occur in the conclusion ∀xF(x). Conditions (1) and (2) ensure that one universally generalizes thoroughly on the parameter a within the sentence F(a), and that it is the the dominant quantifier prefix ∀x in the conclusion ∀xF(x) that binds the variable x at every one of its occurrences replacing an occurrence of the parameter a in F(a). These syntactic conditions ensure that the property F( ) shown to hold for the `arbitrary individual' a (conditionally upon the undischarged assumptions on which F(a) rests) has been appropriately attributed to all individuals by the universally quantified conclusion (again, conditionally upon those same undischarged assumptions).

Compare the corresponding elimination rule for ∀

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[¬I] [∧I] [∨I] [→I] [∀I] [∃I]
[¬E] [∧E] [∨E] [→E] [∀E] [∃E]
[EFQ]
[LEM] [Dil] [CR] [DNE]

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