Universal instantiation { Philosophy Index }

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Universal instantiation

Universal instantiation or universal elimination (∀E, sometimes UI) is a basic rule of inference in first-order logic by which a universal statement is removed in a proof by applying its conditions to some constant. The universal instantiation rule allows you to apply the conditions of a universal claim to any object in a proof, including any new object.

The universal instantiation rule may be formally presented as follows:

∀xφ(x) proves that φ(c)

… where c is any constant.

The universal instantiation rule is essentially the affirmation that if something is true for everything, then it is true for some particular thing. As such, universal instantiation is a move from the general to the particular (or specific).

Example of universal instantiation

The following is an example of the universal instantiation rule in a proof. In this case, we are attempting to prove the following argument form:

∀x(Px → Qx), Pa proves that Qa

1 ∀x(Px → Qx) Pr.
2 Pa Pr.
RTP Qa
3 Pa → Qa ∀E, 1
4 Qa MP

On line 3, we have invoked the universal elimination rule. We want to prove that something is true for the constant a. Since the premise on line 1 states that (Px → Qx) is true for every x, that is, everything, then we know that the same is true for a, thus (Pa → Qa).