by Brian Bosse, Copyright July 23, 2008, all rights reserved. 55 views
Now that we have identified what constitutes a proper substitution for variables and name letters we are able to define three new inference rules in the predicate calculus. This post will deal with the first one: universal instantiation.
Universal Instantiation: "Everything is mortal" leads to "Brian is mortal." In other words, ∀x(M(x)) leads to M(B). Let's now present the abstract rule -
∀α(φα) leads to φζ where α is a variable, ζ is symbolic term, φα and φζ are symbolic formulas, and φζ comes from φα by proper substitution of ζ for α.
The key point here is that the resulting φζ comes from φα by proper substitution of ζ for α. That means the resulting formula only substitutes ζ for α in those cases where α is free within φα. Notice, it does not say within ∀α(φα), for then α is not free. Do not confuse φα with ∀α(φα) - they are not the same. Let's go back and illustrate this using our previous example.
"Everything is mortal" translates to ∀x(M(x)). Now our rule is that ∀α(φα) leads to φζ. In this case, α stands for 'x', φα stands M(x). We now let ζ stand for 'B', which is itself the name place for 'Brian'. Since we want to do a proper substitution of ζ for α in φα, we are doing a proper substitution of 'B' for 'x' in M(x). This means our φζ becomes M(B). M(B) says, "Brian is mortal." As such, we can go from "Everything is mortal" to "Brian is mortal" by the universal instantiation of 'Brian'.
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