Proof: Universal Instantiation

Instantiation rules allow us to replace variables with names, thereby transforming quantified statements into statements about particular instances.

Generalization rules allow us to move in the opposite direction, replacing names with variables and adding quantifiers to bind the variables.

Let's start with the rule of universal instantiation (UI):

(x) (...x...)


Consider the argument:

All humans are mortal. (x)(Hx Mx)
Socrates is a human. Hs
Therefore, Socrates is mortal. Ms
The statement (x) (Hx Mx) says that the expression Hx Mx is true of any x. Thus, we can replace the x with the name of anything, and the resulting singular statement follows from the universal statement.

Thus, the proof would be:

1. (x)(Hx Mx) Premise
2. Hs/Ms Premise/Conclusion
3. Hs Ms 1 UI
4. Ms 2,3 MP

Equivalence rule |
Universal instantiation | Existential generalization |
Existential instantiation | Universal generalization


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