What is Existential Instantiation?
Existential instantiation is a logical rule used in predicate logic. It allows us to take a general existential statement and infer a specific instance of it. This is helpful in logical proofs where we need to work with a particular example from a general statement.
Definition
From the statement “∃x P(x)” (There exists an x such that P(x)), existential instantiation lets us assume P(c) for some constant c. However, this c is arbitrary and should not have been used elsewhere in the proof before.
Example
Given: ∃x (x is a philosopher)
By existential instantiation, we can say: Let c be a philosopher.
Now we can reason with the specific example of c.
Rules
- c must be a new term not used earlier in the proof.
- We cannot generalize about c unless further justified.
Conclusion
Existential instantiation is an essential tool in symbolic logic. It helps in translating general existential claims into a specific form, which allows for further logical analysis or proof construction.