A More Complex Example: A Supplement to Frege's Logic, Theorem, and Foundations for Arithmetic
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Stanford Encyclopedia of Philosophy

Supplement to Frege's Logic, Theorem, and Foundations for Arithmetic

## A More Complex Example

### Example of an Inference Using the Definition of Membership

If given the premise that
1 + 2^{2} = 5

one can prove that
1
( + 2^{2}
= 5)

For it follows from our premise (by
-Abstraction) that:
[*z* *z* + 2^{2}
= 5]1

Independently, by the logic of identity, we know:
([*z* *z* +
2^{2} = 5]) =
([*z* *z* +
2^{2} = 5])

So we may conjoin this fact and the result of
-Abstraction to produce:
([*z* *z* +
2^{2} = 5]) =
([*z* *z* +
2^{2} = 5]) &
[*z* *z* + 2^{2} = 5]1

Then, by existential generalization on the concept
[*z* *z* + 2^{2} = 5],
it follows that:
*G*[([*z* *z* +
2^{2} = 5]) =
*G*
& *G*1]

By the definition of membership, we obtain:
1
([*z* *z* +
2^{2} = 5])

And, finally, by our Rewrite Rule, we establish what we set out to prove:
1
( + 2^{2}
= 5)

Return to Frege's Logic, Theorem, and Foundations for Arithmetic

Copyright © 1998 by

Edward N. Zalta

*zalta@stanford.edu*
*First published: October 12, 1998*

*Content last modified: October 12, 1998*