Agrippa's Trilemma and Foundationalism

[Leontiskos]

@Banno

Logic doesn’t reject x=x. It’s a common shorthand for U(x)(x=x),

You continue to conflate “logic” with first-order predicate logic.

@Banno

Stipulations can be used in justifications. That the rules stipulate that four aces beats a flush of any sort is justification for taking your money.

This is of course sophistry

@Banno

I suppose I should be flattered that you are following me from thread to thread. In the other thread you mentioned, I did not say that quantification cannot occur in a domain containing everything.

Here is what you said:

@Banno

Sider acknowledges the situation in footnote 45 but thinks he can get around this by stipulating that he can have a domain of just “any quantifier wholly unrestricted in its application to individuals (nonsets)”. Can he? I think not;

So again, if Leibniz is talking about identity in a way that is not restricted to a particular domain, then your attempt to translate his thesis into first-order logic won’t work on your own terms.

@Banno

Leibniz could not have been thinking in terms of an unrestricted domain of quantification, since the modern model-theoretic notion of a domain did not exist in his time. Rather, he took logic itself to be universal in scope, to apply to everything that is.

So Leibniz’ law means a quasi-first-order thesis when it suits you, and when it doesn’t suit you it doesn’t. That’s how you do philosophy.

The basic point is not at issue, which is why I put ‘domain’ in scare-quotes when I first related it to Leibniz. But your approach is ad hoc in that you have no principled way of translating Leibniz into your favored logical system.

@Banno

We can choose the one that best suits our needs.

This is the voluntarism I spoke of.

@Banno

We could have written instead U(φ)(∀x ∀y (x = y → (φ(x) ↔ φ(y)))).

Wrong again. That would be second-order, not first-order.

@Banno

Doing so would be to change the way “1” and “2” work, or to change the use of “=”.

If it’s just stipulation, as you say, then there is no normative way that it should “work”.