The argument about semantics of higher-order logic

Joe Shipman joeshipman at
Thu Jul 9 09:24:56 EDT 2020

The paper
On the Semantics of Higher-order Logic
by José Ferreirós,

makes a case that “standard semantics” for second order logic is an improper mixing of mathematics and logic, and they ought to be separated because logic ought to be universal and subject-matter-independent.

But I wonder about this. It doesn’t bother me that there might be universally valid logical statements which we can’t know are universally valid, and the author doesn’t give a deductive calculus which he proposes as sufficient for generating all genuine logical validities.

As in my earlier post (on Universe vs multiverse views of set theory), I would like to bring an argument down to earth and find as concrete a disagreement as possible. So here is my question:

Can the people who object to “standard semantics” for second-order logic provide an example of a specific sentence which ZF proves is a validity of SOL using “standard semantics“ (leaving AC out of it), which in their opinion is NOT a logical truth and is not derivable in any SOL deductive calculus they would accept?

— JS

Sent from my iPhone
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