Re: The simple essence of Proof Theoretic Semantics

On 2026-07-02 23:02, olcott wrote:
> On 7/1/2026 9:03 PM, olcott wrote:
>> Q cannot do the ∀x without an infinite sequence of steps.
> 
> So your phrasing is good: Q would need something like an infinite 
> sequence of steps (or a single principle that summarizes them) to get 
> the ∀x. Since formal proofs must be finite, and Q lacks the tool 
> (induction) that would allow a finite proof of the infinite claim, the 
> universal statement remains unprovable.

I'm not sure why you are responding to yourself nor who 'your phrasing' 
refers to since you don't quote anyone. But, assuming we're still 
talking about ∀ x, S(x) ≠ x in Q, your reasoning is simply off.

You *can* prove universally quantified claims in Q, just not that 
particular claim.

And there isn't an infinite sequence of steps that will get you from the 
axioms of Q to ∀ x, S(x) ≠ x. There's *no* sequence of steps, finite or 
infinite.

The issue here is that there are models of Q in which ∀ x, S(x) ≠ x is 
true, but there are also models of Q in which it is false.

For any given model of Q, it will either be true or false, so your claim 
that ∀ x, S(x) ≠ x is somehow 'not a truth bearer' is simply ludicrous. 
It's simply the case that this particular statement cannot be derived as 
a theorem of Q nor can its negation. Thus Q is incomplete.

André

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