I have spent 28 years of primary research on this. The proof theoretic halt prover H gets stuck in infinite recursion when it attempts to prove the halt status of input D. H does recognize that its proof would never terminate for this reason and rejects its input as ungrounded within the proof theoretic semantics foundation. Within PTS an expression only acquires semantic meaning through a finite sequence of inference steps.
Proof-theoretic semantics is inherently inferential, as it is inferential activity which manifests itself in proofs. It thus belongs to inferentialism (a term coined by Brandom, see his 1994; 2000) according to which inferences and the rules of inference establish the meaning of expressions Schroeder-Heister, Peter, 2024 “Proof-Theoretic Semantics” Proof-Theoretic Semantics (Stanford Encyclopedia of Philosophy)
Yes. PTS defines meaning in terms of proof. That’s a choice between theoretical approaches. It simply declares the Liar meaningless because it cannot be assigned a truth value.
We also know from Gödel that truth separates itself from provability. There will always be truths that are unavailable to PTS. If meaning is constituted by proof conditions, then what is the semantic status of Gödelian sentences? This is a real challenge.
Why does the Liar feel meaningful in natural languages? What exactly goes astray when we evaluate it? How do the different responses mesh?
Engineers regularly drop by the Forums to explain the answer to various philosophical issues. Almost ubiquitously, what they demonstrate is their failure to understand the problem.
That is great you are the first person that I could ever have a meaningful dialogue with on this. The semantic status of Gödelian sentences within PTS is meaningless. This reframes things such that the only truth in PA is whatever can be derived from the axioms of PA. Truth defined in an external model of arithmetic is simply outside the scope of PA.
Because in English it is simply too slippery so see that it is meaningless. When we evaluate in in a formal language like Prolog we can directly see that is has a cycle in the directed graph of its evaluation sequence. This essentially causes the evalution to get stuck in an finite loop.
Sure, if you accept PTS, the sentences are meaningless. But why should we accept PTS? The reason can’t be “because it gets rid of the Liar,” unless you are satisfied with begging the question.
“This sentence is false” is a plain English sentence about a plain English sentence. It’s not obviously meaningless.
So what does it tell us about truth, and about truth values, and about proof?
Simply assigning it to the naughty corner, as PTS does, is to ignore the issues, not to address them.
It is not the “circularity” of my position. It is the “recursive” nature of the data relative to its inference steps
Because PTS is the only foundation that accurately represents the actual underlying semantic relations between the finite strings that cause them them to gain semantic meaning from these relations.
It need not be obviously meaningless to be meaningless. It does seem to obviously not resolve to a truth value.
Some expressions of language such as: “What time is it?” do not have truth values. Through both Dag Prawitz and Saul Kripke ungrounded in a truth value means not truth apt. Within PTS the lack of a finite proof essentially means not a truth bearer.
When logic is bigger than grounded though PTS then this aspect of Logic becomes unfounded.
“G is not provable in PA” has only been syntactically expressible in PA, it has only been meaningful in an external model of arithmetic, never directly meaningful in PA. PTS directly sees this. All of knowledge expressed in language can be encoded as stipulated relations between finite strings. PTS seems to take this as an implicit assumption.
When there is no back-chained sequence of inference steps from expression X of system F to the axioms of F, then X is not meaningful in F.
A mere assertion without any support. Show this to be the case. Same for the rest of that post - mere assertion. Is your argument no more than from a supposed authority? Demonstrate your case, not the self-sealing goop.
You define meaning in such a way that G is meaningless. It is open to others to treat this as a reductio that shows your definition of meaning is wanting. You have not demonstrated your case.
Unless it is disguised as the: (a) Liar Paradox (b) Incompleteness Theorem (c) The Halting Problem proof, then it becomes utterly unfathomable.
The proof is that a counter-example cannot possibly exist. At this point it seems to only be a thesis.
Meaning expressed in language always has been defined such that expressions of language are only defined in terms of other expressions of language. “cats” [are] “animals” is simply the stipulated relation type of [are] between the otherwise arbitrary finite strings of “cats” and “animals”
The algorithm of PI is within the body of knowledge that can be expressed in language. The value of PI is outside of this body.
So you have no actual counter example?
I assume that the basis is something like a written list of “atomic facts” such as “cats are animals”. From this basis every member of the body of knowledge expressed in language can be derived.
Is semantic gibberish the same as:
“What time is it (yes or no)?”
The value of Pi is the ratio of the diameter of a circle to its circumference. Now “The value of Pi is the ratio of the diameter of a circle to its circumference” is expressed in language.
So, again, it ain’t so clear what you mean.
Yes - extensional and ostensive definitions.
How very Tractarian. I’m more an Investigations type.
…only if you presume that PTS is true; your task is not to assume, but to prove. Again, we understand what “This sentence is false” means - if we didn’t there would be no consternation.
The infinite digit sequence of the value of PI is outside the body of knowledge.
A 100% totally concrete example of knowledge that can be expressed in language that is not (a) Stipulated to be true such as {“cats” are “animals”} or derived from other expressions of language.
I never have never been boxed in to any conventional school of thought. I always reverse-engineer on the basis of first principles for 28 years.
Infinitely recursive semantic gibberish that never resolves to a truth value.
Yep. And? It remains that the algorithms that calculate Pi are fathomable, yet infinite and recursive. You were saying that there is a problem inherent in infinite recursion. But it works for Pi. So what exactly is the problem you see?
Because, you see, as talked about above, recursion does work in some situations, and there are variants of the liar that do not use recursion, hence recursion, even infinite recursion, does not appear to be the source of the problems with the liar.
You seem to have moved your opinion so as to include stipulation; but a definition is just a stipulation. You had said
That’s not so. We can define terms by pointing, or by listing the items to be included. That is my cat Lilly; the planets of the Solar System are {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}.
Lilly is defined not by an expression but by pointing. The planets are defined not as the list of names but as the very things those names pick out. So it remains very unclear what you might mean here.
Now it’s important, in a long discussion such as this, to go back to the original point. So again, it appears to me that all you have done is to specify a framework - Proof Theoretic Semantics - that does not permit the Liar.
And the reply is, so what? What we are after is to understand how the Liar works.
My 28 year goal has been to make
“true on the basis of meaning expressed in language”
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.
is my cat Lilly; the planets of the Solar System are {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}.