Speranza's Paradox

by J. L. Speranza
for the Grice Club

JONES, in "The syntax-semantic interface", this blog:

"Forster [2003] seems to me to have things
the wrong way round, for often enough once the semantics
is in place you are scuppered on getting a complete notion
of derivability within the usual constraints (essentially that
proofhood is decidable and hence the set of theorems effectively
enumerable). If the true sentences are effectively enumerable
then people will expect to see a complete deductive
system and hence "⊢" = "⊨"".

Excellent. This reminds me of Speranza's Paradox, which I conceived one hour ago when browing for what Jones refers to as 'the currency' of the rather otiose term, 'syntactic entailment'.

For google-hits were getting pretty boring about this: "semantic entailment entails syntactic entailment". They would not say this, but rather: "semantic entailment implies syntactic entailment" or vice versa, "syntactic entailment implies semantic entailment". Which is ONE step removed from Foster's rather vague reading about the 'identity' of the 'two notions' -- As Jones suggests, Foster is somehow minimising the intension-extension distinction here -- and hence Jones's apt formulation above,

"⊢" = "⊨"

--- which seems it applies to 'utterance-part' meaning (WoW:v!) we may like to expand in the proper metalogical terms, complete with phi and psi, to read as per belowed and be referred to as the "Identity Thesis" or perhaps something weaker in terms of iff, which I read as ')(', where each represents a horseshoe (I cannot find online the inverted horseshoe sign, i.e. the ().


φ⊢ψ )( φ⊨ψ

----

Then, I would like to oppose this "Iff" Thesis to what I may call the Semantic-Primitiveness Thesis and the Syntactic Primitiveness Thesis. For the Syntactic Primitiveness Thesis (which SEEMS more 'primitive'), we would have to replace the 'iff' (or )( above, in my rewriting of Jones's cited thesis) by 'syntactic entailment' to read:


φ⊢ψ ⊢ φ⊨ψ

There is a sub-version of this, which says quite the opposite, and reads:


φ⊨ψ ⊢ φ⊢ψ

---

Then we have the two variants of the Semantic Primitiveness thesis, which read


φ⊨ψ ⊨ φ⊢ψ


φ⊢ψ ⊨ φ⊨ψ

----

Informal analogues:

Syntactic primitiveness, version A:
"Syntactic entailment syntactically entails semantic entailment"

Version B:
"Semantic entailment syntactically entails syntactic entailment"

Semantic primitiveness, version A:
"Semantic entailment semantically entails syntactic entailment"

Semantic primitiveness, version B:
"Syntactic entailment semantically entails semantic entailment".

If only for euphonic reasons, I propose as true:
"Syntactic entailment syntactically entails semantic entailment"
(Thesis of Syntactic Primitiveness, Version A).

But then... I hope I have not myself implicated into something sordid...









φ⊢ψ ⊢ φ⊨ψ








lowercase φ or math symbol ϕ), pronounced [ˈfi] in modern Greek and /ˈfaɪ/ or sometimes /ˈfiː/ in English, is the 21st letter of the Greek ...

en.wikipedia.org/wiki/Phi - Similar
Psi (letter) - Wikipedia, the free encyclopedia
Psi (uppercase Ψ, lowercase ψ; pronounced in English as /ˈsaɪ/, sigh) is the 23rd ... with the Schrödinger equation and