-- by J. L. S.
--- for the GC.
----- IN REVISING SOME OF THE SYMBOLS used by Grice in his extended oeuvre (some of which I have posted to this blog under "Grice on ..." headings) we don't think we find an apt distinction made to me by R. B. Jones, viz. the assertion sign ('... yields ...', or logical-syntactic consequence) from
⊨
which does look like the Frege double (complex) sign of the turnstile, for the assertion or judgement stroke and the content stroke.
⊨
is best understood thus as involving a given interpretation of, say, Grice's model for a formal system in his contribution to Davidson/Hintikka, "Words and objections" -- never so far reprinted elsewhere but partially, in "Definite descriptions", MIT).
I wouldn't be sure what vernacular counterpart to give for
⊨
-- and this would NOT be a problem for Grice. Recall that when he cares to list some of the LOGICAL or object-language 'logical' "constants" or operators -- hardly just connectives since they include 'negation' and the three quantifiers -- 'all', 'some' and 'the' -- he first notes the formal symbol, and THEN the vernacular.
The assertion sign is often used, when in its syntactic guise as:
"Jack is an Englishman; he is, therefore, brave"
Assuming this is a *deductive* -- rather than abductive or inductive -- enthymeme, we would assume to depend on the implicit premise:
"If x is an Englishman, x is brave"
Yielding thus
Jack is an Englishman, If x is an Englishman, x is brave ⊨ Jack is brave
---
We have eliminated the vernacular semicolon, and the parenthetical deictic, originally, 'therefore' -- and we allow for premises to be just iterated by commas, rather than the more cumbersome 'conjunction' sign. Etc.
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I already had a message in my drafts folder (intended for hist-analytic) giving my understanding of both "|-" and "|=" (can't figure out how to get the symbols in here).
ReplyDeleteThe usual distinction is that the first is syntactic and the second semantic (consequence).
Though the first looks closest to the Frege judgement and content strokes, I don't know which is closest in meaning.
My account ot these will be out on hist-analytic as soon as Bayne gets to it.
RBJ
Excellent. I just copied and pasted them from online sources. And I will looking forward to your posting with Hist-Analytic. Incidentally, this may provide a reply, wouldn't R. B. Jones think, to issues like 'changing the topic' -- e.g.
ReplyDeleteRELEVANT LOGIC:
The dog went to the right or to the left.
The dog did not go to the right.
------------
Therefore the right went to the left.
This is valid in classical logic, but not in relevant logic. Wouldn't this entail that we need the very idea of the 'consequence' in semantic terms to deal with such divergence?
I think S. Haak would say as much. While the classical logician and the relevant logician are using the same syntax they abide by a different semantics and it's the combo that makes up for a language, right? Or so I would think.
There are two pragmatically desirable connections
ReplyDeletebetween syntactic and semantic consquence.
The first is soundness. Syntactic consquence is sound if it entails semantic consequence.
The second is completeness. Syntactic consequence is complete if it is entailed by semantic consequence.
Valid as a property of derivations would typically be defined something like this:
A |- B is valid iff A |= B
So a disagreement about "validity" of derivations is a diagreement about semantics.
However, you might just have been talking about the correctness of some proof.
So your point might have been
A |- B & ~(A |'- B) => ~(|= = |'=)
i.e. if two notions of syntactic consequence disagree then so must the corresponding semantic consequence relation.
This will only be the case if the syntactic consequence relations are both is sound and complete.
In that case syntactic and semantic consequence are the same.
I think Greg is disagreeing with Carnap about what a language is, and not noticing perhaps that in the later presentations ("Empiricism, Semantics and Ontology") Carnap he talks about "language frameworks" thus creating a technical term which is clearly intended to encompass the rules of derivation.
RBJ
Further on ⊢ and ⊨. As R. B. Jones comments: two
ReplyDeletepragmatically desirable connections
then:
(a)
'being sound':
""φ ⊢ χ" is sound" ⊢ "φ ⊨ χ"
In Jones's wording: "Syntactic consquence is sound if it entails semantic consequence."
(b)
being complete (or 'compleat' as I may prefer):
"φ ⊨ χ" ⊢ ""φ ⊢ χ" is complete".
In Jones's wording: "Syntactic consequence is complete if it is entailed by semantic consequence."
Then we have 'being valid', which as Jones notes is best defined as:
""φ ⊢ χ" is valid" ≡ φ ⊨ χ
As distinct from
φ ⊢ χ & ~ (φ |'- χ) => ~(|= = |'=)
(Something to be completed in formula above):
As Jones puts it:
if two notions of syntactic consequence disagree, ... so must the corresponding semantic consequence relation
Jones goes on to consider how this deals with definitions of 'language':
As Jones notes it:
In "Empiricism, Semantics and Ontology") Carnap talks about "language frameworks" thus creating a technical term which is clearly intended to encompass the rules of [syntactic] derivation
And I'll think about Grice, revising, I hope, his "Vacuous Names". In any case, felt like symbolising the things using metalogical signs and all.