Talk:Logical consequence/Archive 1

Threrfore redirects to this page

But Implies redirects elsewhere. Should we merge? Can someone explain the difference? (OR is there a better Math dictionary somehwere?)

1. $X\supset \ Y$
2. $X\to Y$
3. $X\implies Y$
4. $X\therefore Y$

NevilleDNZ 09:37, 7 June 2007 (UTC)

syntactic consequence

Where does this term originate?—Preceding unsigned comment added by Philogo (talk • contribs)

Regardless of that [good] question, the example given in the lede:

"... For example, Kermit is green is a logical consequence of All frogs are green and Kermit is a frog."

Is most certainly NOT an example of syntactic consequence, rather semantic. Zero sharp (talk) 23:59, 19 May 2008 (UTC)

Back to Philogo's question, a cursory google search only gives this [1] reference for the phrase 'syntactic consequence' which is a well attested concept (though perhaps not by this exact name) -- I think it's clear enough that what's meant is consequence with respect to some system of formal (syntactic) deduction (a.k.a. proof calculus, etc.). This is _different from_ semantic consequence, in fact if they were the same thing then Godel's Completeness Theorem wouldn't be a big deal at all. It's not clear to me what exactly this article is meant to cover but the semantic/syntactic distinction that I'm used to seeing in discussions of first-order mathematical logic is not evident here. Zero sharp (talk) 00:06, 20 May 2008 (UTC)

The lede as was implied 'Logical consequence' was synonymous with 'syntactic consequence', that's what I was querying, but this has now been fixed.—Preceding unsigned comment added by Philogo (talk • contribs)

there four their fore

I scribe, therefore, um,....

Is therefore sometimes therefor ??

Thank you,

[[ hopiakuta | [[ [[ %c2%a1 ]] [[ %c2%bf ]] [[ %7e%7e ]] ~~ -]] 02:28, 26 August 2007 (UTC)

There is ongoing discussion in philosophy about the nature of logical consequence

In contemporary mathematical logic, logical consequence defined semantically, not syntactically. See, e.g., Barwise, "An introduction to first-order logic", Handbook of mathematical logic p. 10:

Let us write $T\models \psi$ to indicate that ψ is a logical consequence of T in the sense that ψ is true in all models that make all the axioms of T true.

Or Hinman, Fundamentals of mathematical logic, p. 96, at the beginning of the section "Basic semantics", who defines the logical consequence relation $\models$ the same way. I don't have other texts here, but I'm confident they agree about this.

This definition of logical consequence was the subject of Tarski's 1936 paper "On the concept of logical consequence". It is true that, in philosophical logic, the nature of "logical consequence" is a very interesting and active area of discussion. However, it would be incorrect for this article to claim simply that logical consequence is "syntactic consequence" given the large number of texts where it is defined semantically and the philosophical arguments in favor of a semantic definition. — Carl (CBM · talk) 01:04, 20 May 2008 (UTC)

syntactic consequence

A formula A is a syntactic consequence within some formal system of a set Г of formulas iff there is a formal proof in the formal system of A from the set Г. Syntactic consequence does not depend on any interpretation of the formal system. Pontiff Greg Bard (talk) 01:39, 20 May 2008 (UTC)

Wait, does Hunter actually use the term "logical consequence"? Regardless whether he does, as I pointed out above (and as a few minutes of research on Google will convince you), there are numerous different ways of defining logical consequence within philosophical logic, including both syntactic and semantic ones. This is why it's simply incorrect to claim that logical consequence is a synonym for syntactical consequence. — Carl (CBM · talk) 01:44, 20 May 2008 (UTC)

Sounds all right. I just wish I could make that "FS" look like a subscript of that turnstile. Pontiff Greg Bard (talk) 02:06, 20 May 2008 (UTC)

One option is to move the subscript into the TeX formula. — Carl (CBM · talk) 02:20, 20 May 2008 (UTC)

Since I have Hunter's book in my office, I was able to look up the reference today. Hunter doesn't use the term logical consequence on the cited page (p. 75). I looked in the index to see whether Hunter uses the term logical consequence at all, and I found this:

logical consequence: see semantic consequence

So Hunter is another example of an author who would use the term logical consequence to mean semantic consequence (except I think Hunter avoids the term logical consequence completely, and just uses the term semantic consequence).

As I said above, this is actually the most common definition of logical consequence in symbolic logic texts. — Carl (CBM · talk) 17:06, 20 May 2008 (UTC)

Just quotes

The concepts of consequence and validity are semantic concepts, defined in terms of relationships between our formulas and the extralinguistic world. The concept of derivability and of theorem, on the other hand are syntactic. Their definitions refer only to the shapes of the expressions, not to what they may denote when interpreted. ...it is possible to specify [for FOPL] a set of performable inference rules which are such that the theorems of logic will coincide with the valid sentences and consequence will coincide with derivabity.

. Mates 1972 page 164.

--Philogo 20:49, 20 May 2008 (UTC)

A sentence Φ is a truth-functional consequence of a set of sentences Γ if an only if Φ is assigned the truth value T by every normal assignment that assigns T to all sentences of Γ.

Mates 1972 page 89

--Philogo 22:26, 20 May 2008 (UTC)

A sentence Φ is a consequence of a set of sentences Γ if an only if there is no interpretation under which all the sentences of Γ are true and Φ is false.

Mates 1972 page 65

--Philogo 22:30, 20 May 2008 (UTC)

semantics

Hello. I noticed that the article "semantic consequence" is redirected to this article. In fact. there is even a link to "semantic" from here, what is quit strange.

I noticed this discussion up there... So, are those the same thing or not? If not, this redirection should be much better justified. Myself, I run away from the word "semantics", except in a Tarskian context... -- NIC1138 (talk) 04:52, 5 June 2008 (UTC)