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According to Wikipedia:

A logical consequence is the relationship between statements that holds true when one logically "follows from" one or more others.

So,

A ⊨ B

B is a logical consequence of A when in all cases of A being true, B is true as well.

However, to my understanding, that is also what implication means.

A → B

B should be true whenever A is true. Isn't that the same as entailment?

ninesalt
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  • I only know the very basics of logic. But I think that the symbols live on different levels. Symbol $A \vDash B$ denotes a relationship between axioms $A$ and a formula $B$. On the contrary, $A \to B$ is a single formula. Please correct me if I'm wrong. – Matias Heikkilä Oct 20 '15 at 14:31
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    As said in the above comment, $A \to B$ is a formula in the language (e.g. propositional calculus) whlie $\varphi \vDash \psi$ is a relation between formulae, and thus it is an expression in the meta-language. – Mauro ALLEGRANZA Oct 20 '15 at 15:04

1 Answers1

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As said in the above comment, $A \to B$ is a formula in the language (e.g. propositional calculus) while $\varphi \vDash \psi$ is a relation between formulae, and thus it is an expression in the meta-language.

It is true that $A \vDash B$ iff $\vDash A \to B$, but still the difference is important.

We may have, e.g., a language with only $\lnot$ and $\lor$ conncetives; in it the definition of well-formed formula changes ($A \to B$ must be introduced as an abbreviation) while the definition of $\vDash$ does not.

In addition, the relation $\vDash$ holds also with a set $\Gamma$, possibly infinite, of formulae :

$\Gamma \vDash \varphi$,

while $\gamma \to \varphi$, being a formula, must be a finite string, and thus the antecedent $\gamma$ can be at most a finite conjunction.

Mauro ALLEGRANZA
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    Note that, in some cases, the deduction theorem does not hold. If you define $\Gamma \models \phi$ as "if $\mathcal{A} \models \Gamma$, then $\mathcal{A} \models \phi$, where $\phi$ may be a formula and $\mathcal{A} \models \phi$ is defined as $\phi$ is true in $\mathcal{A}$ under every variable assignment, the corresponding deduction theorem will only hold if $\phi$ is a sentence. See this post for more details: http://m-phi.blogspot.com.br/2013/07/two-notions-of-consequencededucibility.html – Nagase Oct 20 '15 at 23:03
  • @Nagase - sure... but due to the "level" of the question, I would prefer not to go too much in deep. I prefer to stay at the "ground level" of propositional calculus :-) – Mauro ALLEGRANZA Oct 21 '15 at 07:01