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It is common practice to use $\models$ both for the satisfaction relation between models and sentences, and for the corresponding semantic consequence relation.

Question. Suppose I don't want to use $\models$ for semantic consequence (personally, I think this particular convention causes more confusion than its worth), what should I use instead? In particular, is there a standardized alternative?

goblin GONE
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1 Answers1

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In Ben-Gurion University, where I did my B.Sc. and M.Sc. we used $T\implies\varphi$ to denote logical implication, which was really a semantic property:

$T\implies\varphi$ if and only if for every interpretation for the language $M$ and assignment $s$ for $M$, such that every formula in $T$ is true under that assignment; $\varphi$ is true under $s$ as well.

This makes all the more reason to pay attention as to what is $\implies$ and what is $\rightarrow$ (statement about propositions vs. a connective in the language).

Asaf Karagila
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  • Thanks. Is this considered standard in Israeli mathematical circles? – goblin GONE Jan 12 '14 at 14:20
  • I'm not sure, really. I actually don't know what's the common notation in my current university, The Hebrew University, and to add insult to injury I was the TA (and in five weeks will be again) in the Logic 2 course. :-) – Asaf Karagila Jan 12 '14 at 14:22