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Usually ${\mathcal{M}}$ $\models$ $\phi$ means that the sentence $\phi$ is satisfied in model $\mathcal{M}$.

2 Answers2

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The symbol $\models$ is heavily overloaded: it has (at least) three common and closely related meanings.

  1. As you note in your question, when $M$ is a structure and $\varphi$ is a sentence, $M\models \varphi$ means "$M$ satisfies $\varphi$".
  2. When $M$ is a structure and $T$ is a theory, $M\models T$ means "$M$ is a model of $T$": $M\models \varphi$ for all sentences $\varphi\in T$.
  3. When $T$ is a theory and $\varphi$ is a sentence, $T\models \varphi$ means "$T$ entails $\varphi$": For all structures $M$, if $M\models T$, then $M\models \varphi$.

Often, when people write $\models \varphi$, the symbol $\models$ stands for entailment (the third meaning), and the empty left-hand-side represents the empty theory. So $\models \varphi$ means $\varnothing \models \varphi$, the empty theory entails $\varphi$. So all structures satisfy $\varphi$ (since $M\models \varnothing$ vacuously), i.e. $\varphi$ is valid.

Similarly, $\vdash \varphi$ means that $\varphi$ can be proven with no assumptions.

As Mark Kamsma notes in his answer, this notation (like any notation) can have other meanings in other contexts. So it's important to make sure that the way you understand the notation makes sense in context.

Alex Kruckman
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    I've also seen people use $a\models p$ , for $p$ a type and $a$ an element of some model, to mean $a$ realizes $p$. – Reveillark Apr 15 '20 at 16:11
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Usually when we just write $\models \phi$, we mean that $\phi$ holds in the monster model of the theory we are studying.

I will give a brief description of what this means, but if you have never heard of the monster model then I suggest you read up on that first.

We are working with a fixed theory $T$ throughout. Then the monster model $\mathfrak{M}$ is a model that is very saturated and very homogeneous. This is a bit vague, but it is not too important how to make this precise. Here are a few options.

  1. Fix some large cardinal $\kappa$, then we can build a $\kappa$-saturated and strongly $\kappa$-homogeneous model $\mathfrak{M}$.
  2. Let $\kappa$ be larger than any set or model we will be interested in, and then take some $\kappa$-saturated strongly $\kappa$-homogeneous $\mathfrak{M}$.
  3. In Bernays-Gödel set theory we can construct a class-sized model $\mathfrak{M}$ that is $\kappa$-saturated and strongly $\kappa$-homogeneous for all cardinals $\kappa$.

As you can see, the point is that the monster is saturated and homogeneous enough for all the sets and models we will ever be interested in.

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Mark Kamsma
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    This is a common notational convention in model theory, when we are working in the context of a fixed complete theory (and especially when $\varphi$ includes parameters from the monster model). In a wider logic context, though, $\models \varphi$ often just means $\varnothing \models\varphi$, i.e. $\varphi$ is entailed by the empty theory, and hence true in every structure. – Alex Kruckman Apr 15 '20 at 14:53