Models, Truth, Logical Implication, and Validity

We are almost ready, finally, to say what it means for a sentence to be true in a structure. First, we define what it is for a structure to be a model of a formula.

Definition: If $\phi$ is a formula in the language and is an -structure, we say that is a model of $\phi$, and write $\ensuremath{\mathfrak A}\models \phi$, if and only if $\ensuremath{\mathfrak A}\models \phi[s]$ for every assignment function $s$. If $\Phi$ is a set of -formulas, we will say that models $\Phi$, and write $\ensuremath{\mathfrak A}\models \Phi$, if and only if $\ensuremath{\mathfrak A}\models \phi$ for each $\phi$ in $\Phi$.

The symbol `$\models$' is rather versatile! The right-hand side can be either a formula or a set of formulas. Much more versatility is yet to come.

Informally, the basic idea is that a structure is a model of a formula if the formula must be true in the structure, i.e. if the formula comes out true no matter what terms you substitute for its free variables.

Of course, sentences are special cases of formulas. In the case of sentences, it makes no difference what assignment function $s$ we use. Why? Because sentences have no free variables, and assignment functions have no effect on bound variables. So if there is any assignment function $s$ for which $\ensuremath{\mathfrak A}\models \phi[s]$ in the special case in which $\phi$ is a sentence, then, since it makes no difference what assignment function we use, $\phi$ will be true for every assignment function, and hence we can simply say that $\ensuremath{\mathfrak A}\models \phi$, period.

To make it obvious when we are dealing with formulas that are also sentences, we will use a different Greek variable for sentences, `$\sigma$'. (Notice the alliterative choice of Greek letters: phi, with its initial `f' sound, for formulas, and sigma, with its initial `s' sound, for sentences.)

Definition: If $\sigma$ is a sentence, then we say that $\sigma$ is true in if and only if $\ensuremath{\mathfrak A}\models \sigma$, which in turn is the case if and only if there is any assignment function $s$ for which $\ensuremath{\mathfrak A}\models \sigma[s]$.

Notice that a structure models a formula if and only if the sentence that results from prefixing to the formula universal quantifiers binding all the free variables of the formula results in a sentence that is true in the structure.

Finally, we turn to Leary's definitions of logical implication and validity. Logical implication is defined in Definition 1.9.1 on p. 43:

Definition: Suppose that $\Delta$ and $\Gamma$ are sets of -formulas. We will say that $\Delta$ logically implies $\Gamma$ and write $\Delta \models \Gamma$ if for every -structure , if $\ensuremath{\mathfrak A}\models \Delta$, then $\ensuremath{\mathfrak A}\models \Gamma$.

If we restrict ourselves to sentences for a moment, we can say that one set of sentences $\Delta$ logically implies another set of sentences $\Gamma$ if and only if every structure in which all the sentences $\Delta$ are true is a structure in which all the sentences in $\Gamma$ are true. Equivalently, we can say that $\Delta$ logically implies $\Gamma$ if and only if every model of $\Delta$ is also a model of $\Gamma$.

Still another way to say the same thing: sometimes structures are called interpretations, and a model of a sentence is called an interpretation in which the sentence is true. So we can say that $\Delta$ logically implies $\Gamma$ iff every interpretation that makes all the sentences in $\Delta$ true also makes all the sentences in $\Gamma$ true.

It is a short step to the notion of a valid argument: an argument from a set of premises $\Gamma$ to a conclusion $C$ is valid iff $\Gamma \models C$.

The term `valid' is also used for a property of individual formulas. This usage is defined in Leary's Definition 1.9.2:

Definition: An -formula $\phi$ is said to be valid if $\emptyset \models \phi$, in other words, if $\phi$ is true in every -structure with every assignment function $s$. In this case, we will write $\models \phi$.

Notice the relation between this definition and Barwise and Etchemendy's more informal definition in Language, Proof, and Logic. B & E informally charactize a valid sentence as one that remains true regardless of which predicates are used. The idea of substitution of predicates is an informal way to accomplish what the formal definition accomplishes by considering different structures (which will have different interpretations of the predicates).