On truth and satisfaction

Now that we know how to assign values to our terms relative to a structure $\mathbf{S}$ and a variable assignment $\mathbf{a}$, it is easy to see how each quantifier-free formula gets its value in the language $\mathcal{L}_1$. In particular, the calculations are essentially identical to those of $\mathcal{L}_0$, except if you encounter a variable (all of which will be free by assumption of the formulas being quantifier-free), you need to calculate with the assignment $\mathbf{a}$ instead of the interpretation function $I$.

In particular, for atomic formulas, of form $P^n(t_1, …, t_n)$, their semantic value, as determined relative to $\mathbf{S}=\langle\mathbf{D}, I\rangle$ and the variable assignment $\mathbf{a}$ in $\mathbf{D}$, will just be: $$\mathbf{S}\models P^n(t_1, …, t_n)[\mathbf{a}] \textit{ iff } \langle I(t_1)[\mathbf{a}], …, I(t_n)[\mathbf{a}]\rangle \in I(P)$$ In other words, for each term in our atomic formula, we check if their value under $\mathbf{S}$ and $\mathbf{a}$ is in the interpretation of $P$ or not. Notice that this means most of the time, the value of an atomic formula of $\mathcal{L}_1$ with free variables relative to a structure $\mathbf{S}$ will depend on the particular variable assignment we are calculating with.

One crucial thing regarding our terminology is that technically, if the atomic formula $P^n(t_1, …, t_n)$ is open because it has some (trivially, free) variables occurring in it, then what $\mathbf{S}\models P^n(t_1, …, t_n)[\mathbf{a}]$ says is not, in general, that the formula is true. Rather, what it says is that the formula $P^n(t_1, …, t_n)$ is satisfiable in $\mathbf{S}$, and specifically, satisfied in $\mathbf{S}$ under $\mathbf{a}$. The formula cannot be called true because relative to some other assignment $\mathbf{b}$, it may not be the case that $\mathbf{S}\models P^n(t_1, …, t_n)[\mathbf{b}]$.

Indeed, truth in a structure $\mathbf{S}$ will be defined just as satisfaction in $\mathbf{S}$ under every variable assignment $\mathbf{a}$ in $\mathbf{D}$. And incidentally, sentences will get their value independent of any particular assignment $\mathbf{a}$ so they will all be truth-apt; either true or false in a structure. This is partly why sentences are so crucial. Because we are interested (as of now) in truth, and not satisfaction under an assignment.

Accordingly, to say that a formula $X$ is true in $\mathbf{S}$, we will suppress the notation $[\mathbf{a}]$ (as by definition, it is irrelevant), and write: $$\mathbf{S} \models X$$ If we want to say that a formula $X$ is satisfied in $\mathbf{S}$ under $\mathbf{a}$, we write: $$\mathbf{S} \models X[\mathbf{a}]$$

You can think of the distinction between satisfaction under an assignment and truth as follows. If you say something like “$x$ is tall”, you cannot really say that this is either a true or false statement as it is. Clearly, if we understand ‘$x$’ as former professional basketball player Yao Ming (height: 7’ 6″), it would be ‘true’, and if we understand ‘$x$’ as movie star Danny DeVito (height: 4’10″), it would be ‘false’. But in itself, it is neither true nor false, because $x$ may stand for anything! On the other hand, the sentence “Yao Ming is tall” is true, because he is tall independent of how we understand ‘$x$’ (because it is not even part of the sentence).

For example, let $\mathbf{S}=\langle\mathbf{D}, I\rangle$ be such that $\mathbf{D}=\mathbb{N}$, $I(\mathfrak{c}_{n})=n \times 2$, and let $\mathbf{a}(\mathbf{x}_{n})= n + 7$ ($n \in \mathbb{N}$) as before. Let $I(\mathfrak{P}^{3}_{1})=\{\langle n, j, k\rangle\mid n+j=k\}$, and $I(\mathfrak{P}^{3}_{2})=\{\langle n, j, k\rangle\mid n\times j=k\}$. Then, consider an atomic formula, such as: $$\mathfrak{P}^{3}_{1}(\mathfrak{c}_{1}, \mathfrak{c}_{3}, \mathbf{x}_{5})$$ Is the above formula satisfied in $\mathbf{S}$ under to the assignment $\mathbf{a}$? Well, $I(\mathfrak{c}_{1})[\mathbf{a}]=2$, $I(\mathfrak{c}_{3})[\mathbf{a}]=6$, and $I(\mathbf{x}_{5})[\mathbf{a}]=12$. On the other hand, $2+6\neq 12$, so this triple is not in $I(\mathfrak{P}^{3}_{1})$. Thus: $$\mathbf{S} \not\models \mathfrak{P}^{3}_{1}(\mathfrak{c}_{1}, \mathfrak{c}_{3}, \mathbf{x}_{5})[\mathbf{a}]$$ Of course, as we discussed above, a well-chosen $\mathbf{x}_{5}$-variant of $\mathbf{a}$ may very well satisfy this formula in $\mathbf{S}$ relative to it. Of course, this is easy to specify with our explicit notation: $$\mathbf{S} \models \mathfrak{P}^{3}_{1}(\mathfrak{c}_{1}, \mathfrak{c}_{3}, \mathbf{x}_{5})[\mathbf{a}^{\mathbf{x}_{5}}_8]$$ Clearly, we can also consider: $$\mathbf{S} \models \mathfrak{P}^{3}_{1}(\mathfrak{c}_{1}, \mathfrak{c}_{3}, \mathbf{x}_{1})[\mathbf{a}].$$ And in fact, with a different change in our formula, we get: $$\mathbf{S} \models \mathfrak{P}^{3}_{2}(\mathfrak{c}_{1}, \mathfrak{c}_{3}, \mathbf{x}_{5})[\mathbf{a}]$$

As far as the connectives go, there is no change in how their meaning is specified relative to their less complex constituents, except again, we are calculating with both $I$ and $\mathbf{a}$ at each turn.

Thus, we get that:

1. $\mathbf{S} \models \neg X[\mathbf{a}]$ if, and only if, $\mathbf{S} \not\models X[\mathbf{a}]$;

2. $\mathbf{S} \models (X \wedge Y)[\mathbf{a}]$ if, and only if, $\mathbf{S} \models X[\mathbf{a}]$ and $\mathbf{S}\models Y[\mathbf{a}]$;

3. $\mathbf{S} \models (X \vee Y)[\mathbf{a}]$ if, and only if, $\mathbf{S} \models X[\mathbf{a}]$ or $\mathbf{S}\models Y[\mathbf{a}]$ (or both);

4. $\mathbf{S} \models (X \rightarrow Y)[\mathbf{a}]$ if, and only if, if $\mathbf{S} \models X[\mathbf{a}]$, then $\mathbf{S} \models Y[\mathbf{a}]$.