## Truth evaluation in SL

Truth evaluation in general is a matter of determining the truth value of a **sentence** in a given language. A sentence is a formula that can have a meaning.[^1]

[^1]: It happens that this distinction doesn't matter for SL, since all formulas in SL are sentences. We'll revisit this distinction in our discussions of FOL.

Suppose $\Phi_{\text{SL}}$ is the set of all sentences in SL. Then the truth function assigns to each sentence $\phi \in \Phi_{\text{SL}}$ a truth value:

$$ v_i: \Phi_{\text{SL}} \to \{1, 0\}$$

That is, every sentence in SL is either true or false under an interpretation. In application:

$$ v_i(\phi) = 1 \quad \text{or} \quad v_i(\phi) = 0$$

A **truth evaluation** is a determination of the truth value of a sentence in a given language. Remember that the meaning (truth value) of a sentence in a language is the confluence of:

- an interpretation of the non-logical symbols in it, and
- the truth-functional semantics of the logical symbols in it.

## Technical application

A device useful for depicting truth-evaluative relations between logical and non-logical symbols in SL is the **truth table**. A truth table is a table depicting, optionally, all possible interpretations and truth values of a given sentence or set of sentences in SL.

A truth table is a table of columns and rows:

- There are two sets of **columns**:
    - one set of columns for all non-logical symbols in the target sentence or sentences, and
    - another set of columns for all target sentences.
- Each **row** corresponds to:
    - an interpretation, or combination of assigned truth values to all non-logical symbols in the target sentences(s), and
    - a truth value for the target sentence(s), as determined by the interpretation and the truth-functional semantics of the connectives in the target sentence(s)

### Example

The following is a complete truth table for the set of sentences: $\{(P \wedge Q), (Q \to \neg P)\}$.

$$
\begin{array}{|c|c|c|c|}\hline
P & Q & (P \wedge Q) & (Q \to \neg P) \\\hline
1 & 1 & 1 & 0 \\
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 \\\hline
\end{array}
$$