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# Semantics of Predicate Logic

In propositional logic, the key semantic notion is that of a truth-value assignment. A truth-value assignment involves the assignment of truth-values (truth and falsity) to the atomic wffs. After assigning truth-values to atomic wffs or sentences, we determine the truth-values of complex wffs by looking at how the complex wffs were built out of atomic wffs and logical connectives. In predicate logic, our atomic wffs are themselves analyzable into sub-sentential components. So, for example, the atomic wff '$F\phantom{\rule{0}{0ex}}a$' is analyzable into the predicate '$F$' and the constant '$a$'. These components are not assigned truth-values directly, as was the case with the sentences of propositional logic. Instead, the subsentential components of atomic wffs (constants, predicates, quantifiers, and variables) are given interpretations, and these interpretations are used to determine the truth-values of sentences, or wffs, of predicate logic.

In what follows, we will lay out an informal sketch of the key semantic notion of predicate logic, that of an interpretation. For a more formal discussion, please see Formal Semantics for Predicate logic.

## Informal Semantics for Predicate Logic

The basic semantic notion for the semantics of predicate logic is the notion of an interpretation. An interpretation interprets every individual constant, predicate, and sentence letter of predicate logic.

### Discourse

An interpretation takes place relative to a domain, or universe, of discourse. A domain is a non-empty set. In first-order logic, a domain is a set, or collection, of objects. We can specify a domain either by giving the defining characteristics of the members of that set or by enumerating the elements of that set. So, for example, the domain that equals $\left\{x|x$ is a natural number less than 2 $\right\}$ is the same domain as $=\left\{0,1\right\}$.

### Constants

We interpret a constant by specifying which object the constant refers to. If one is interpreting a sentence or argument that contains a constant, one should be sure to include the constant in the domain. We do not interpret the variables and, so, we do not assign them objects in the domains as referents.

### Predicates

Predicates can be interpreted in one of two ways. One can either say that a predicate '$P\phantom{\rule{0}{0ex}}x$' means '$x$ has $P$' or one can specify the set of things that have the property P ( $\left\{x|x$ has $P\right\}$). The second is called the 'extension of the predicate $P$.

### Quantifiers

We interpret quantified sentences relative to the specification of a domain and relative to the interpretations of the predicates involved in the sentence. So, given the following interpretation and domain,

Domain: $\left\{x|x$ is a living creature $\right\}$

```        $F$: $\left\{x|x$ is human $\right\}$
$G$: $\left\{x|x$ is kind $\right\}$,
```

the sentence $\forall x\left(G\phantom{\rule{0}{0ex}}x\to F\phantom{\rule{0}{0ex}}x\right)$ will be false, since presumably there are kind creatures that are not human. The sentence $\exists x\left(F\phantom{\rule{0}{0ex}}x\wedge G\phantom{\rule{0}{0ex}}x\right)$ will be true, assuming that there is at least one kind human.

### Examples

For example, if we have
Domain: $\left\{x|x$ is a living being $\right\}$

```        $F$: $\left\{x|x$ is a car $\right\}$

```

Then $F$ will not hold of anything in the domain. If we change the domain to $\left\{x|x$ is involved in NASCAR $\right\}$, then $F$ will hold of things in the domain (since there are a lot of cars involved in NASCAR).