Predicate

A predicate is a logical function that can take in zero or more terms (called the subjects) and outputs truth values (true or false). However, a key distinction between predicates and propositions is that predicates (that have at least one subject) do not have a truth value until their variables have been specified.

A predicate can be considered an $n$-ary (n=1: unary, n=2: binary, etc.) function for $n\ge 0$ that takes in $n$ terms. A 0-ary predicate degenerates into a proposition (since a proposition returns truth values with no inputs).

A predicate that does not contain any sub-predicates is considered an atomic formula.#todo extend Atom definition to include this!

Definition §

#todo function that takes in terms and returns truth values.

#todo move to atom

Truth (Atomic Predicate) ^definition-truth

The truth of an atomic predicate (a predicate having no sub-predicates), $P$ in a model, $M$ (having an interpretation, $\mathcal{I}$ and universe, $\mathcal{U}$) with a variable assignment $v$ is defined as:

$$P(t_1,t_2,\dots t_n)\text{ is true iff }(v(t_1),v(t_2),\dots ,v(t_n))\in I(P)$$

• $v(t)$ is the term $t$ under an extended variable assignment
• $I(P)$ is the interpretation of the predicate $P$, and is a set of elements that make $P$ true

Examples §

• $A(x)$: $x$ is an animal
• $G(y)$ : $y$ is greater than 3

Predicates are conventionally notated with a capital letter, $P(x)$.

Once a variable has been defined, the predicate becomes a proposition:

• Dog is an animal (True or False)
• $4$ is greater than 3 (True or False)
• etc.