In first order logic, is $(\text{True}\lor P(x))$ considered $\text{True}$ (where the variable $x$ is free)?

Solution 1:

P(x) is not actually a proposition and hence it shall not have any truth value associated with it.

It's more accurate to say that predicates typically have varying truth values.

$Q(x): \text{True} \lor P(x)$

we cannot assign a truth value to the predicate Q(x) , it is not a proposition...

Do you agree that the predicate $$(x=y\land y=z)\to x=z$$ is true? If so, then the predicate $Q(x)$ is likewise true.

We don't need to know the actual formulation of $Q(x)$'s left disjunct "True" to assert that it is a validity; likewise, we don't need to know $Q(x)$'s full formulation (its right disjunct is unspecified) to assert that $Q(x)$ is a validity.

Solution 2:

The issue is: are we approaching the question formally or intuitively?

From a formal point of view, truth can be ascribed to a formula when we specify an interpretation, i.e. a domain $D$, a "meaning" for the predicates, i.e. for a unary predicate $P$ a subset of the domain, and a "temporary" meaning to free variable, for example with a variable assignment function $s : \text {Var} \to D$.

Thus, assume for example an interpetation $\mathcal I$ with domain the set $\mathbb N$ of naturals, as meaning for predicate $P$ the property "$x \text { is Even}$", which means that $P^{\mathcal I} = \{ n \mid n \in \mathbb N \text { and } n \text { is Even} \}$, and as variable assignment the function $s(x)=3$.

Now we have all we need to compute $\mathbb N, s \vDash (\text T \lor Px)$, i.e. the truth value of the formula for the given interpretation.

In plain text we have to check if the statement "either (True) or (3 is Even)" is true in the domain of naturals.

According to the truth table for "or" the answer will be: YES, and the obvious fact is that this will hold irrespective of the value that we choose for $x$.

If instead we approach it from an intuitive point of view, a free variable acts as a pronoun. Thus, "$Px$" can be read as "it is (a) $P$".

The truth value depends of course on the meaning of $P$ but also on the way we "understand" the pronoun: are we pointing with our finger to something on the table when we utter the statement?

If so, we have simply to check if that object is "a P".