In first order logic,is $(\text{True}\lor P(x))$ considered $\text{True}$ (where the variable $x$ is free) ?
One we can approach is as follows, since there is a $\text{True}$ in a dis-junction with $P(x)$ where $P(x)$ is a predicate, then irrespective of the what value $P(x)$ takes the first order logic (FOL) expression could considered to be $\text{True}$.
But what also bugs me in the above is that, if $x$ is free, then $P(x)$ is not actually a proposition and hence it shall not have any truth value associated with it. As such assuming $P(x)$ to be $?$ in this case, [$?$ is something unknown], is it quite right to apply a logical connective to it and get a truth value, such as $\text{True}\lor ? \equiv \text{True}$.
I mean, we could just write $Q(x): \text{True} \lor P(x)$, as $x$ is free, we cannot assign a truth value to the predicate $Q(x)$ , it is not a proposition…
Please help me.
Best Answer
It's more accurate to say that predicates typically have varying truth values.
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.