# Translating a sentence with a time element into first order logic

The sentence I want to translate into FOL = "Some people lie sometimes"

I know that Ex(Px ^ Lx) = "Some people lie"

And Ex(Tx) = "There exist some times" (or "moments in time")

Where I am struggling is how to combine them to say...

"There exist some moments in time in which some people lie"

Opinions welcome! Looking to learn from this interesting sentence. Perhaps it can't be translated into FOL but I am far from an expert.

You would need a predicate L(x, t) that says that x lies at time t. Then you could write something like

Ex Et P(x) ^ T(t) ^ L(x, t)

which say that there is a person which at some point lied.

The issue here is more to do with understanding what you are trying to express, rather than with turning it into formal logic.

The word 'some' is vague, but it is usually treated in formal logic as meaning 'at least one'. So, "some people lie sometimes" could be taken to mean, "at least one person has told at least one lie" or simply, "at least one person is a liar". Since we usually understand 'liar' to mean a person who sometimes lies rather than a person who always lies, we do not need any further qualification.

Since telling a lie always happens at some time, the time element is somewhat irrelevant. If you wished, you could state that the telling of a lie happens at some time. "At least one person has told at least one lie at some time t." This might be formalised as:

(∃x)(∃y)(∃t)(Person(x) ∧ Utterance(y) ∧ Lie(y) ∧ Time(t) ∧ Stated(x, y, t))

When we say 'some' or 'sometimes' then depending on the context there may be an implicature that the speaker intends "not all". If you wish to state this explicitly, then you could express it as, "there is at least one person who has said at least one thing that is a lie and who has also said at least one thing that is not a lie". As before, the time element isn't really needed, though you could put two timestamps in.

You are trying to do something out of scope. Words such as 'yellow', 'easily', 'Wednesday', etc don't have special representations in logic, and as far as I know, 'sometimes' is no different. You can content yourself with saying there are people who sometimes lie. Alternatively, you might try to express it along the lines of there being some people who make statements some of which are deliberately untrue, and you can take the 'sometimes' as read, since people only make statements one at a time. However, if you are happy to include the explicit adverb 'deliberately', then why not just include the explicit adverb 'sometimes'?