I have been reading Predicate Logic couple of days and while everything has been pretty intuitive so far I understood that I do not exactly understand what the predicate is. This became clear after I tried to understand the computation of truth conditions for the following sentence:
"Every person works."
Starting computing truth conditions from expression:
∀x(person(x) -> work(x))
[[ ∀x(person(x) -> work(x)) ]]M,g = 1
[[ person(x) ]]M,g[x/a] = 0 OR [[ work(x) ]]M,g[x/a] = 1
g[x/a](x) ⊄ Vm(person) OR g[x/a](x) ⊂ Vm(work)
we eventually come to the following part:
iff a ⊄ Vm(person) OR a ⊂ Vm(work)
The moment where my confusion became obvious is this part, what is person
here exactly?
I understand predicates as functions. If predicate is a function it should accept some value and return some other value and predicates in FOL are functions that return True OR False. In the example above person
is a predicate thus a function. It should accept some value and return True OR False like person(a)
for example.
So I understood this line as "iff a
returns False for person
and returns True for work
" then this whole statement stands True. However my classmate told me that here person
is a set in our Universe. How can a function represent a set, which is a collection of elements? Moreover how can a set return True and False values? So now I am confused how can a predicate be a set?
Best Answer
In first-order logic, a predicate is a symbol of the language.
According to Gottlob Frege - one of the "founding fathers" of modern logic - the meaning of a predicate is exactly a function from the domain of objects to truth-values : "the True" and "the False".
Thus, the predicate $philosopher(x)$ denotes a function such that :
and :
In modern view of logic, the meaning of a predicate is a subset of the domain, i.e. the set of all objects of the domain such that the predicate holds of them. In "traditional" terms, an (unary) predicate corresponds to a property.
Thus, the meaning of the predicate $philosopher(x)$ is the set $Philosophers$, i.e. the set of all philosophers, so that :