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Using the common definition of a propositional function $P(x)$ as "a WFF which would be either true or false were it not for a variable $x$, with other variables also allowed to be free".

For example, if we have a propositional function $P(x)$ with $x$ and $y$ free, then if we only specify $x$, $P(x)$ can still take either "true" or "false" value depending on $y$.

What is the reason for this notation and definition?

Why not define $P(x)$ as "a WFF which would be either true or false were it not for a variable $x$, with no other variables allowed to be free". It would seem much more intuitive if the truth value of $P(x)$ depended on $x$, the truth value of $P(x,y)$ on both $x$ and $y$, etc.

What are the advantages of the former definition over the latter?

Andrés E. Caicedo
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cxzcxzcas
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1 Answers1

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From :

Consider the proposition (expressed by the sentence) "Socrates is a man". The part of this proposition (expressed by) "— is a man" or "$x$ is a man" is a predicate; "Socrates" is a subject. Read "$x$ is a man", using the mathematical notation of a variable, the predicate is seen to be a propositional function, i.e. for each value of the (independent) variable "$x$", it becomes (or takes as value) a proposition, true for example when $x$ is Socrates, false in Greek mythology when $x$ is Chiron, and in the Kleene household when $x$ is Fleck.

Of course, a propositional function may have more than one variable free, like a mathematical function of more than one variable, but it will have a definite truth-value only if a value (or reference) is assigned to all its free variables.

"$x$ is father of $y$" is a propositional function of $x$ and $y$, while "$x$ is father of John" is a propositional function of $x$ alone.

Only "Tom is father of John" is a proposition, i.e. has a truth-value, because it has no more "unsaturated" slots.

Mauro ALLEGRANZA
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