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From MIT notes:

A proposition excludes statements whose truth varies with circumstance such as, “It’s five o’clock”.

And:

A predicate is a proposition whose truth depends on the value of one or more variables.

If a proposition excludes statements whose truth varies with circumstance, why is a predicate considered as a proposition whose truth depends on the value of one or more variables?

In other words, why can't I just write a statement like "It's 5 o'clock" as:

$P(x) ::=$ "$x$ is equal to 5"

Thanks!

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The notes seem doubly confused, firstly because they use the notion of proposition in two different ways.

Of course, the notion of a proposition is used in somewhat different ways by different authors, and there is no agreement on correct usage. But is a Bad Thing to mix different usages in the same notes.

One usage, as exemplified in the first quotation, has propositions as bearers of determinate truth-values, being the contents of statements as said in a given context. So "it is five o'clock" said in different contexts expresses different propositions (in this first sense). In this use of the notion, an open sentence with a variable neither is nor expresses a proposition. So we can't say, using the first sense of proposition, that an open sentence is a proposition whose truth depends on the value of one or more variables.

We could use proposition more broadly to mean declarative sentence and allow for open sentences to count as propositions; that would allow something like the second claim, but then the first quoted claim will be false with this broader use of the term proposition.

But there is another confusion. A predicate is not a proposition, even in the second broad sense. And a predicate need contain no variable (it seems that the notes confuse a predicate with an open sentence). Thus compare the formal expressions $Fx$, an open wff, and its predicate $F$. Or compare the ordinary language expressions "She is a logician"' corresponding to an open sentence, and its predicate "... is a logician".

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