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My teacher said that $x=5$ and $x^2=25$ are not propositions but they are logical predicate with $x$ as a free variable (I don't know what are logical predicate though).

My question is whether "If $x=5$ then $x^2=25$" is a proposition or not?

I think it shouldn't be because the atomic statements aren't propositions themselves.

(And please don't mind any mistake because it's my first time asking a question on StackExchange.)

Righter
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sanyam
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  • "If $x=5$ then $x^2=25$" is rather informal, but people commonly use to really mean "For all $x\in U$, if $x=5$ then $x^2=25$", where $U$ some set, for example the real numbers. The latter is a proposition. – plop May 16 '21 at 13:09
  • @plop So this means if I just say "if x=5 then x^2=25" and don't assume that x belongs to U, then the latter should not be a proposition. Right? – sanyam May 16 '21 at 13:14
  • Hard to judge. Formalism helps avoid ambiguities like this. I don't know if they really indented to consider the implied $U$ as free. If I were to bet, they didn't indent that and they want want you to read "If $x=5$ then $x^2=25$" as "For all $x\in \mathbb{R}$, if $x=5$ then $x^2=25$" and then say that yes, it is a proposition. – plop May 16 '21 at 13:19
  • It's technically still a logical predicate if it doesn't lock down all its variables with quantifiers. In an "if-then" sentence, however, quantifiers are often implied. – Arthur May 16 '21 at 13:20
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    Does your text have an official definition of "proposition"? If so, compare this to that definition. To ask here, quote that definition for us, then tell us why you still cannot answer the question. (Of course, if there is no definition of "proposition", then there is no way to answer the question.) – GEdgar May 16 '21 at 14:18
  • See the post Formal definition of proposition as well as Sentence vs proposition – Mauro ALLEGRANZA May 16 '21 at 14:47
  • See also Meaning of free variables and Basic question concerning free variables – Mauro ALLEGRANZA May 16 '21 at 14:49

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Usually, in logic, "proposition" has the same meaning of "declarative sentence".

A sentence:

is a meaningful group of words that express a statement, question, exclamation, request, command or suggestion.

In philosophy, there is a sharp contrast between a sentence (a linguistic entity) and the corresponding proposition, i.e. some non-linguistic entity that is expressed by the sentence.

The proposition is taken to be the thing that is in the first instance true or false, while a declarative sentence is true or false derivatively, in virtue of expressing a true or false proposition.

Having said that, a declarative sentence is an expression stating a fact, like: "The rose is red", and we can assign to it a definite meaning and a truth value.

Thus, according to this point of view, an expression with a free variable is not a sentence (proposition) because it lacks a definite meaning.

We can follow Frege's analysis, starting from a sentence: "Socrates is a Philosopher", that we can symbolize with: "$\text{Philosopher}(\text {Socrates})$".

Then erase the name "Socrates", replacing it with a place-holder (a variable) to get:

"$\text{Philosopher}(\xi)$".

The result is an incomplete expression: an open formula.

The incomplete expression has no meaning, unless we replace the variable "$\xi$" occurring in it with a name, and thus it is not a sentence.

The same with the arithmetical sentence "$2 < 5$", from which we get open formula "$x < 5$" replacing the numeral "$2$" (the name of the number two) with the variable "$x$".

As you said, an open formula, i.e. a formula with a free variable (also called propositional function) expresses a predicate.

Mauro ALLEGRANZA
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