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I am beginning to read Mathematical Logic by Chiswell and Hodges. In Chapter 2 Informal Natural Deduction, the authors don’t give a definition of statement. They say to use a test. I’m confused why they use f(x) > g(x) as a statement since there are free variables. They also use examples later in the chapter that have free variables. In my discrete math class a sentence that had a free variable would not be considered a statement. For example, in this book “x is prime” would be considered a statement, but x is free so I’m thinking it wouldn’t. If this is a statement how do quantifiers come into play? I thought they helped make these sentences become statements. Can someone clarify my confusion and explain what Chiswell and Hodges are doing in their book? Thank you!

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Dr. J
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Don't get too hung up on the local usage, in particular books, of the likes of "statement", "sentence", "proposition".

Yes, when it comes to formal syntax, we want sharp definitions for "wff", "wff with the variable 'x' free", "closed wff" and the like. But in the pre-formal motivational chat before we get down to technicalities (and Chiswell and Hodges are here at this stage) it's enough to settle on some way of marking basic distinctions e.g. between declarative claims and questions and imperatives.

Still, you might say, isn't it odd to call e.g. '$x$ is prime' a statement? I'm not sure it would be my preferred usage, but it isn't that odd.

For example, I guess you wouldn't balk at calling Jack is tall a statement. Yet I haven't told you who Jack is. You just think Jack is tall is the sort of expression that could be used (in a context which fixes the referent of the name) to make a contentful assertion. And what about he is tall? Again, this is the sort of expression that could be used (in a context which fixes the referent of the pronoun) to make a contentful assertion. For some purposes, then, we might reasonably count them together -- and on the other side of the divide from the likes of "is Jack tall?", or "Give him the ball".

Well, likewise, it might be said in the spirit of Chiswell and Hodges, $x$ is prime is again the sort of expression that could be used (in a context which fixes the referent of the variable) to make a contentful assertion.

So maybe Jack is tall and '$x$ is prime' aren't so different after all, and at least for certain purposes can be co-classified.

Peter Smith
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  • I see. If we consider “x is prime” a statement in propositional logic that wouldn’t cause problems when we go to FOL? For example in this book in the chapter on FOL the authors say “x is prime” is a predicate and says $\forall x$ x is prime is a statement. But we just said x is prime is a statement. Also, in section 7.2 the authors also say “A sentence of LR(σ) is a formula of LR(σ) in which no variable has a free occurrence”. What is this meaning? Thank you! – Dr. J Jun 30 '24 at 20:44
  • It does look as if C+H may not be consistent in their usage here. I'd rather say that the predicate is "$\xi$ is prime" where "$\xi$" is a placeholder, whereas "$x$ is prime" is an open sentence with "$x$" a variable. But it is very common to elide the two in informal chat. – Peter Smith Jul 01 '24 at 16:24
  • How would the placeholder and the variable be different? To me they seem to be the same thing but I might be confusing something. Thank you! – Dr. J Jul 01 '24 at 23:48
  • Think about e.g. the two place predicate we use to express the relation of loving. We might represent the predicate "--- loves ... " Here, of course, the dashes and dots aren't part of the predicate: they are place-holders indicating where we are to insert referring our other expressions if we want to make a sentence from our predicate. Now, compare "--- loves ... " with e.g. "he loves her". The (gappy) predicate is not a sentence, while "he loves her" is. The predicate doesn't evaluate as true or false, the sentence does, relative to an assignment of referents to the pronouns. – Peter Smith Jul 02 '24 at 15:24
  • Placeholders aren't pronouns, then. Now, in more formal contexts it is conventional following Frege, to use the likes of $\xi$ as placeholders in certain contexts. While of course variables like $x$ are the correlate of pronouns. Again, the two have different roles. Hope that helps! – Peter Smith Jul 02 '24 at 15:28
  • Thank you! This helps! – Dr. J Jul 02 '24 at 15:44
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In logic, a sentence is a formula having no free occurrences of variables, like $\forall x (0 \le x)$ and $(0 < 1)$.

In the context of the discussion above, a statement is considered in contrast with imperatives, questions, etc that have no truth value, neither in principle. See statement: "something that someone says or writes officially, or an action done to express an opinion"

Unfortunately, in logic we use many related terms, not always consistently.

See John Corcoran, Sentence, Proposition, Judgment, Statement, and Fact.

The words sentence, proposition, judgment, statement, and fact are ambiguous in that logicians [and philosophers] use each of them with multiple normal meanings.

A judgment is a private epistemic event that results in a new belief and a statement is a public pragmatic event, an act of writing or speaking. Both are made by a unique person at a unique time and place.

In contrast, propositions and sentences are timeless and placeless abstractions. A proposition is an intensional entity; in some cases it is a meaning of a sentence: it is a meaning composed of concepts, a complex sense composed of simpler senses. A [declarative] sentence is a linguistic entity.

Mauro ALLEGRANZA
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