Talk:Algebraic sentence

Confused

The definition makes no sense as stated. In mathematical logic, a sentence does not have any free variables. According to the definition given here, an algebraic sentence has no quantifiers and therefore not bound variables. Therefore an algebraic sentence would have no variables at all.

That's certainly possible. E.g. in the theory of the natural numbers with signature (0,1,+ *,<) the sentence (1+1)*(1+1)=(1+1)+(1+1) would be an algebraic sentence. But I doubt that this is meant.

I found some uses of this term in Google Books, but nothing really clear. It appears that some people use "algebraic sentence" as a synonym for "identity" or "equation" in the sense of universal algebra. That's very misleading, and if this is what we are discussing here it must be made clear. Hans Adler 18:09, 19 November 2009 (UTC)

$x+0=x$ is an algebraic sentence. It has one free variable, $x$. AshtonBenson (talk) 23:46, 22 November 2009 (UTC)

It is a formula, but not a sentence. In logic a sentence is always defined as a formula with no free variables. This article doesn't give a proper context and is completely unsourced. Could you please say where you have taken the definition from. I am sure you have found or learned it in some specific context. Hans Adler 23:48, 22 November 2009 (UTC)

Look, if you want to pick that nit, go head over to Propositional_calculus and fight the battle there. Once the word "sentence" has been completely removed from that page, come back here. AshtonBenson (talk) 23:50, 22 November 2009 (UTC)

Or, change "sentence" to "[sentential formula]" in this article if you like. AshtonBenson (talk) 23:52, 22 November 2009 (UTC)

It seems to have escaped your notice that all the "variables" that appear in propositional calculus are boolean variables that can only take the value true or false. It's possible in some weird and rarely discussed extensions to quantify over them. But that's pretty pointless because you can just as well emulate it with finite conjunctions or disjunctions.

When passing from propositional logic to the more general first-order logic there is a weird shift: What's called a "variable" in propositional logic becomes a "nullary relation" (a.k.a. "nullary predicate"), and many often don't even admit these. Basically, propositional logic is many-sorted first-order logic with 0 sorts, i.e. no elements at all. From this point of view it's clear why formulas in propositional logic are called "sentences" even if they contain "variables": The "variables" aren't actually variables in the usual sense.

Changing "sentence" to "sentential formula" is certainly not an improvement unless you want to write about Carnap's idiosyncratic terminology. It doesn't make much sense to mix it with anything else. But perhaps you were writing about Carnap all the time? Then you should have written it in the article. Hans Adler 00:32, 23 November 2009 (UTC)