TonesInDeepFreeze

I think of mathematical logic sub-subject of formal logic. I guess because logic and formal logic have under philosophy, and mathematical is a part of...

Category theory centers on arrows, and as involving functions, composition of functions and morphisms and things.

Yet, often censoriously regarded as bad philosophical sense in The Philosophy Forum.

Absolutely sure. If x in 0 then x in 0. That's an instance of P -> P. I don't opine as to 'because'. I meant 'is proved by'. I find that the word 'hol...

Another friendly picking of nits: Some writers use the word 'contained'; it is not wrong. But sometimes I see people being not clear whether it means ...

The duals run all through logic and mathematics. The main result concerning propositional logic is that there is an isomorphism between propositional ...

I'm not expert either. But my understanding is that yes, category theory couches mathematics in different terms from set theory, and thus provides a d...

0 subset of 0 holds by P -> P.

Then let me nitpick that. You didn't mean 'nitpick' pejoratively, but subset vs member is not a nitpick, and the point about circularity is a good cat...

Your probability exploration is interesting. I think there's probably (pun intended) been a lot of work on it that you could find. Indeed, logic and m...

You mean it is a subset of every set (the empty set is not a member of every set). Not a correction, but a reminder: We prove that the empty set is a ...

Maybe you mean by analogy? For example: Check (1) If Churchill was English then Churchill had a stiff upper lip Churchill had a stiff upper lip theref...

More information and explanation I specified exactly what a sentential logic interpretation is. To add to that, here is what is meant by "true (or fal...

Of course, reduction ad absurdum. But how is that "checking the validity of one argument using another"?

"respectful discussion" Respect includes not intentionally or carelessly putting words in the mouth of a poster, especially after the poster has dropp...

No, that is not what I mean. In a post, I spelled out in detail what an interpretation is.

It is not valid since there are interpretations in which the premises are true but the conclusion is false.

That gives the impression that I opt for the latter more than the others. But that is not the case: I started in the thread by pointing out that the a...

I don't need to read a whole article that I've read before. I'm just wondering whether you'd give a particular example.

Good list. (1) P -> Q ... is a sentence that is interpreted as true if and only if either P is interpreted as false or Q is interpreted as true. Indee...

No, I do distinguish between what is object-language and what is meta-language. '->' is in the object language. Or if the object-language is English, ...

That is equivalent with my argument. But my argument did not mention consistency. There is a conceptual reason for that. Though, it is not incorrect t...

Yes. Wrong. (Even considering the difficulty with the definite description 'the present King of France'.)

I don't know what you mean. Example?

Correct that I didn't mention inconsistency. But "never both true" implies inconsistency. It is a theorem: If as set of sentences is not satisfiable t...

That is one way of looking at it. But we don't need to refer to inconsistency (which is syntactical) as we can also just note that semantically, there...

What a stupid thing to say. The original argument was symbolic. Of course, that could be taken as symbols meant to stand for natural language sentence...

It is a relation. It is the relation whose members are all and only those arguments that are such that there is no interpretation in which all the pre...

Explosion is related, but I didn't mention it or need to mention it for the purpose at hand. There are both semantical and syntactical versions of pri...

What in the world? There is no interpretation in which both a statement and its negation are true.

I mentioned it lately only because the matter was raised. It is taken for granted that in such contexts, the material conditional is used. But since t...

English as a meta-language regarding formal logic. In that meta-language, 'if then' is taken in the sense of the material conditional. Indeed, we coul...

That doesn't make sense and it is not how interpretations and validity work. An interpretation assigns one and only one truth value to each sentence l...

They are different but related. Anyway, yes, my point was about question begging in everyday polemical discourse. But I also contrasted it with the si...

No, they are not. But (1) is a consequence of (2).

We interpret by assigning a truth value to each sentence letter. In sentential logic, that's all there is to it. Each row of a truth table represents ...

They are different, but (1) follows from (2). Df. An argument is valid if and only if here is no interpretation in which all the premises are true and...

It is what I'm saying. The above is not the definition of 'valid argument' but it is a consequence of the definition. (1) Two equivalent definitions: ...

Thanks.

In an interpretation, a sentence is either true or false and not both, and has the same truth value no matter where it occurs in the formulas. With an...

It was an interesting idea, though.

A choice of three ways to figure it: (1) Prove it in the sentential calculus. (2) Show it as an instance of an already proven theorem schema (as @"Mic...

I don't mean that sense of 'tautology'. I mean the sense: a tautology is a sentence that is true on every row of the truth table. (If we are confined ...

And also an example of modus ponens.

What reductio? A premise is not sound or unsound. An argument is sound or unsound. Df. An argument U is sound if and only if U is valid and all the pr...

Thanks for that post. It is helpful and I need to look more into the subject.

The validity relation is: {<X Y> | X is a set of sentences & Y is a sentence & there is no interpretation in which all the members of X are true and Y...

In this instance, the use of quote marks made it look like a quote, and not just a paraphrase. And even as a paraphrase, it would be incorrect. Oh, pl...

Another one: "a major topic in the study of deductive logic is validity. This is a relationship between a set of sentences and another sentence; this ...

I ignored nothing. The bolded part is another way of saying the unbolded part: "An argument is valid if it would be contradictory (impossible) to have...