Also, if an opinion is a belief, then an opinion can neither be true nor false. This is because people can believe something without sufficient evidence.

Thanks for the question! There are a few ways I can interpret your question. I'll try to answer every interpretation, but if I miss the mark, just let me know, and I'll be happy to give it another shot. First, you might be asking (1) whether there's anything wrong with mapping propositions to truth values in this way, and (2) given that there isn't, why this wouldn't suffice for truth valuation. As far as I can see, you can definitely map propositions to truth values. So, set theory gives you the tools to take the set of all propositions and define some function such that, for every proposition, it gets paired with one truth value or another. However, this wouldn't mean that set theory is evaluating these propositions. Consider: you could just as easily create mappings that map onto errant truth values, or anything else at all. There's nothing about the tools of set theory that would provide any guidance on which mapping is correct - so, set theory remains silent on this issue. Second, you might be asking what, exactly, it is that deductive systems have that set theory lacks that allows for truth evaluation. Technically, no formal system evaluates for a semantic notion of truth. However, deductive systems like first order logic contain truth functional connectives (negation, conjunction, disjunction, the conditional, and the biconditional (or other fun ones like the sheffer stroke)) that are defined in the semantically rich metalanguage that creates the system. Unlike in set theory, these connectives have their truth mappings built in - they evaluate for truth in the following impoverished sense: they provide a definite set of truth-value outputs for truth-value inputs. We then hope that these artificial connectives mirror the natural language ones closely enough to tell us something useful. I think you're trying to create something like this in your example, though the mapping wouldn't so much compute truth values as just have each proposition assigned a truth value from the outset. Finally, you might be asking a more general question about the role of truth in these formal systems. The truth is, truth plays no role in formal systems. By 'truth', I mean the semantically rich notion of truth - the one that means that some representation of a state of affairs (e.g., a proposition) actually obtains. In deductive systems, "T" and "F" are just formal symbols, we could have them stand for anything. The rules then tell the system how to manipulate those symbols to get new ones. Set theory is a little different in that the members of its sets are objects. But set theory does not contain truth functions - it contains only membership relations. I hope this helps!

@@philologick6175 that was INCREDIBLY HELPFUL KIND SOUL! May My feeble mind ask one more question as I trek thru intro to logic: I’ve noticed a trend that depending on the source, some say these are near equivalent, some say they are fundamentally different, but can you give me an indisputed so to speak definition of each because it’s confusing the hell out of me: “interpretation” “structure” “semantics” “model” “theory”. Thanks so much!!

Thanks for the comment! Unfortunately, this inference would be invalid for E- and I-type statements as well. This can be proven through the use of Venn diagrams (which I hope to make a video about in the future). For now, though, we can stick to coming up with counterexamples. Let's say, for "No A are B," that A stands for "dogs" and B for "cats" such that the statement is "No dogs are cats." The statement "No nondogs are noncats" wouldn't follow. This can be tricky to see because of the complements, but I think it's a bit clearer if we rephrase it as such: "There are no things that are not dogs that are also things that are not cats." But there are plenty of such things. For instance, my washing machine is a nondog that is a noncat. The "no nondogs" bit can't be double negated because the "no" just serves as a universal quantifier indicating the relationship between both categories - it isn't serving to negate the complement. As for I-type statements, this one threw me for a loop! That's because I found it impossible to think of any categories for which "Some non-A are non-B" would be false. There might be an example that I'm just not creative enough to think of. But even here we can prove with the use of Venn diagrams that the inference would be invalid. Even without, if inversion is defined as just swapping each term with its complement, then it should be equally possible to get from "Some non-A are non-B" to "Some A are B," and here we can easily find counterexamples. Consider: "Some nonparrots are nontrees." This is true, some things that aren't parrots are things that aren't trees. If we grab each term's respective complement, we get "Some parrots are trees," which serves as a counterexample.

Thank you! I'm happy to hear that I was able to help.

I would recommend watching the whole video.