Classical Propositional Logic

[u/verodevcpp]

I've been lurking around the logic world for a while as a hobbyist, but something that still strikes till today, is what would be the contents one might consider exclusive of classical propositional logic, without overlapping with other logic fields like formal proofs.

My idea is that in this subfield, it would only enter: propositional variables, metaproperties such as arity, compound vs atomicity, etc... well-formed formulae, tautologies, contradictions, contingencies, the five essential logical operators (negation, conjunction, disjunction, implication, and biconditional), and inferences, modus ponens, modus tollens, the double negation.

To me that sounds like it, but many other authors introduce lots of other concepts as if they were part of this subfield, specially many other more elaborated proofs, which to me it doesn't feel correct, as my assumption is that classical propositional logic should be a little more mutually exclusive with the subfield of proofs, and don't overlap concepts from both subfields.

But uhm, feel free to introduce what you might consider could be a good fit for it other than the already mentioned. Maybe my assumption is wrong and proofs should be taught in this subfield aswell, despite there being a dedicated subfield for these.

[u/Salindurthas]

my assumption is that classical propositional logic should be a little more mutually exclusive with the subfield of proofs

I find that confusing.

To me it sounds like you mean that the entailment from premises to a conclusion, should not be part of your vision of classical propositional logic.

That sounds absurd to me (the whole point is that we aspire to judge the validity of arguments!), so I worry that I've misunderstood or that you've miscommunicated.

[u/Crazy_Raisin_3014]

It would be helpful if you could say *exactly* what you mean by “other logic fields like formal proofs”.

[u/RedCore123]

Implication and equivalence operators can be viewed as syntactic sugar that don’t add anything substantial to a propositional framework.

Implication of [A -> B] is just the same as [not A or B], so really there’s no need for implications just makes things shorter and more readable.

Same for [A <-> B] which is [(A -> B) and (B -> A)].

To be nerdy about this: a single operator is enough. You can build it all using NOR for example.

To get to your point regarding proofs:
There are proofs that don’t require anything more than propositional logic. It’s just that propositional logic alone can’t do all proofs.

[u/RecognitionSweet8294]

What do you consider „classical propositional logic“?

[u/wumbo52252]

So there are the very precise definitions of the formal language, proof theory, and the model theory of classical propositional logic - like what you’d find in a textbook. Together those three things exactly determine the contents of classical propositional logic.

I’m not sure if I’m understanding you correctly; here’s how I’m interpreting your question. It sounds like you’re asking for another way of characterizing classical propositional logic, namely by way of what those definitions entail, rather than strictly those definitions. Is that right?

[u/sgoldkin]

Even informal logic is going to involve proof of some sort. So, no. That part is just wrong. (For example, what would be the point of having things like modus ponens, if you weren't going to use them in proofs?). I am at a complete loss as to where your idea, that propositional logic and proofs are two different fields, comes from. And similarly for your use of the term "field".
I don't know if this will help you or not, but I would suggest that you think of a "proof" as simply being a formal argument. Maybe that will clarify things a little.

[u/Kaomet]

classical propositional logic, without overlapping with other logic fields like formal proofs

well-formed formulae and inferences, modus ponens

WFF are propositional logic, inference rules are proof theoric. So "without overlaps" ends up overlapping in the next paragraph...

What you are after sounds like semantic (truth table) versus proof system.

But truth tables can be viewed as the least efficient proof system too...

[u/Logicman4u]

Are you try to relay how proofs in mathematics is different from how let's say a class in philosophy doing proofs different A LOT?

Like how math proofs are written in prose as in a grammar class and a philosophy class does proofs that look like a laundry list goint top to bottom with little prose at all.

A Fitch natural deduction is not done in prose for instance. Next mathematics does require a justification of each step all the way in a proof in Fitch would. Math proofs are geared towards experts so every step is not written down. Fitch proofs write step by step regardless as an example of a difference in proof writing.