@user21820 Are Hilbert/Gentzen style the main two flavors of classical prop logic?

How would you demonstrate that a set of three propositions are inconsistent?

@user525966 I distinguish between the logic and the deductive system. There are many kinds of deductive systems for propositional/first-order logic, each of which has many variants. The main kinds are Hilbert-style, sequent-style, Fitch-style, tableaux-style.

@jo_1 What's the definition of "inconsistent"?

That the propositions cannot all be true.

@jo_1 No that's wrong. Are you sure you were taught that as the definition?

@user21820 I am not sure I understand. How are you using "logic" and how are you using "deductive system" or "variant"?

@user525966 The deductive system is the syntactic system that you use. For example the Fitch-style system I taught you and ProofMood's system are two different deductive systems for propositional logic.

I use the term "logic" for the underlying thing we intend to capture via a deductive system, which is admittedly vague, but I hope you get the idea.

Just like "computation" is a notion that can be captured by many different things such as Turing machines, Python programs and so on.

As for "variant", there is a basic idea behind each style. Fitch-style has explicit contexts and indentation. Hilbert-style has only one inference rule, namely modus ponens. Tableaux-style is always done by constructing a tree. But within each style there are many possible variants. ProofMood's system is also Fitch-style, but different from my variant.

I see

@user21820 What do you think of Tableaux style? I get confused there too because sometimes I see it referred to as a tree and sometimes I see it referred to in this way: proofwiki.org/wiki/Definition:Tableau_Proof_(Formal_Systems) (the "line by line" style we see with e.g. Hilbert style proofs)

@user525966 Tableaux-style refers to this (which seems not at all the one you linked to), which is also called tree-style. Note that ProofWiki is usually less reliable than wikipedia.

@user21820 What is the "better name" for the line-by-line style linked on proofwiki? or does it not really have a name?

No idea.