Types of Logic

Perhaps surprisingly, there is not one kind of logic but many different kinds.Of all the things people disagree on, you'd thought that we at least all agree on this one fundamental thing. Hell, no. There are so many kinds of logic that the sheer attempt to enumerate them all, from those split by substantial factors, to the ones with all their tiny variances, would deplate the little interest you're cultivating to learn logic.

The major dividing factor that split the logic into the two most popular main-stream branches today, was the movement in mathematics called constructivism. Constructivist demand proofs as well, but they require having a witness - at least one instance, one element to bear witness to the mathematical proof in question. Constructivism bore intuitionistic logic, which has discarded the laws of logic that don't produce a witness, primarily, it doesn't recognize the law of excluded middle which is vital to classical logic. Apart from the things the intuitionistic logic excludes, it remains compatible with classical logic - it is its subset.

Classical logic (CL), like any logic, deals with declarative statements to which you can assign a truth value. You assign a truth value to a statement and then you make your argument as to why it should be so. However, it seems that CL takes a shortcut of sorts, because it equates true (proposition) and the not false (and vice versa). This also entails the corollary that the truth twice nagated still comes out as the truth, i.e. not not true equals true again (double negation is best read as: "it is not the case that...").

The law of excluded middle (LEM) states that a proposition is either true or false, not both and not neither. Informally, it means there's no third option, no alternative, nothing but the whole truth or the complete contradiction.

Both intuitionistic and classical logic are bivalent, having two truth values, however classical logic insists on upholding the LEM. Of course, adhering to a principle as if it were written in stone only inspires people to come up with an example (a witness!) that CL couldn't deal with. The canonical example, called the Liars Paradox, that stabs this stern position in the groin is the statement: "This proposition is false". Although being a declarative sentence, no truth value could be assigned to this statement without yielding a paradox.

Another factor that has produced many divisions of logic, is the number of truth values. There are logics with 3 or more truth values, but there are also logics where the notion of truth values is a lot more granual; for example, fuzzy logic has the truth values spread in the 0-1 interval.

https://en.wikipedia.org/wiki/Category:Systems_of_formal_logic

A complete assignment of truth values to each variable at each world is valuation.

A logical system is essentially a way of mechanically listing all the logical truths of some part of logic by means of the application of inference rules recursively (i.e. rules are repeatedly applied to their own output). All of the axioms must be logical truths, and the rules of inference must preserve logical truth.