What is logic?

Part of the fun of logic is realizing that there are multiple forms of it which change one's language and ability to express ideas. They also don't dominate one another—some statements' truth depends upon what logic they're formalized within.

Model theory is good here in that it studies the connection between logical language and more "physical" models. Some of these models can give a relatable account for what a logic "means". For instance, Boolean logic correspond with set operations and therefore we can use that correspondence to portray a meaning for statements in Boolean-like logics.

Intuitionistic logic is an interesting one if you're not familiar with it. It can be seen as the logic of working with "demonstrations", building and analyzing them. For instance, if I have an object "X" such that witnessing it immediately convinces someone of the truth of "1+1=2", and if I have a whole collection of such items, then intuitionistic logic talks about how to compose and decompose collections of these things. Another interesting operation is "not" such that "not X" is a tool for using any X you happen to have to show that the world is inconsistent. This acts as a refutation of the existence of values like X!

This differs from set-like logics because, for instance, "not (not X)" is not the same as X: being able to show that if you had (a way to prove the world inconsistent if you had an X) you could show the world as inconsistent... it's a far cry away from actually having an X!

Intuitionistic logic is interesting for two reasons. First, it models communication and persuasion by being an algebra of "persuasive items" like X. Second, it's "constructive" and thus works similarly to how programming languages express things through their creation.

Gödel's arithmetization of logic is one hell of a beautiful mathematical idea, up there with linear algebra or probability theory. One of its big selling points for me was how easily it solves the unexpected hanging paradox. (The simplest arithmetization of the judge's self-referential statement is a statement about numbers that can be shown to be self-contradictory.) It's also cool how it turns Russell's paradox into Gödel's theorem, Curry's paradox into Löb's theorem, etc. The connections to algorithms and computability theory are also neat (Gödel's idea of "effectively axiomatized system" is any computer program that can print sentences, which has just the right amount of generality and connects to the halting problem in the obvious way). To me arithmetization is simply the right approach to logic, which easily subsumes everything that makes sense and rejects everything that doesn't.

My take: it's a human means of systematically exploring implications of already accepted beliefs.

There are implicit and explicit forms of it.

The implicit form is something our brains do automatically and is a consequence of its structure. Perhaps the way it models things intrinsically does not allow for what we would call contradicting statements (though it contains many separate models which if unified would contain contradictions); then, the space of desirable implications is narrowed by a motivation to find a certain kind of implication (i.e. there is typically a goal when engaging in reasoning—we aren't often indifferently interested in all implications) combined with the fact that contradictions aren't allowed. It seems like implications are usually discovering that some entity belongs to a class it wasn't previously known to belong to, at which point it inherits the attributes of other things in that class.

The explicit forms seem like attempts to model the implicit form in mathematical language, in order to bring logical processes more under control of conscious thought, or 'executive function' (plus reasoning on paper extends working memory). It seems possible that with the 'correct' formulation, a logic could viewed as a theoretical science describing properties of the structure of the human brain that give rise to reasoning.

Speaking of classical logic, I came across a book on Fred Sommers' work entitled "The Old New Logic" (David Oderberg is editor). I haven't read it yet, so I can't comment, but based on some excerpts and reviews, it sounds like a good read. I thought I'd mention it here for those interested. Link: https://mitpress.mit.edu/books/old-new-logic

Also, those interested in the fallout of Russell's paradox are encouraged to look into Lesniewski's mereology (this is a topic that occurs in my master's thesis where I attempted to formalize the structure of Aristotelian living things mereologically using many-sorted logic). In short, Lesniewski held that the axiomatic set theory of the day betrayed the intuitions of naive set theory and resulted in paradoxes. He attempted to dispense with the these paradoxes by proposing a formal system known as mereology that replaced the primitive element relation in set theory with the parthood relation (other relations can be defined in terms of parthood, e.g., overlap, disjoint, proper/improper parthood depending on which one you assume, and vice versa). Link: https://plato.stanford.edu/entries/lesniewski/

"The idea that ordinary language is expressively inadequate to account for mathematical (or even logical) reasoning became a recurring theme in the ensuing tradition of mathematical logic, so much so that the term ‘symbolic logic’ became synonymous with this tradition. Doing logic came to mean simply working with special symbols, not with ordinary words. In this respect, it is worth noting that the humanist authors had criticised the Latin of scholastic logicians precisely as ‘too artificial’, and even the Greek language that Aristotle relies on for syllogistic logic is regimented and removed from ordinary ways of speaking at the time. In a sense, perhaps a certain degree of ‘artificiality’ is at the core of logic throughout history, as it operates at levels of abstraction that are at odds with ordinary language usage."

True that, but were/are there alternatives? Difficult to say.

> “It’s logical, but logic has nothing to do with reality.” Logic is the art or skill of non-contradictory identification. Logic has a single law, the Law of Identity, and its various corollaries. If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. If so, then: a. things are not what they are; b. things can be and not be at the same time, in the same respect, i.e., reality is made up of contradictions. If so, by what means did anyone discover it? By illogical means. (This last is for sure.) The purpose of that notion is crudely obvious. Its actual meaning is not: “Logic has nothing to do with reality,” but: “I, the speaker, have nothing to do with logic (or with reality).” When people use that catch phrase, they mean either: “It’s logical, but I don’t choose to be logical” or: “It’s logical, but people are not logical, they don’t think—and I intend to pander to their irrationality.”

http://aynrandlexicon.com/lexicon/logic.html

From what I know, pure logic or even math are useless without assumptions. This is what reason and science give us: a standard set of axioms from which logic and math can lead us to useful conclusions.

I've been surprised to not find anything for ctrl-F "Bayes". It turns out that traditional deductive logic is simply a special case of applying Bayes rule in order to find out the state of some binary variables given the other variables in which the probabilities are only 0 or 1. See for example these examples from Barber's "Bayesian Reasoning and Machine Learning" (p. 39):

https://i.imgur.com/W1b0Wem.png

https://i.imgur.com/JOFvd9C.png

"We just don't know"

https://www.youtube.com/watch?v=Kh0Y2hVe_bw

Interesting article.

I would add that Frege, Russell and Whitehead did not just want to logicize mathematics. They thought that reality itself followed this logic, and so were doing metaphysics. The later Wittgenstein is a decisive refutation of this idea, and this is a key reason logic has gone out of fashion.

Logic is a tool. Like all tools, they hold no value until given a purpose, then they take on the value of the task they solve.

(Just my opinion. Agree or disagree. That's cool.)

Great summary/history lesson, but the title is a bit misleading. The author spends very little time discussing the nature of logic per se.

Just a tool to embrace the unknown.

An article titled ”What is logic?” with no reference to computing? Does the author still live in the 19th century or what?

A set of general rules for not losing a connection to reality.

Could be used in reasoning or computer programming.