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Logic is at the intersection of philosophy and mathematics, and is the theory and practice of determining what must be true (a conclusion) given certain things we already accept as true (premises). By the classic example, if we accept as premises that 1. Socrates is human, and 2. If someone is human then they are mortal, it follows by the rule modus ponens that Socrates is mortal.
A logical argument is valid if it proceeds only according to the accepted rules of inference. A valid argument guarantees that if the premises are true then the conclusion is true. But a valid argument won’t necessarily produce a true conclusion unless its premises are true. A valid argument might propose as premises that 1. Socrates is human, and 2.) All humans love the music of Taylor Swift, and deduce that therefore Socrates loved Taylor Swift’s music. While valid, this argument depends upon a false premise, the second. An argument that is both valid and proceeds from true premises is sound.
Mathematician Eugenia Cheng captures the magic of logic in the first chapter of The Art of Logic In an Illogical World. Each step in a logical argument provides no new information, nothing you didn’t in a sense already know from earlier steps. And yet, such arguments—and especially the subset known as mathematical proofs—are often sources of deep insight and enlightenment. As Cheng puts it:
I am really fascinated by this mental version of an optical illusion, where you can take tiny steps that don’t seem too surprising and get somewhere that is extremely surprising and a very long way from where you started. This is how logic works. Each step you take…should really be just an unpacking of some definitions and should seem rather obvious and perhaps even trivial. But when you stick them together in succession, you can arrive somewhere that seems quite new and very far from where you started. […] Long chains of logical implications can also be virtuosic and masterful. They are how mathematics progresses, and I will later argue that I think they are an essential skill of a powerfully rational person.
For me this has echoes of mental models and the way that looking at the world in a different way can itself produce new, actionable insight, even if your raw information about the world hasn’t changed.