I'm curious as to how rules of inference are established. Is this an empirical act? For example if I know the premises "If I jump, then I fall" and "I jump" is it truly valid to conclude that "I fall"? If rules of inference are empirical observations is it possible that there are many unknown rules of inference?
This question in the philosophy of logic is studied as the nature of the relation of Logical Consequence (SEP Article). Broadly speaking, a construction relating a number of premises to a conclusion is a valid argument if and only if for every interpretation in which the premises hold, the conclusion/s hold. But what kinds of interpretation are we interested in? Perhaps an argument that seems logically valid to one person who only accepts a certain model of acceptable interpretations might not to someone who accepts a more narrow delineation of the possible ways things might be. (perhaps some of our core concepts are Intensional ones, which invites a much more complex connection between how things are and how things might be in our interpretive understanding)
Formal work tries to use mathematical fields like Model theory and Proof theory to help explore how we might split up our domains and our arguments in a structured way so as to help explore what a logical consequence might look like. Intuitively we do want to use certain connectives with well-established behaviour (like And, Or, Not, All, Some etc.) to help standardise some of what we do in fact appeal to in our practice of logical reasoning and inference. The mathematics of these connectives can be very informative as to what makes them tick, but this is more of a scaffolding than an explanation of what, exactly, you might mean in a given validity claim.
From Wikipedia (so take with a pinch of salt but this one seems accurate)
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true.
Greek philosophers defined a number of syllogisms, correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with the most famous of them all:
All men are mortal Socrates is a man Therefore, Socrates is mortal.
The reader can check that the premises and conclusion are true, but Logic is concerned with inference: does the truth of the conclusion follow from that of the premises?
To infer truth from an argument it must be both valid and sound. A valid argument is one where it is impossible for the premises to be true and the conclusion to be false in the above example, this is the case so the syllogism is valid and sound.
A sound argument is one where all the premises are actually true. This prevents the garbage in/garbage out effect that computer programmers will be familiar with where a valid piece of code gives an incorrect response because the variable are wrong.
Here are a couple of examples:
- All men are mortal
- Socrates is a mortal
- Therefore Socrates is a man
This is invalid but sound. All the premises are true but the conclusion is false. Socrates could be an animal or a strangely named woman.
- All men are immortal
- Socrates is a woman
- Therefore Socrates is mortal
This is unsound and invalid. At least one of the premises is false (P1) Socrates may be a woman (we only have Plato's word otherwise) and even if both premises were correct, there is nothing to infer that being a woman makes one mortal.
This is an entire field of study in itself with a lot of bearing on Mathematics. If you are interested, check out Matt dilahunty's YouTube video Foundations 1.0 on the Logical Absolutes for a beginners guide.