Proof, mathematics and logical fallacies.

Hi Jason!

I will just talk from the philosophical perspective, not the mathematical perspective. “What branch of logic is proof”.

          Proof is not a branch of logic, rather [a] proof is an abstract object that logicians and other philosophers frequently give (or attempt to give) in the course of defending their positions, evaluating the logical logical status of arguments (that is to say, assessing whether an argument is valid or invalid), exploring the implications of a set of propositions, and so on. This is similar to how experimentation is not a branch of science but is something that scientists undertake as part of their work.

“Are logical fallacies considered to be a proof”.

          This is an interestingly worded question. Interpreting this question at face value the best I can, the answer is no. Logical fallacies qua logical fallacies are not proofs. A fallacy is a mistake or flaw in an argument that prevents it from fulfilling its rational persuasive task.

“What is the difference between evidence and proof”.

          A wonderful question. To get started on an answer, I will try to provide you with at least a basic understanding of what a proof is, although the idea of proof is complex. A proof is an argument with certain properties. In philosophy, there are at least two notions of “proof”. There is the notion of a proof in logic , and there is the notion of a proof that extends our knowledge . A proof in logic is a sound argument—an argument that is valid and has only true premises. It is not always the case that such proofs are epistemically useful to us, however. This is because a proof in logic may not be known to be sound, or its premises may be less well-established from our perspective than the conclusion, and so on. That is to say, a proof in logic is such that although it is a proof, it might not give us knowledge relative to its conclusion. Thus, some philosophers doing work on this issue developed another notion of a proof that is understood as a proof that extends our knowledge. A proof that extends our knowledge involves a bit more than a mere proof in logic. After all, it must have further properties in virtue of which our information is increased by it, relative to its conclusion. Here is a proposal made by some philosophers regarding seven conditions necessary for an argument to be a proof that extends our knowledge.

For any argument A, A is a proof that extends our knowledge relative to conclusion C only if*:

(i) A is a proof in logic;
(ii) we know that A is valid (that its premises entail its conclusion);
(iii) we know that A is sound (that its premises are true);
(iv) for each premise P of A, we can know whether or not P is true
without having to know whether C is true (our knowledge of each of the argument’s premises is independent of our knowing whether the argument’s conclusion is true);
(v) for the conjunct of all of the premises of A (premise one and premise two and premise three, etc.) we can know whether or not that conjunct is true without having to know whether C is true (our knowledge of all of the argument’s premises together is independent of our knowing whether the argument’s conclusion is true);
(vi) for each premise P of A, our knowledge of P is better founded than our knowledge of C;
(vii) for the conjunct of all of the premises of A (premise one and premise two and premise three, etc.), our knowledge of that conjunct is better founded than our knowledge of C.

          We are still left wondering: what is evidence? I am not sure that “evidence” has a technical meaning. When I personally use the word “evidence”, I am usually referring to concrete objects rather than arguments or propositions. I would call a footprint a piece of evidence within context, and this is a physical thing out there in the world. At the end of the day, "evidence" is such a flexible and broad term. Whatever you mean by the term, Jason, just compare it to the basic outline I gave you of what a proof is. By doing so, you can get a fuller answer as to the difference between proof and evidence.

Thank you, Jason.

From, Kaiden

*Yandell, Keith E. Philosophy of Religion: a Contemporary Introduction . Routledge, 2016.

I don't think I know enough about all the branches of logic to answer your first question, but as to

Are logical fallacies considered to be a proof? 

 I can say that some are - deductive fallacies can prove an argument to be fallacious (not necessarily wrong). Consider Affirming the Consequent 

If P then Q.
Q.
Therefore, P.

If one argues

If I sleep with Trixie, then I will get herpes. I got herpes, so I must have slept with Trixie.

I would be fine with anyone saying they can logically "prove" this to be fallacious referring to the fallacy Affirming the Consequent . But remember, this isn't "proving" the conclusion false... I could have still slept with Trixie (I didn't... and for the record, I don't have herpes).