What is the philosophical equivalent of mathematical proofs?

Question

In mathematics, there seem to be five standard methods of proving or refuting an argument: a proof by induction, contradiction, counter example etc.

Are there some typical proof methods that exist in philosophy as well (in some ways analogical to what is used in pure mathematics, and and not analogical in other ways)? If yes, then what are they?

Edit: What I am trying to ask is what are the general patterns/structures used in philosophy. It is not required that the methods/structures actually resemble those used in mathematics.

Answer 1

Specifically, which branch of philosophy are we talking about here? Since you've tagged the question logic, is it reasonable to assume that you're talking strictly about formal logic?

Because in that case, the proofs look an awful lot like mathematical proofs, and use many of the same basic patterns mentioned in your question. For a more thorough introduction to each of the types and the symbolic notation, see this page. Also see this complete sample proof for a fairly simplistic problem.

But if you're talking about moral philosophy or any of the more "touchy-feely" branches of philosophy, then no. The best you get is an analytically-reasoned argument. And sometimes you don't even get that.

Answer 2

If I can show that a given position is self-contradictory then (modulo that the paraconsistent folk may well object) I have proven the position false.

In terms of positive proof for a position, then it does not seem that there is anything available quite equivalent to mathematical proof in either rigour or epistemic certainty. However, that need mark no failing of philosophy. Rather, the message might well be that different standards are appropriate to different disciplines and enterprises. (What historian ever proves anything in a fashion "equivalent to mathematical proof"?)

John Stuart Mill discusses this point in Utilitarianism in what I think is a helpful way:

[B]ut what proof is it possible to give that pleasure is good? If, then, it is asserted that there is a comprehensive formula, including all things which are in themselves good, and that whatever else is good, is not so as an end, but as a mean, the formula may be accepted or rejected, but is not a subject of what is commonly understood by proof. We are not, however, to infer that its acceptance or rejection must depend on blind impulse, or arbitrary choice. There is a larger meaning of the word proof, in which this question is as amenable to it as any other of the disputed questions of philosophy. The subject is within the cognizance of the rational faculty; and neither does that faculty deal with it solely in the way of intuition. Considerations may be presented capable of determining the intellect either to give or withhold its assent to the doctrine; and this is equivalent to proof.

Answer 3

Cody and Joe gave good answers, but I'd like to add that a common form of argument that you find in philosophy and not so much other places is a Syllogism. This is an argument of the form "P implies Q, P, therefore Q."

As an example, a simple form of the argument from marginal cases goes like:

1. There is no morally relevant difference between some non-humans and infants
2. Infants have direct moral status
3. Therefore, these non-human animals must have direct moral status

Answer 4

A philosophical analysis, work or exposition is roughly equivalent, in the sense that as a mathematical proof constructs new functions and analyzes their behavior, a philosophical work or exposition produces and experiments with new concepts.

Within a work there will generally be many inter-related speculative arguments. These arguments will generally be concerned with weighing consequences of other arguments -- evaluating the depth and rigor of the coherence between premises and conclusions. Some of these arguments will be primary and deal with key concepts or experimental conjectures; other arguments may include critical analyses of related arguments and ideas.

As @Xodarap notes, many but not all arguments will be formal or informal syllogisms. (There are many variations on the "standard" syllogism, and also a number of fallacies related to improper use of syllogistic reasoning.)

Answer 5

2 basic methods are

1. Searching for a proof (e.g. inductive proof or a constructive proof)
2. Providing a counterexample (which proves a "nonexistence" of the consequence so the assumption must be false and both proof by contradiction and proof of negation are these types of proofs)

4 basic rules of logic are ∀x (introducing true for any x) and ∃x (introducing true for an x) and the 2 corresponding elimination rules when you eliminate ∀ or ∃.

Answer 6

Try this, where at extremes, is where ends meet.

The proof of the theorem is that there is no difference in terms of conviction of belief between a religious fanatic and an atheist. They both cling to and defend their positions beyond reasonable argument to where by simply transposing “god” and “no god” the outcome is indistinguishable.