There are two categories of things that can be proved in philosophy:
- That a thinking thing exists;
- The trivial truths of logic.
I'll cover these in order. In fact, there are philosophical arguments you'll find against them both. The basic idea that 'a thinking thing exists' comes to us via the Ancient Greeks but became widely known and was made popular by Descartes in his Meditations.
In this work he doubted everything he possibly could until he reached a base, the truth of which he could be absolutely certain. He thought it important to have a solid foundation to build his philosophical system on.
Archimedes used to demand just one
firm and immovable point in order to
shift the entire earth; so I too can
hope for great things if I manage to
find just one thing, however slight,
that is certain and unshakable.
(Unfortunately he very quickly lost his way and went from this solid foundation to a very questionable argument for the existence of God.)
Cogito ergo sum ("I think therefore I am") is the famous phrase from Descartes' Meditations. "I think therefore I am" is a stronger statement than "A thinking thing exists" so I have put the second forward for this answer.
It is in the class of truths that are self-evident. Thinking about it proves its truth. In philosophy we can't do physical experiments to disprove our theories so we need to rely on thought experiments instead. This is an example of a thought experiment. I can't conceive of any logically possible way of this being self-contradictory, i.e. false. By simply doing that thinking I have proved the proposition's truth.
The second class of provable things are the trivial truths of deductive logic. I'll divide this into two parts:
- The Laws of Thought - axiomatic laws that we should agree on before we can start discussing philosophy.
- The truths of propositional logic - these follow after we set up an axiomatic system.
I'll cover these in turn, very briefly. I'll leave others to tear them down.
The Laws of Thought as a collection are attributed, like so much in philosophy, to Aristotle. They are:
- The Law of Identity - an object is the same as itself - (A ≡ A).
- The Law of (Non)-Contradiction - "the same attribute cannot at the same time belong and not belong to the same subject and in the same respect"1 - ¬(P ∧ ¬P).
- The Law of the Excluded Middle - "it will not be possible to be and not to be the same thing... there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate."2 - (P ∨ ¬P).
There are arguments against each of these.
Propositional logic is a simple formal system. We define what is and isn't true via truth tables before we enter into discourse about it.
A simple truth, by definition, in propositional logic is found in logical conjunction. Here's the truth table (from wikipedia):
If both of its operands (p, q) are true the conjunction of them (p ∧ q) is also true.
1Aristotle, Metaphysics. Aristotle claimed this as the most secure and unshakable of all principles.