This argument is for Sye, who posed this question in his latest reply in our ongoing debate on the Transcendental Existence of God. To continue following that debate, feel free to scroll down and put your thoughts in the "peanut gallery" section. Otherwise, take your time to read over and comment on what I say here (including Sye!), unless making sure the world you perceive actually exists isn't a question that you feel is important.
Note that this argument proves the existence of the outside world by starting from the view of solipsism and leading to a necessary logical contradiction, and additionally, since my account for the laws of logic does not itself presume the existence of an outside world, I do not ultimately beg the question with this argument.
Logical Format of the proof
1. If solipsism is true, then everything that begins to exist has a cause, and this cause must be my perceptions or by an internal faculty able to cause perceptions.
2. The first state of my perceptions began to exist.
3. Therefore, my perceptions must have a cause.
4. (From 1) This cause must be my internal faculty that is able to cause perceptions to begin to exist.
5. My internal faculty that causes perceptions must have a content.
6. This content itself began to exist.
7. Therefore, this content has a cause.
8. This cause cannot be itself and cannot be from perceptions.
9. Therefore, solipsism is false.
Discussion and Justification
One of my philosophy professors, Dr. Stewart, used to pose hilariously to his classes the scenario of killing himself to end all of existence by pointing his forefinger to his head, hand in a gun-shape, and screaming "STOP OR I'LL KILL YOU ALL!"
Throughout almost all of professional philosophy, the question of whether one can absolutely verify the existence of a world outside our consciousness has been considered open and perhaps even unanswerable for thousands of years. Indeed, an absolute proof of the impossibility of solipsism by a professional philosopher would throw the philosophical community - or, more likely, bring that professional under extreme ridicule, since many philosophers (unlike Dr. Stewart, who regards it as a thoughtful joke) enjoy hanging this over the heads of others as a verification of the old canard that "you can't prove anything!" A proof is long overdue, and since professional philosophers cannot afford the risk of giving an irrefutable one, I thought I would present one myself.
For those of you unfamiliar with the A-time theory/B-time theory split, a brief explanation is necessary here before my disproof is given. A-time theorists, to paraphrase Dr. Craig, hold that things and events aren't equally real - only the present is metaphysically true; the past is no longer the case and the future will be the case, but it is absolutely true from every vantage in the universe that they are not currently the case. Only the present is such.
B-time theorists hold that there exists a point of reference by which the past, present, and future as experienced illusory by men all are true metaphysically. They are ordered by a temporal relation "before" and "after," where "a is before b" means simply that the truth of b requires the truth of a, but that the truth of a does not require the truth of b (simultaneous occurrence would require both).
To illustrate the difference more clearly, A-time theorists simply state that the universe exists in the temporal progression of one state to another. Thus, the notion of "x begins to exist," loosely, is that x exists over a timespan of finite measure. Yesterday has happened; tomorrow will happen; today is happening. B-time theorists hold that something's temporal existence (i.e. the whole of "yesterday") is simply extended finitely, exactly like a ball is finitely extended in space, when the universe as a whole is taken under consideration. This is what I think is usually meant by referring to time as "the fourth dimension."
That we perceive the "A-theory" of time is granted by the B-time theorist, but the B-time theorist states that this is due to our role as part of the universe, and that ultimately, this is, as stated previously, an illusion, as from the perspective of the entire universe all the past, present, and future simply exists in terms of ordered extensions in a dimension of space representing temporality. Thus, nothing begins to exist, even if its existence is finite in the time-dimension; it is simply finite in time-extension and no more begins to exist by virtue of this fact than a yardstick begins to exist by virtue that it has edges.
Note that if solipsism is true, then A-time theory is absolutely true, since the fact that we perceive a progression of events means that this is exactly the case, due to no other outside point of perception being at all possible. Therefore, all of our perceptions began to exist.
Now, under A-time theory, everything that begins to exist has a cause (under B-time theory, it needen't have a cause; it just *is*). A "cause" is the action of one entity upon another entity with the "effect" being the result of this action in accordance to the nature of the interaction of these entities. The nature of a bat and the nature of a ball, for example, entail that if the ball is hit by the bat, it goes flying, all things in context being fairly granted. Since "nothing" has no properties, i.e. no identity, and thus does not exist, then everything that begins to exist in A-time theory must be caused by other things or must be caused by itself, and that action must occur before the effect or in simultaneous relation to the effect (i.e. quantum entanglement).
Given solipsism, first note that I must begin to exist, because my perceptions are in motion and because this motion has not always been occurring. Thus, I have a cause, and if solipsism is true, I must have caused myself to begin to exist. This leaves three possibilities: I caused myself to begin to exist before I began to exist, a contradiction; or, that I caused myself to begin to exist at a later state of my existence, which would entail that the effect precedes the cause, an impossibility by definition; or, most plausibly, that I caused myself to begin to exist simultaneous to the moment of my beginning.
The latter statement is all that's left for solipsism at this point, so refuting it renders solipsism literally impossible. Note first that since my perceptions themselves began to exist, they must have a cause (they can't exist as "brute fact," as people positing solipsism love to say, since A-time theory is absolutely true under solipsism and thus it is absolutely certain they must have a cause). This cause must either be due to other perceptions, or due to a creative ability inherent in myself. At the moment when I began, my first perceptions cannot have been caused by previous perceptions, or else it was not the first moment when I began. Therefore, my creative faculty must have (simultaneously with my beginning) caused them to begin to exist. Furthermore, my creative force must itself begin to exist with a content with which to create, since it would otherwise be empty (as I have not perceived at the moment of my beginning) and thus would lack any means of creative power, since there would be nothing by which it has means to create - even God has an idea of what He creates if we presume theism and creation out of nothing on that account. So, I must begin to exist with some preconceived notions of what I cause to exist at my beginning. Therefore, these notions must themselves have a cause.
Since no other outside force (evolutionary processes, a nasty god, etc.) exist to supply these notions, and since these notions cause the perceptions to begin to exist (and thus, by logical precedence, cannot caused by these perceptions), we have nothing left to supply their necessary cause. Therefore, I could not have caused myself to begin to exist simultaneously at the beginning of my existence.
Therefore, we have nothing left to supply the necessary cause of the self under the assumption of solipsism. But this is a logical contradiction.
Therefore, an outside world exists absolutely, 100%, objectively, irrefutably, and inarguably exists by the impossibility of the contrary. So quit worrying that it could possibly be "all in your head". =P
By the way, if one does not accept my view of causation, and asserts that things may still begin to exist without a cause whatever their definition of cause may be (the loosest form is that "a causes b" means "not-a entails not-b"), then the solipsism assumption renders this impossible - if, assuming solipsism, a percept A began to exist uncaused, then there is necessarily no connection between it and my consciousness. At most, A can only have the property of being coterminal with my consciousness, since if it had dependence on my consciousness, then my consciousness causes its existence continuously even if it began to exist uncaused.
But even if A simply is coterminal with my consciousness, this means that the fact that it has no necessary dependence on my consciousness entails that it must exist independently of my consciousness, refuting solipsism. So under the assumption of solipsism, everything in the content of my consciousness must be caused by my consciousness, including their first state at their beginning, entailing my consciousness caused their beginning to exist. The argument thus still follows.