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There are a lot of philosophical arguments that are pretty short but end with a strong conclusion. All of the arguments of this type I have ever come across are more or less easily refuted, and held at best by very few philosophers. These seem to either contain a well-hidden fallacy rendering the argument invalid, or contain an unwarranted premise making the argument unsound.

A great example is St Anselm's ontological argument, which is almost universally accepted as bad (though not everyone agrees on where the argument fails exactly).

I am interested in more or less short arguments having a pretty strong conclusion that are considered by the majority of modern philosophers to be both valid and sound.

By "strong" I mean "extremely counterintuitive, challenging beliefs that seem to you absolutely correct [when looked at from the eyes of an ordinary person, not a professional philosopher]".

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    You've posed this question in a way that makes it unanswerable for most philosophers. If most philosophers accept the premises and the logic of the arguments, they should also accept their conclusions, on pain of irrationality. Therefore, most philosophers will not think the denials of these conclusions to be "absolutely correct"! – ChristopherE Jan 11 '17 at 17:08
  • @ChristopherE I agree with you. Hopefully my latest edit elaborates further on what I mean exactly. – Constantine Jan 11 '17 at 17:22
  • This is one of the most interesting questions I've encountered on StackExchange – Lothrop Stoddard Jan 13 '17 at 0:39
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There are many, many simple arguments with strong conclusions that are accepted by the majority (>50%) of philosophers, for example Euthyphro's dilemma or the Third man argument.

Then, many simple arguments with strong conclusions which are not accepted by the majority of philosophers, like Frank Jackson's knowledge argument against physicalism (is even rejected by Jackson himself) or John Searle's arguments against computationalism (not the Chinese room, the other arguments), are far from obviously fallacious and they surely do not contain a controversial premise, too.

Consensus is another question. This is notoriously difficult to achieve in philosophy, but it happens. For example, there is consensus that self-interested rational action can lead to non-optimal outcomes for everybody, as proven by the prisoner's dilemma. Does this qualify as a strong conclusion?

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    wow the greeks had it man haha – user6917 Jan 11 '17 at 15:25
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    I'd be interested how you prove that >50% of philosophers accept Euthyphro's or the 3rd man argument. I'm not saying that it's wrong that >50% of philosophers do, I'm merely saying that you stated an empirical assertion but no empirical evidence. Is there a kind of hidden survey you are automatically asked to participate in once you join the faculty of a phil department, regardless of the country/school/time you live in? – Felix Emanuel Jan 12 '17 at 19:08
  • @FelixEmanuel there is a survey, the PhilPapers survey, though they don't ask for Euthyphro's dilemma specifically here, it gives you some plausible statistics where philosophers would stand on the issues. Only 14% are theists but the majority are (56%) are moral realists, so with extreme plausibility the intersection of those groups must accept Euthyphro's dilemma as sound. And by all likelihood the atheist moral anti-realists won't believe that the existence of God would make moral realism true, so they will also accept Euthyphro's dilemma. => Both groups in sum give you >50% sure. – wolf-revo-cats Jan 12 '17 at 19:32
  • @FelixEmanuel: similarly the 3rd man argument. Yeah, really hard statistics are lacking, but the general tone of textbooks or specialized encyclopedias, who are supposed to be neutrally written, makes it a bizarre idea that the majority of philosophers don't agree with this argument. The IEP calls the 3rd man "devastating". And in the philosophical literature its seen as a decisive reductio ad absurdum if the 3rd man argument can be applied to a theory. The defenders of such a theory won't argue if the 3rd man argument is sound per se, they'll try to show that the 3rd man does not apply. – wolf-revo-cats Jan 12 '17 at 19:37
  • @FelixEmanuel: also the OP doesn't give any statistics or justification for zir claims, too, so.... I guess, I have the liberty to not supply super hard empirical evidence that withstands all doubts. And 3rd man again: in that paper the author believes he gives a strong counter-argument against mathematical structuralism just by applying the 3rd man. Why do you think that? – wolf-revo-cats Jan 12 '17 at 19:49
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One can represent the mind-body problem as an argument with a counter-intuitive conclusion.

  1. Bodies are physical things.
  2. Minds are non-physical things.
  3. Minds and bodies interact.
  4. Non-physical things and physical things cannot interact.

C. Therefore, since many of us believe all of the above sentences, but they are inconsistent, many of us have false beliefs about minds and bodies.

Many individual philosophers over several centuries have denied premises 1, 2, and 4. But all of these denials are based on controversial reasons, and none is universally accepted. So most philosophers accept the conclusion that there is a deep problem with some pretty basic beliefs most of us hold.

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