A transcendental argument is a deductive philosophical argument which takes a manifest feature of experience as granted, and articulates that which must be the case so that experience as such is possible. Transcendental arguments may have additional standards of justification that are more demanding than those of traditional deductive arguments.
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Transcendental arguments explained
Typically, a transcendental argument starts from some accepted aspect of experience, and then deduces what must be true for that type of experience to be possible. Transcendental arguments are often used as arguments against skepticism, usually about the reality of the external world or other minds.
So-called progressive transcendental arguments begin with an apparently indubitable and universally accepted statement about people's experiences of the world, and use this to make substantive knowledge-claims about the world, e.g., that it is causally and spatiotemporally related. They start with what is left at the end of the skeptics process of doubting.
Regressive transcendental arguments, on the other hand, begin at the same point as the skeptic, e.g., the fact that we have experience as of a causal and spatiotemporal world, and show that certain notions are implicit in our conceptions of such experience. Regressive transcendental arguments are more conservative in that they do not purport to make substantive ontological claims about the world.
An example is used by Kant in his refutation of idealism. Idealists believe that the experience of objects independent of our mind is not legitimate. Briefly, Kant shows that
He has not established that outer objects exist, but only that the concept of them is legitimate, contrary to idealism.
Not all use of transcendental arguments are intended to counter skepticism, however. The Dutch philosopher, Herman Dooyeweerd, used transcendental critique to establish the conditions that make a theoretical attitude of thought (not just the process of thinking, as in Kant) possible. In particular he showed that theoretical thought cannot be neutral, rather, must be based on presuppositions that are psychological in nature.
Kant
It was Immanuel Kant who gave transcendental arguments their name and notoriety. It is controversial, though, whether his own transcendental arguments should be classified as progressive or regressive.
In the Critique of Pure Reason (1781) Kant developed one of philosophy's most famous transcendental arguments in 'The Deduction of the Pure Concepts of the Understanding'. In the 'Transcendental Aesthetic', Kant used transcendental arguments to show that sensory experiences would not be possible apart from our contributing to them their spatial and temporal form.
Criticisms of transcendental arguments
As stated above, one of the main uses of transcendental arguments is to use one thing we can know, the nature of our experiences, to counter skeptics' arguments that we cannot know something or other about the nature of the world. One need not be a skeptic about those matters, however, to find transcendental arguments unpersuasive. There are a number of ways that one might deny that a given transcendental argument gives us knowledge of the world. The following responses may suit some versions and not others.