according to Cary

Transcend means to be or go beyond.
Transcendence is a property of something that transcends.
Transcendent means transcending, or describes something that transcends or transcended.

Unfortunately none of this has anything to do with transcendental, which describes a necessarily pre-existing condition to something.  When you Google the term, this can cause confusion, for which we have Kant to thank.  Kant [mis]named the concept, and the world is stuck with his terminology.

The term is only used in the abstract sense, but some concrete illustrations may be helpful.  In the context of existence:  Bricks are transcendental to a brick house.    But no particular bricks are transcendental to any brick house, because any brick house could have been built with other bricks.  Your parents are transcendental to you.  In the context of motion, moving 1 meter is transcendental to moving 2 meters.

Getting more abstract:  Geometric axioms are transcendental to geometry.  Any axiom is transcendental to anything based on it.

Getting totally abstract:  If A is transcendental to B, then A must exist in order for B to exist.  If A and B are propositions, then A must be true in order for B to be true.

There are levels of transcendentality:  e.g.  The laws of logic are transcendental to the laws of physics, which are transcendental to matter, which is transcendental to bricks.

Premises are transcendental to arguments.  A non-transcendental argument might say, "A = B because of this evidence."
A TA might say, "A = B because if A did not = B, an impossible or inconceivable state of affairs would be the case."
I define a TA as one which does that, and assert this definition to be clearer than the Wikipedia definition.

There are levels of transcendental arguments.  On the first level, a transcendental argument is one that argues from prior necessity rather than posterior evidence.  e.g.  An axiom can't be proven, but must be assumed in order to prove anything based on it.  Therefore an axiom is transcendental to any premise or argument based on it.

On a higher level, a transcendental argument cannot argue from anything, because the transcendental is necessarily prior to argument itself.  e.g.  Objective truth is transcendental to any arument about what is true.  e.g.  "X is true because if it were not true, we wouldn't know true from false." It may go so far as to say, "X is true because if it were not true, there would be no such thing as truth."  Taken to its extreme, a transcendental argument self stultifies, because there is ultimately no such thing as a transcendental argument - only transcendental inference.

In any case, a TA is the height of question begging chutzpah, unless it is followed with, "And here's why..."

A TAG argues that a personal Supreme Being is a necessary precondition for something the arguer or the arguer's opponent is not in a position to deny.  It is not an argument for the Christian version of that Being, as many Christians try to sell it.  TAG is poorly understood, as evidenced by its much contested Wikipedia article, which when last updated by its proponents is either incoherent, or appears so because of its complexity.  When updated by its detractors, it becomes an obviously begged question.  Therefore there are many versions (and again levels) of TAG.  Here are some arbitrarily chosen versions on 4 levels, which I find more intellectually satisfying than the following atheist objections, which I've never been able to refute to their satisfaction:

Here's a TAG based on the prior assumption of objective morality paraphrased from Joseph Dowd:

  1.   I can't help but believe that some things are objectively right and wrong.
  2.   This belief doesn't make sense unless there's an objective standard of morality.
  3.   The only possible objective standard of morality is a Supreme Being.
  4.   Therefore, I need to accept the premise that a SB exists (whether or not I strongly believe—or believe at all—that a SB exists).

Atheist objection:  Then a TA can be used to argue for virtually anything.  e.g.  Someone can't help but believe that their dead grandmother is in a better place.  Therefore they conclude that a heaven must exist.

Agreed, but a TA merely asks people to be logically consistent:  If you believe X, and X implies Y, then if you don't accept Y, you're not logically consistent.

The next is just an extension of the Cosmological Argument:

All systems require something outside of themselves to get them started.  Therefore something eternal must have started the first created system.  I agree with this because I don't know of any exceptions.  That's an argument from ignorance, but it's not a deductive fallacy, unless I claimed to know all systems require something outside of themselves.  Deductive fallacies disprove deductive syllogisms, not probability judgments.

Atheist objection:  An eternal multiverse may be an exception.

I can't prove otherwise, so it's reduced to subjective probability judgment.

The one I learned from Greg Bahnsen:  (And these are his words, not mine.)

Without the existence of God it's impossible to prove anything, because in the atheistic world, you cannot justify (account for) laws in general, laws of thought in particular, laws of nature, the human mind, and the fact that it's more than electrochemical complexes.

Atheist objection:  Laws are just the way things are, and they no more need accounting for than a Supreme Being needs accounting for.  Why is God theory any more sound than "just-the-way-things-are" theory?

"Just-the-way-things-are" is just another way of saying there is no accounting for it, which is what Bahnsen said.

Atheist objection:  Then account for the existence of God !

I can't, but if the atheist view entails that there is no accounting for things, then any accounting for things, must be theistic.  If only one possibility can account for the existence of something that obviously exists, that possibility is necessarily true, even if its own existence cannot be accounted for.

Bahnsen's version expanded on:

In order for knowledge to exist as something that accurately distinguishes true from false, something must connect brain cells and their activity to the concept of true vs. false.  That thing must be in a position to know true and false without reliance on something above itself.  A personal Supreme Being is the only possible candidate for that position.  So if I claim to know anything at all, I automatically claim to know that all necessary prerequisites of knowledge exist - including a personal Supreme Being.

Atheist objection:  What we call knowledge may be nothing more than an illusion that facilitates the survival and procreation of organisms that believe it.  In order to facilitate survival & procreation, the illusion must have some degree of correspondence to reality.  Approximate truth recognition is all we need in order to say we know stuff.  Once we learn logic, then logic compels us (if we are rational) to admit we "know" other things that do not facilitate survival and procreation.  But it all averages out such that our species continues to survive and procreate.
Also the above argument only works if one assumes an eternally omniscient Supreme Being.  Otherwise the SB may learn at least some things.  If so, then human minds may learn all of the things they "know" by the same means.  And there is no reason to assume the SB is necessarily omniscient.

Again, I'm only saying the above TAG arguments are sufficient to convince me.  None of them objectively refutes, or is refuted by, its atheist rebuttal.  It's just a matter of which is subjectively easier to believe.  I prefer the TAG arguments, because they appear more reasonable than "shit happens", which is all I'm left with otherwise.  And if I say I know something, I mean I'm sufficiently convinced of its truth to assert it dogmatically (on whatever level of knowledge I'm talking about), not that I like it because it increases my chances of survival and/or procreation.

There is another version by a person who prefers to remain anonymous:

There are no levels or versions.  There is only TAG and erroneous attempts to state TAG by people who don't understand it.  There is no transcendental argument, but only transcendental inference.  That statement, however, does not self-stultify.  TAG exists, but cannot be stated in a coherent paragraph or even an essay.  It must be absorbed gradually.  Nevertheless, a few things can be said:  If rationality exists, then all things that must be true of the world for rationality to exist must also be true.  Its not that I have a rational argument for the existence of God, but that there is no non-theistic account for rationality.  If the world is intelligible, all the preconditions of its intelligibility are true - but we don't know if it's intelligible.

Atheist objection:  WTF ?! ?! ?!

Yeah, I know, but this guy appears to understand it better than anyone else I'm aware of.

Afterthought:  It is my experience that in general, atheists are more intellectually honest than theists, but less rational than those Christians who understand TAG.  But TAG has never been clearly stated, because when clearly stated, its limitations become clear.  Christians assert that it proves a Supreme Being, which it doesn't;  it just shows a SB to be highly probable - in my subjective judgment of probability.  Christians have also asserted that it proves Christianity, which it doesn't even begin to do.  So, they tried to patch up TAG by proposing TACT (Transcendental Argument for Christian Theism).  But TACT has never been more than a bogus wannabe, and easily refuted.