Well, the problem is at the inference from 3 to 4. The contrapositive of a statement has its antecedent and consequent inverted and flipped. Therefore, the contraposition of (P->Q) is (¬Q->¬P).
Step 3 states (¬A => A). (Note that i put the antecedent in italic and the consequent in bold, to facilitate). Thus, the contrapositive of (¬A => A) is (¬A => A), which is the same thing.

Good reply.

Therefore, the contraposition of (P->Q) is (¬Q->P). — Nicholas Ferreira

(¬Q->¬P) you mean for the latter.

#2, It's false to claim that (A V A) is a tautology unless A is a tautology
(A v A)<=> A, is tautologous.

I don't intend to be mean but what you are doing is inherently wrong and shows a lack of understanding of the concepts of logic.

1) A is not a premis. Instead A is a (atomic) logical formula phi. This formula contains an assignment of the binary Values True T or False F.
A specific assignment of a given Formula models a interpretationto to be True or False.
2) AvA is NOT a tautology. A tautology means that no matter what assignment we choose the Formula is ALWAYS True. An example herefore would be ¬AvA.
We can assign the Formula AvA with the value False. False or False is always False therefore the formula AvA is falsifiable. If we assign True we see the formula is also verifiable. Note: the opposit of tautology is unstatisfiable.Example A&¬A.
3) Is a reformulation of 2).
i) At this point it is worth mentioning that from a False antecedens everything follows. This is why we require the Antecedens to be True for the implication to be meaningfull. If the consequence holds True as well the implication holds.
ii) You correctly state that:

(C v D) = (¬C => D) — Emils

.
However (¬C => D) ≠ (¬A => A).
The difference is the number of assignment you can do and that there is a codependence between antecendes and consequence in (¬A => A). One could illustrate this by drawing the truthtables.
While we can make 2 assignments in the first case we can only do 1 assignment(one per letter) in the latter (creating truthtables of size 4 and 2). This leads to the situation where the implication as you use it can only result in True if we assume a False antecedens.
iii) Regarding the equality of (P => Q) = (¬PvQ).
This equivalency strongly predicates on using the assignment True for P since then ¬P is False and the final Truthvalue of the or-formula depends soley on Q. If Q is True the entire or-Formula is True if not it isn't.
However if P is False then obviously any or-formula containing ¬P is True and Q is not important. So to observe the relationship of P and Q we have to choose a method that is sensitive to P and Q. This is done by demanding that P is True and then looking at the result of the or-formula.
4) Is already explained. Basically it doesn't make sense to formulate a implication and/or a contraposition for a formula with only one truth assignment. Thats why after applying the contraposition you are at the same place you where in 3)
5) Translates to (AvA)&(¬Av¬A) which is unstatisfiable meaning no interpretation exists under which this would be true.
6) The result of 5) translates into 6)

, as mentioned by , you can't get A ⇒ ¬A

3. ¬A ⇒ A
4. ¬A ⇒ ¬¬A (contraposition of 3)
5. ¬A ⇒ A (double negation elimination)

so your 4 is wrong.

Oh, yeah, misstype. Thanks!

As I am a beginner, and don't know how to translate the notations into real world examples, and as I do much of my thinking visually, please can you give some examples of this "in the round"?

I think tautology in the Wittgensteinian sense is a source of much knowledge though active mental contemplation, which is too much neglected by the public. I am glad to see there are indeed equations for it!

To answer the original concern, are you referring to the Liar's Paradox (propositions of self-reference)?

I'm not exactly sure, but I think those would break the rules...