Can a tautology break the law of non-contradiction?

Emils April 04, 2019 at 10:47 4750 views 9 comments

I know that this argument must be invalid, but I can't quite put my finger on what exactly makes it so.
My question: which line makes this argument invalid and why?

1. A _ _ _ _ _ _ _ _ _ _ _ _ _ Premise
2. A v A _ _ _ _ _ _ _ _ _ _ _1. Tautology
3. ¬A => A _ _ _ _ _ _ _ _ _ 2. Implication
4. A => ¬A _ _ _ _ _ _ _ _ _ 3. Contraposition
5. (¬A => A) & (A => ¬A) _ _3. and 4. Conjunction
6. ¬A = A _ _ _ _ _ _ _ _ _ _5. Equivalence

Surely this argument can't be valid, yet I can find no fault in the reasoning, as it seems to follow logical laws of equivalence (Q = (Q v Q); '(C v D) = (¬C => D)'; etc.). I suppose the contradiction of the law of non-contradiction arises from line 3., but I could find no specific law that would forbid this move
(since 'P => Q' truly is '¬P v Q' and vice versa). I'm positive I'm missing something silly and rudimentary here.
Your input is greatly appreciated!

Comments (9)

Nicholas Ferreira April 06, 2019 at 05:36 #273060 0 likes

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.

Banno April 06, 2019 at 06:01 #273063 0 likes

Reply to Nicholas Ferreira Good reply.

Terrapin Station April 06, 2019 at 10:54 #273100 0 likes

Quoting Nicholas Ferreira

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

(¬Q->¬P) you mean for the latter.

Owen April 06, 2019 at 13:32 #273130 0 likes

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

CaZaNOx April 06, 2019 at 16:21 #273199 0 likes

Reply to Emils 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:Quoting Emils

(C v D) = (¬C => D)

.
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)

jorndoe April 06, 2019 at 17:35 #273234 0 likes

Reply to Emils, as mentioned by Reply to Nicholas Ferreira, 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.

Nicholas Ferreira April 07, 2019 at 01:15 #273407 0 likes

Reply to Terrapin Station Oh, yeah, misstype. Thanks!

Fine Doubter September 18, 2019 at 15:02 #330424 0 likes

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!

3017amen September 18, 2019 at 16:26 #330440 0 likes

Reply to Emils

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...