## The Law of Non-Contradiction as a theorem of Dialectical Logic

• 76
A topic that has really been on my mind as of late is the question whether the Law of Noncontradiction, viz. ~(A & ~A) should be a theorem of a Dialectical Logic. For our purposes, let us define a Dialectical Logic as a logic which is Paraconsistent, contains Adjunction in rule form i.e. A, ~A / A & ~A, and which is simply inconsistent (i.e. contains at least one thesis of the form A & ~A.

Now clearly there are a range of logics which fit within this criteria, particularly Deep Relevant Logics, Paraconsistent Many-Valued Logics, and the Logics of Formal Inconsistency. The question under consideration is whether the LNC should be a theorem of a suitable Dialectical Logic? Note that the LNC is by default not a theorem of the Logics of Formal Inconsistency, while is it by default a theorem in the Paraconsistent Many Valued Logics. Deep Relevant Logics are a mixed bag; some of them contain the LNC as a theorem, while others do not.

To make things fully transparent, I should say that I am of the opinion that a decent Dialectical Logic will include the LNC as a theorem. But it is something I am still working through. But more importantly, why might someone think otherwise?

More precisely, why might someone think that the LNC should not be a theorem of Dialectical Logic? Well there is one obvious reason; namely, if we have already accepted theses of the form A & ~A, then it would seem to be the case that the LNC is already thereby rejected. Moreover, if we are interested in formalizing dialectics of the Hegelian or Meinongian varieties, then it would only seem natural to reject the LNC as a theorem, since that is what these men did.

But I have 2 overriding reasons why I think the LNC should be maintained as a theorem. Firstly, if the LNC is not a theorem, then this must mean that the negation operator under consideration is something radically different from what it is standardly taken to be; namely, a contradictory-forming operator. If the negation operator in our logic does not adhere to the contradictoriness relation familiar from the Traditional Square of Opposition, then just what exactly is it supposed to mean?

Secondly, the idea that if we have theses of the form (A & ~A) then we should reject ~(A & ~A) is based upon a Consistency Assumption. More precisely, the claim presumes that if a given proposition A is true, then it follows that ~A must be false. But this is exactly what we have rejected in formulating a Dialectical Logic. So it follows by Reductio Ad Absurdum that the original claim is false.

But I am aware that this is a complex issue and the Dialectical Logics which do not contain LNC as a theorem certainly have a lot going for them. What do you guys think?
• 4k
Law of Noncontradiction, viz. ~(A & ~A) should be a theorem of a Dialectical Logic (DL).
It is - or rather, it isn't, in DL. See Aristotle, Rhetoric, or any many modern versions such as Corbet's
https://www.amazon.com/Classical-Rhetoric-Modern-Student-4th/dp/0195115422/ref=cm_cr_arp_d_pdt_img_top?ie=UTF8
The idea is that in DL you consider both sides of a question. The sides may be contradictory in content, but not in significance-for-discussion. Examples: "Shall we attack at dawn?" or, "Shall we build ships or walls?" At dawn you shall have either attacked or not attacked, or, you shall either build a wall or build triremes. But in consideration - what DL is all about - both are legitimate occupants of the possibility space. In a sense, formal logic does not have a possibility space, because that's always about what either is, or is not.

For example, in logic: 2+2=4, 2+2=5. We already know the right answer to this one. but underlying - coming before - this knowledge of ours, is/was the "entertainment" of both as possibly correct. Which gives ground to the insight that DL is the senior to all the formal logics. A distinction that survives in the recognition that rhetoric and logic are different tool sets for different materials, and both ignorance of or confusion about this distinction has buried many a discussion.
• 76

These are interesting remarks. While I do recognize that you are describing one traditional way of characterizing Dialectical Logic, I am coming at it from another angle. I should have been clearer about this in my OP, but I am referring to DL in the specific sense of a logic in which some sentences are both True and False at the same time and in the same respect.

Now, with that being said, should the LNC be a theorem of that sort of logic?
• 4k
logic in which some sentences are both True and False at the same time and in the same respect.
Some logic, some sentences. Sample? And what of the fact that in a logic in which both sides of a contradiction can stand, anything is provable?
• 76

For instance, Graham Priest's Logic of Paradox is logic in which sentences can be both true and false at the same time and in the same respect. Priest likes to use the Liar Sentence as an example of one such, viz. "This sentence is false."

As for your second question, I specified in the OP that DL must be Paraconsistent. Paraconsistent Logics are ones in which contradictions do not imply everything. Thus, in a Paraconsistent Logic, you can have a sentence of the form A & ~A, but it will not follow that some arbitrary sentence B can be proved from this.
• 4k
in which sentences can be both true and false at the same time and in the same respect. Priest likes to use the Liar Sentence as an example of one such, viz. "This sentence is false."

But his is a bespoke understanding.
in a Paraconsistent Logic, you can have a sentence of the form A & ~A, but it will not follow that some arbitrary sentence B can be proved from this.
Presumably according to some paraconsistent rule, but how would that work? (New to me; please educate.) Perhaps the question is, if you're going to be both consistent and inconsistent, then how is that decided?
• 76

I'll show you how it works in LP. In LP, we have 3 truth-values: T, P, F. 2 of these values, i.e. T and P, are designated, while F is undesignated. T and F are the normal values of truth and falsity, while P represents a new value, viz. 'paradoxical'. Essentially, sentences with the value P are both true and false at the same time and in the same respect. In LP, an argument is considered valid if and only if there is no semantic interpretation wherein all the premises are designated an the conclusion is undesignated.

Now, let's see how negation and conjunction work in LP. If a sentence A has the value T, then ~A has the value F, and vice-versa (as in Classical Logic). But if a sentence A has the value P, then ~A also has the value P. A conjunction will be designated just so long as each of its conjuncts are designated.

Say we have a sentence A with the value P. Since ~A will also have the value P, it follows that A & ~A has the value P too. Now say we have a sentence B with the value F. This being the case, is the following argument valid in LP?:

1. A & ~A
2. Therefore, B

No. It is invalid because the premise is designated, but the conclusion is undesignated.
• 76
Perhaps the question is, if you're going to be both consistent and inconsistent, then how is that decided?

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That is partly what I am trying to get at with this discussion.
• 4k
In LP, an argument is considered valid if and only if there is no semantic interpretation wherein all the premises are designated and the conclusion is undesignated.
My origami skills are not great, but I'm pretty good at unfolding. It appears, on unfolding, that 1) any argument with designated premises and a false conclusion is not-valid in LP. How about otherwise?

Hmm. Cases. v=valid argument
1) t=>t, v
1a) f=>t, v
1b) f=>f, v

Now with p
2) p=>t, v
3) t=>p, v
4) p=>p, v
5) f=>t, p, v

This look right so far?
• 76

Yep, looks good to me :smile:
• 4k
But we all know that validity and truth are different. Can you asign truth values to the arguments? (Assuming that the premises t=true are true, f are false, and P are as you describe.)
• 76

No, not truth-values to the arguments. Just validity, invalidity, soundness, etc.

As in classical logic.
• 5.3k
I did a cursory read of the wikipedia article on paraconsistent and it seems negation has a different meaning in it than in classical logic; you mentioned this somewhere towards the end of the OP. If that's true then,

1. what does negation mean in paraconsistent logic?

2. If negation has an altogether different meaning than its meaning in classical logic then (A & ~A) in paraconsistent logic is NOT a violation of the law of noncontradiction. Equivocation?
• 4k
It seems to me then that at least by the light of regular logic that LP, is not sound, and only valid under its own rules. Yes?
• 76
1. what does negation mean in paraconsistent logic?

2. If negation has an altogether different meaning than its meaning in classical logic then (A & ~A) in paraconsistent logic is NOT a violation of the law of noncontradiction.

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It depends on the type of paraconsistent logic. Some of them understand negation to be a subcontrary-forming operator (subcontraries, recall, are pairs of propositions that cannot both be false, but can both be true). Other paraconsistent logics understand negation to be a contradictory-forming operator (contradictories are pairs of propositions that cannot both be true or both be false).

Now if we understand negation to be a subcontrary-forming operator, then I agree with you that A & ~A is not a violation of the LNC. But if we understand it as a contradictory-forming operator, then A & ~A is a violation of the LNC.

I personally do think that some sentences violate the LNC, but I accept the LNC too :smile:
• 76

I don’t think it’s correct to say that a logic is either valid or sound. To be sure, most logics do have valid arguments in them, but the logic itself is not valid.
• 4k
It seems to me then that at least by the light of regular logic that LP is not sound, and only valid under its own rules. Yes?

I don’t think it’s correct to say that a logic is either valid or sound. To be sure, most logics do have valid arguments in them, but the logic itself is not valid.

Hmm. Ok, point taken. Let me try this. There are arguments from LP, it seems, that cannot be validly made in ordinary logic. Question: Is there any reason to use LP? What would it be? What does it produce that ordinary logic cannot?

Let me add to this for clarity: I could build a machine that requires a special tool to make it work. Another mechanic might ask what use that tool had beyond its bespoke use. And the answer would be, no other use at all. Is that how it is with LP? Or do the conclusions that LP can reach find a home in the ordinary world?
• 76

Well, one good reason to use it is that LP can deal with contradictory theories. Classical and other non-paraconsistent logics cannot do this.

I don’t think LP is the right logic myself (I was just using it as an easy example), but I do think that the actual world is contradictory. So whatever system of DL is the correct one, its conclusions will find a home in the ordinary world.
• 5.3k
Other paraconsistent logics understand negation to be a contradictory-forming operator (contradictories are pairs of propositions that cannot both be true or both be false).

How is this done? The answer probably has to do with the implications of the LNC. In classical logic contradictions make proving anything possible with the aid of the disjunction introduction rule. The wiki article says paraconsistent logic doesn't allow disjunction addition which I see as a measure to prevent ex falso quodlibet. Here's what bothers me: if paraconsistent logic wants to roadblock ex falso quodlibet then the only reason, as far as I can tell, for that is to stop proving contradictions and that's exactly what the LNC is there for. It's like firing an employee only to give a job to a different person with the same job description.
• 76

The reason that I personally want to block ex falso quodlibet is because I think that some contradictions are true. Therefore, I don’t want my theories to explode into triviality.
• 76

But I should say that not all Paraconsistent Logics block disjunction introduction. Some of them block Disjunctive Syllogism. These are the ones I favor, since I don’t think DS is a valid rule of inference.
• 4k
Well, one good reason to use it is that LP can deal with contradictory theories. Classical and other non-paraconsistent logics cannot do this.

Sure they do. It's like saying a chef can't deal with a rotted fish. Of course he can; he throws it out.

How does LP deal? What does it get done, using them?
• 76

LP in particular deals with contradictions by inferring only those propositions that have interconnections with them; thereby allowing us to reason about them. Classical logic, and all other non-paraconsistent logics, infer everything from a contradiction. Thus, we cannot reason when we come across a contradiction using these logics.

If you want a recent example of how LP has been applied in metaphysics, Graham Priest in his recent book “One” has used it to formulate his Gluon Theory, which is a new way of answering the Problem of the One and the Many.
• 4k
Looks like an interesting book. But in the preface - as far as Amazon shows it - he makes clear his is an exercise in metaphysics. It can be a useful exercise, for example, to wonder whether a bicycle is a one or a many, or how. But the use of it, it's usefulness, is far removed from any metaphysical consideration of it.

Is there anything within that Plato and Aristotle did not cover?
• 76
Is there anything within that Plato and Aristotle did not cover?

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Well sure. For one thing, Plato and Aristotle never seriously considered dialectical responses to this problem. Moreover, Priest connects his Gluon Theory to interesting themes in Buddhist philosophy, such as nothingness, impermanence, etc.
• 850
I've never seen a good explanation of what a contradiction even means
• 76

A contradiction is just a true sentence with a true negation. Or, in other words, a sentence that is both true and false at the same time and in the same respect.
• 850
in the same respect.

I think that part there is undefinable
• 76

Let’s see what this means through 2 examples. Suppose we have an object that is round on the front side, but not round on the back side. We can understand that, while the object is both round and not round at the same time, it is not both round and not round in the same respect. This is because it is the front side that is round, but the back side that is not round. So this object is not contradictory.

Now suppose we have another object that is both round all over and it is not the case that it is round all over. This new object is both round and not round at the same time and in the same respect. This is because it is the entire object that is both round and not round. Therefore, this second object is truly contradictory.
• 850
Now suppose we have another object that is both round all over and it is not the case that it is round all over. This new object is both round and not round at the same time and in the same respect. This is because it is the entire object that is both round and not round. Therefore, this second object is truly contradictory.

A line segment have infinite parts while being finite. A "contradiction" can always being resolved so I'm thinking it doesn't exist and nothing is impossible
• 76

Surely some seeming contradictions can be resolved, but I don’t think this is true of all of them. For instance, I don’t think the Liar Sentence and other similar semantic paradoxes have any consistent solutions, so these are radically contradictory objects on my view.

Now as for whether nothing is impossible, I am somewhat undecided on this viewpoint; so I don’t think I can give any meaningful comments on it just yet.
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