The Law of Non-Contradiction as a theorem of Dialectical Logic
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?
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?
Comments (53)
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?
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.
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.
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That is partly what I am trying to get at with this discussion.
Yep, looks good to me :smile:
No, not truth-values to the arguments. Just validity, invalidity, soundness, etc.
As in classical logic.
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?
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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:
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.
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.
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.
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.
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.
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.
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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.
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.
I think that part there is undefinable
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.
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
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.
“The mind is in a sad state when Sleep, the all-involving, cannot confine her spectres within the dim region of her sway, but suffers them to break forth, affrighting this actual life with secrets that perchance belong to a deeper one.”
? Nathaniel Hawthorne, The Birthmark and Other Stories
I don't say I'm deeper than anyone. My ideas themselves might abstractly be so though
I've tried, but nothing is working :cry:
PLEASE make it stop!
:lol:
What do you mean here by triviality? Does a theory become trivial if, with it, one can prove any and all propositions? What if every proposition is true? If not, then doesn't it mean that we don't want contradictions? And doesn't that indicate affirming the LNC? In short, that one wants theories to be non-trivial means one affirms the LNC.
Quoting Alvin Capello
That achieves the same purpose doesn't it?
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Precisely; triviality just means that every proposition is true. But I don’t want to avoid this just because I want to avoid contradictions. In fact, I don’t want to avoid contradictions because I think some of them are true. But I don’t think every proposition is true. So I will need a theory in which only some contradictions are true.
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Yes indeed. Blocking disjunctive syllogism achieves the same goal as does blocking disjunction introduction.
Well, since you don't think every proposition is true then suppose you think proposition P is false. Then ~P would be true, right? What do you then make of P & ~P? Is it true or false? Either you accept all propositions true on the basis that LNC false or you deny all propositions are true, like you've done, on the basis that LNC is true
If P is false, then ~P is true and P & ~P is false.
But if P is both true and false, then ~P is both true and false, and P & ~P is both true and false as well.
Now, while I do accept the LNC, it is not the basis for my thinking that not all propositions are true. The reason I think this is simply because it is demonstrable that not all propositions are true. For example, we can demonstrate right now that I am not currently in Indonesia. So it is false that I am currently in Indonesia.
How would you demonstrate that not all propositions are true? Anyway why did you say:
Quoting Alvin Capello
???:chin:
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In order to demonstrate that not all propositions are true, I need only demonstrate that one proposition is false, cf. my Indonesia example. It is like if someone were to say “All dogs are white.” To demonstrate that this is not true, I need only demonstrate that at least one dog is not white.
I said this because I in fact think that some contradictions are true. However, I still accept the LNC (in fact, I think the LNC is both true and false).
Of course, liar paradoxes are only contradictions if their truth is considered to be atemporal; otherwise these contradiction are avoidable using a tensed logic in which every sentence of a proof is temporally indexed according to the moment of it's creation, wherein the only distinction between premises and conclusions is that the latter is constructed after the former.
In such a tensed logic, liar paradoxes of the form P(t) => ~P(t+1) are consistent and only the simultaneous derivation P(t) and ~P(t) is inconsistent.
I'll go with your example ~D = It's false that I'm currently in Indonesia
So, you're saying ~D is true or that D is false
Consider now paraconsistent logic as a system that mustn't descend into triviality which it would be if all propositions are provable as true within it. That means you don't want paraconsistent logic to prove the opposite of ~D, which is D, to be true. Doesn't this amount to saying you don't want (D & ~D) to be true, which it would be if ~D is true (you're not in Indonesia) and D (you're in Indonesia) is also true? Isn't not wanting (D & ~D) to be true just another way of affirming the LNC? In other words the non-triviality of paraconsistent logic is dependent on affirming the LNC.
Or perhaps, contradiction only appears unresolved within logic. Reason, however, can rise above and incorporate the contradiction into a unity (like building a pyramid). Logic could be likened to a prison for the mind (or like stabilisers on a bike). Reason could be likened to a free mind. Plato acknowledged this by highlighting the danger of training philosophers to absolute truth and the "unrestricted" mind of such a person, which he labelled in context of morality, "potentially lawless". He then went on to say that people under 20 should not be given philosophical training because of their tendency to eristic behaviour for amusement. Instead he recommended people of 30 years of age be taught, for 15 years at which point they would potentially be ready to receive such wisdom, depending on how they have incorporated their knowledge to that point.
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I don't agree with your characterization of what's going on here. Surely I think that some (actually most) contradictions should not be provable. But suppose that L is the Liar Sentence. Since I think the Liar Sentence is both true and false, I want both L and ~L, and thus L & ~L, to be in my theory. So, while there are some contradictions I want to avoid (such as D & ~D), there are some that I want to include in my theory.
Also, the LNC cannot be the ground for avoiding triviality, because as I mentioned in my OP, a number of Dialectical Logics do not have the LNC as a theorem. To be sure, I do think that the LNC should be a theorem, but this is not why I want to avoid triviality.
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These are very interesting remarks. Sadly, my knowledge of dynamic logics is sorely lacking at this point in time, but I think dynamic logics at best can only have partial applications; for there are many cases where we need to use a static logic. And it is in these scenarios that the Liar Sentence arises.
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I understand this view, but I don't think anything is off-limits to logic. Of course, the ordinary logic that we learn from the textbooks is woefully limited, but if we turn to a suitable non-standard logic, then there is nowhere we can't go with it.
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This is essentially the view of Paul Kabay. If you aren't aware of him, he is a philosopher who defends Trivialism, i.e. the view that all propositions are true. While I don't agree with this view, it is very interesting indeed.
Certainly the semantic contradiction arises when the meaning of the liar sentence is analysed statically, but there is nothing that necessitates this adoption of tenseless logic in either the construction or analysis of liar sentences.
Indeed the construction of all proofs is a dynamic process over time. In the case of the liar sentence, a typical verbal explanation of the paradox involves alternatively saying "I am telling the truth about my lying, therefore I am lying about my lying, therefore i am telling the truth about my lying...etc". What is static in the construction of this paradox? Isn't the insistence that the liar sentence must be understood statically, the source of the contradiction?
Well, I'm just a novice so bear with me. The whole idea of avoiding the situation thall all propositions are true is to avoid the truth of contradictions. Why should I worry about the possibility that logically unrelated propositions could be true? There's nothing especially concerning about both the propositions "the Eiffel tower is in Paris" and "dogs have four legs" being true is there? The same applies to any logically unrelated set of propositions. Ergo, to me, it seems that the problem with all propositions are true must have something to do with logic itself, something to do with logical relations between propositions.
The logical relations between propositions I know of are:
1. Consistency & Inconsistency
2. Contradictions
3. Logical equivalence
3 isn't a problem because they don't lead to all propositions are true.
1 and 2 however are problematic because if all propositions are true then every set of, in fact all, propositions will be consistent or true
It seems then that the desire to avoid the situation where all propositions are true is to make sure that there's such a thing as inconsistency in a theory of logic, here paraconsistent logic.
The worst kind of consistency problem would result if all propositions and their negations both were true i.e. for any proposition p, both p and ~p were true. Negation would be meaningless then. Ergo, negation should flip the truth value of propositions. So if p is true then ~p should be false and vice versa. However, if the LNC is denied then (p & ~p) would be true. But, if (p & ~p) evaluates to true then consider another proposition q and ~q with opposite truth values. Then [(p & ~p) & (q & ~q)] would be true. Then, if one applies the associative property and the commutative property, which are equivalence rules, implying that the entire statement must always evaluate to true, the following combinations must also evaluate to true: (p & q), (p & ~q), (~p & q) and (~p & ~q). But these compound statements can be true only when each atomic statement is true. In other words, again, all propositions are true or there's something weird going on with either negation or the conjunction operation . Ergo, to prevent the situation that all propositions are true, we must affirm the LNC. I'm missing something aren't I?
I looked up his book. Thanks
I think you’re looking way too deeply into this. The problem with all propositions being true is that it is demonstrable that not all propositions are true. If we are using a classical logic, and we accept that the Liar Sentence is both true and false, then it follows that lions have 700 tongues. But it is empirically verifiable that lions do not have 700 tongues. So we must reject classical logic.
That is really how far the reasoning extends. No need to bring in any complicated machinery.
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These comments are fascinating. We will definitely need to discuss this further in a new thread, I am headed to bed soon, but I should have plenty of time tomorrow and over the weekend :smile:
You’re welcome. The book is amazing, but you can find the dissertation that he based his book on here. Enjoy!
Ok. Thanks.