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; — Alvin Capello
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. — Alvin Capello
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. — Alvin Capello
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.
That achieves the same purpose doesn't it?
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. — Alvin Capello
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. — Alvin Capello
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 — Alvin Capello
How would you demonstrate that not all propositions are true?
Anyway why did you say:
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
— Alvin Capello
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. — Alvin Capello
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. — Alvin Capello
Thus, we cannot reason when we come across a contradiction using these logics. — Alvin Capello
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.
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.
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.
I just want to throw this out there: maybe you reading this are where yo u think you are and in Indonesia at the same time. Maybe it's not about nothing being true, but everything being true. But you experience what you experience. It seems like all the truths should be experienced at once but it's not because if everything is true, than even your experience now is too
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. — Alvin Capello
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. — Alvin Capello
This is essentially the view of Paul Kabay. — Alvin Capello
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?
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. — Alvin Capello
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