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 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 don't think, actually, that being "smart" is even definable either

I should mention as a point of clarification that you don’t need to believe that there are true contradictions in order to use a Paraconsistent or even a Dialectical Logic. Indeed, if you know that a certain philosophical theory is inconsistent, then it would be most reasonable to use a Dialectical Logic when analyzing it.

Please make that cat stop crying.

I've tried, but nothing is working :cry:

PLEASE make it stop!

:lol:

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

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.

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

That achieves the same purpose doesn't it?

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.

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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.

That achieves the same purpose doesn't it?

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Yes indeed. Blocking disjunctive syllogism achieves the same goal as does blocking disjunction introduction.

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

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.

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

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

???:chin:

How would you demonstrate that not all propositions are true?

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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.

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

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).

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

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

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.

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

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.

Thus, we cannot reason when we come across a contradiction using these logics. — Alvin Capello

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.

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.

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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.

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.

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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.

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.

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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.

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

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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.

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

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?

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

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?

This is essentially the view of Paul Kabay. — Alvin Capello

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.

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?

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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!

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

Ok. Thanks.