I meant 'non-contradiction', not 'contradiction'. I meant:

Do you take

"It is not the case that both water can be green and water can be not-green."

as an instance of the law of non-contradiction? — javra

I understood you the first time. The reply I gave still holds as my answer.

At least for my sake, you don't need to link me to a generator for such simple matters.

In the case that ~(A -> (B & ~B)) is true, A is true.

I asked what is your point in asking me these questions.

So when you say you claim the opposite, do you mean you claim the denial of:

"It is not the case that both water can be green and water can be not-green" is an instance of the law of non-contradiction.

If so, I don't understand. I would think that you are claiming:

"It is not the case that both water can be green and water can be not-green" is an instance of the law of non-contradiction.

Or maybe you're not seeing that the scope of 'it is not the case that' is only 'both water can be green and water can be not-green'.

On second thought, so we don't continue going around in endless circles:

E.g.: this water can be green today and blue tomorrow (not a contradiction). Or, this water here can be green and that water there can be blue (again, not a contradiction).

However, if it's affirmed that:

"water can be green and water can be non-green (e.g., blue) at the same time and in the same respect [with "in the same respect" to include its spacial location]" — javra

Then logical contradiction does result.

Again, if A and notA do not occur at the same time and in the same respect, then no contradiction occurs. Only when A and notA do are affirmed to occur at the same time and in the same respect does contradiction obtain.

.

I understand the proviso "in same time in all respects". But that proviso may be given more generally, upfront about all the statements under consideration:

(1) Caveat: We are considering only statements that are definite enough that they are unambiguous as to such things as time, aspects, etc. So we're covered in that regard.

Then we have:

(2) Law: For all statements A, it is not the case that both A and not-A.

Would (1) and (2) suffice for you as the law of non-contradiction?

if A and notA do not occur — javra

Is A a statement? If so, what do you mean for a statement to occur? If not, then what is A and what does it mean for it to occur?

Sure, but that's not really what the example is there to assert, as is clear from the rest of the paragraph. They mentioned replacing the fact about dogs 2+2 = 4 in the next line. It's "if a statement is true, then that statement is implied by any statement whatever," which is straightforwardly counter intuitive. — Count Timothy von Icarus

That's true of classical logic, and more specifically it's fragment known as intuitionistic logic, due to the fact that the respective rule of implication is essentially the logic of functions (including those that ignore their arguments to produce a constant value), rather than the logic of causality - which is described by relevance logic and linear logic.

As for "Modern Symbolic Logic", it doesn't have a well-defined meaning since it refers to a plurality of logics that are separately described in terms of different mathematical categories.

"It is not the case that both water can be green and water can be not-green" is an instance of the law of non-contradiction.
— TonesInDeepFreeze

That looks accurate to me. — Lionino

To be clear, that is not my own claim.

My question is, as we can see from the truth table I posted, (a → (b ∧ ¬b)) is False only when A is True. When we try to convert that to natural language, the result can be something that is evidently untrue — Lionino

Not untrue to me.

just because something does not imply a contradiction, it doesn't mean it is true). — Lionino

You skipped what I said about that.

j is not the case that if A then B & ~B
implies
A.
— TonesInDeepFreeze

does not enlighten me. — Lionino

I didn't intend to enlighten you. You asked me for a translation, so I gave you the most direct translation.

You talked about interpretations/models, and my truth table shows all of them — given LEM. — Lionino

I talked about interpretations so that we are clear about what we claim about these things.

Modern Symbolic Logic", it doesn't have a well-defined meaning since it refers to a plurality of logics — sime

I think the context of the paper is classical logic, or a logic that has material implication.

"if a statement is true, then that statement is implied by any statement whatever," which is straightforwardly counter intuitive.
— Count Timothy von Icarus

That's true of classical logic — sime

I explained why "if a statement is true, then that statement is implied by any statement whatever" is a misleading characterization.

given LEM — Lionino

Usually we have to have LEM to have truth tables. For example, intutionistic sentential logic cannot be evaluated with truth tables with any finite number of truth values.

Yeah, this is weird stuff. Much of it goes back to what I said earlier, "When we talk about contradiction there is a cleavage, insofar as it cannot strictly speaking be captured by logic. It is a violation of logic" ().

A contradiction is a contradiction. It is neither true nor false. It is the basis for both truth and falsity.

Thus one can’t pretend to represent a contradiction in the form of a proposition and then apply the LEM to it as if it makes sense to call a contradiction true or false. As soon as a reified contradiction is allowed to enter logical discourse, a wrinkle is introduced which can never ultimately be ironed out. It is hazardous to try to answer the OP's question with a formal proof, because formal proofs cannot represent the notion of contradiction in its entirety. Granted, natural language also cannot do this, but it is at least capable of not-trying to do this. It is capable of apophaticism.

The tokens "(b∧¬b)" and "¬(b∧¬b)" are neither true nor false if by 'true' and 'false' we are talking about something that goes beyond validity and invalidity (i.e. something which the logical system presupposes and is transcended by). The first is invalid but not false. The second is valid but not true. Too often in logic we conflate validity with truth.

But, as my second phrase in this post, I got confused as to whether ¬(a→(b∧¬b)) means a variable that is in contradiction with (a→(b∧¬b)) or simply that (a→(b∧¬b)) is False. — Lionino

Yes, exactly.

¬(a→(b∧¬b)) is only ever True (meaning A does not imply a contradiction) when A is True. But I think it might be we are putting the horse before the cart. It is not that ¬(a→(b∧¬b)) being True makes A True, but that, due to the definition of material implication, ¬(a→(b∧¬b)) can only be True if A is true. — Lionino

Right, one could approach it from two different directions. This is what I was trying to get at earlier, "I think Lionino was somehow seeing this through the overdetermination of the biconditional with respect to ¬¬A" ().

Material implication is supposed to prescind from the reason why the implication is true or false, and therefore it is supposed to prescind from causal and temporal considerations. But in this case the temporal order in which one evaluates a complicated material conditional affects its outcome,* and I can't see how this would ever happen without having that contradiction in the "interior logical flow" of the argument. Similarly, in my last post:

When are we supposed to reduce a contradiction to its functional truth value, and when are we supposed to let it remain in its proposition form? The mind can conceive of [claims] like (b∧¬b) and ¬(b∧¬b) in these two different ways, the manner in which we conceive of them results in different logical outcomes, and symbolic logic provides no way of adjudicating how the [claims] are to be conceived in any one situation. — Leontiskos

* Or perhaps it is not the temporal order, but rather the simple decision of whether to conceive of the different things as propositions or as truth values. A problem of ordering rather than temporal ordering, as it is the combinations that affect the outcome rather than the succession-order in which they are applied. If this is the nub then the problem is, as I said earlier, that truth-functional logic does not give us any instruction for how to handle contradictions. ...and again, there is no such thing as "handling contradictions," so this is to be expected. The problem is that we are trying to handle something that cannot be handled.

I understand the proviso "in same time in all respects". But that proviso may be given more generally, upfront about all the statements under consideration:

(1) Caveat: We are considering only statements that are definite enough that they are unambiguous as to such things as time, aspects, etc. So we're covered in that regard.

Then we have:

(2) Law: For all statements A, it is not the case that both A and not-A.

Would (1) and (2) suffice for you as the law of non-contradiction? — TonesInDeepFreeze

How does your newly provided caveat (1) added to your previously made statement (2) not fully equate semantically to what I initially explicitly defined the law of noncontradiction to be in full?

If (2) and the now explicitly stated (1) do fully equate semantically to what I initially stated explicitly, then you have your answer. “Yes.”

if A and notA do not occur — javra


Is A a statement? — TonesInDeepFreeze

Quite obviously not when taken in proper given context. ("if a statement both does and does not occur [...]" ???)

If not, then what is A — TonesInDeepFreeze

Anything whatsoever that can be the object of one’s awareness. For example, be this object of awareness mental (such as the concept of “rock”), physical (such as a rock), or otherwise conceived as a universal (were such to be real) that is neither specific to one’s mind or to physical reality (such as the quantities specified by “1” and “0”, as these can for example describe the number of rocks present or else addressed).

and what does it mean for it to occur? — TonesInDeepFreeze

In all cases, it minimally means for it to be that logical identity, A=A, which one is at least momentarily aware of. Ranging from anything one might specify when saying, "it occurred to me that [...]" to anything that occurs physically which one is in any way aware of.

---

I get the sense you might now ask further trivial questions devoid of any context regarding why they might be asked. I don't have as much leisure time as many others hereabout apparently have. If further questions are asked, please provide a context to your questions. I will choose to not further reply without a sufficiently meaningful context being provided.

BTW, general questions about Aristotelian notions of the principle of non-contradiction can be answered in this SEP article.

A contradiction is a contradiction. It is neither true nor false. It is the basis for both truth and falsity. — Leontiskos

This seems to be the source of your difficulties.

As has been explained, in classical logic a contradiction is false. Dialetheism considers what must be the case if some contradictions are considered to be true.The various paraconsistent logics consider what must be the case if A, ~A ⊨ B; that is, if contradictions are not explosive.

All this to say that there are various ways to treat truth values, each with its own outcomes. (the list is not meant to be exhaustive - there are other options)

Each of these systems sets out different ways of dealing with truth values. How the truth value of a contradiction is treated depends on which of these systems is in play.

Asking, as you do, how to treat the truth value of a contradiction apart from the system that sets out how a truth value is to be dealt with makes little sense. It does not make much sense to speak of "the notion of contradiction in its entirety".

The very proposition of "there both a) is a self and b) is no self" has (a) and (b) addressing the exact same thing - irrespective of how the term "self" might be defined or understood as a concept, the exact same identity is addressed — javra

The point is that if there is no determinate entity that 'the self' refers to, if there is only the concept, and if there is no actual entity, then saying that we are speaking about the same thing is incoherent. On the other hand, if you stipulate that the self is, for example, the body, then what would A be in the proposition (A implies B) where B is 'there is a self' ? Let's say that A is 'the perception of the body': this would be 'the perception of the body implies that there is a self". 'The perception of the body implies that there is no self' would then be a contradiction to that.

"the presence of water implies the presences of oxygen"

is not an "if then" statement, since 'the presence of water' and 'the presence of oxygen' are noun phrases, not propositions. — TonesInDeepFreeze

An alternative way of putting it would be 'if water then oxygen'. 'If water then no oxygen' contradicts 'if water then oxygen' according to the logic of everyday parlance.

My point earlier with taking an alternative interpretation, that is with the 'notB' not being interpreted as 'not oxygen' but rather as signifying something other than oxygen, say hydrogen, then the two statements would not contradict one another.

Each of these systems sets out different ways of dealing with truth values. How the truth value of a contradiction is treated depends on which of these systems is in play.

Asking, as you do, how to treat the truth value of a contradiction apart from the system that sets out how a truth value is to be dealt with makes little sense. It does not make much sense to speak of "the notion of contradiction in its entirety". — Banno

See, for example:

If this is the nub then the problem is, as I said earlier, that truth-functional logic does not give us any instruction for how to handle contradictions. — Leontiskos

If you think that I am speaking about, "how to treat the truth value of a contradiction apart from the system that sets out how a truth value is to be dealt with," then pray tell how a contradiction is to be dealt with in classical propositional logic...? You seem to be implying that there are correct answers to the quandaries in this thread. For example, you seem to be implying that, according to the logic, one person is right and one person is wrong when they disagree about whether a given instance of (b∧¬b) should be treated as a proposition/variable or as a simple truth value. What, then, is the right answer?

I freely admit that we could draw up additional rules to avoid the problems that are here arising. I have even proposed that we disallow "contradictions" from logical sentences. But I think the thread testifies to the fact that no such rules are generally acknowledged.

Edit:

As has been explained, in classical logic a contradiction is false. — Banno

I think the thread shows that this is not true. The problem here is that your answer lacks specificity, and contains the very ambiguity that is creating problems. Are we to consider a contradiction as if it were a variable that just happens to be false (e.g. ¬p), or are we to consider a contradiction as if it is a simple (non-complex) falsity (e.g. "If a conditional is false and its antecedent is true, then its consequent is by definition false")? The problem is that we can only be pretending to consult the truth table of a contradiction, and classical logic is premised upon the consultation of truth tables.

So far as I can see, it was you who proffered

the notion of contradiction in its entirety — Leontiskos

I'm puzzling over what this might be.

pray tell how a contradiction is to be dealt with in classical propositional logic? — Leontiskos

As has been explained at length, in classical propositional logic contradictions are false.

...you seem to be implying that, according to the logic, one person is right and one person is wrong when they disagree about whether a given instance of (b∧¬b) should be treated as a proposition/variable or as a simple truth value. — Leontiskos

Another example of your practice of misattributing stuff to your interlocutors - as you did with . What I said is that the disagreement here is as to which system is in play. Hence there is no absolute answer as to which view is "right".

As has been explained at length, in classical propositional logic contradictions are false. — Banno

I've been ignoring Tones, as he is a pill and he inundates me with an absurd number of replies (15 in just the last 24 hours). Presumably he is the only one you believe has "explained this at length"?

As has been explained, in classical logic a contradiction is false. — Banno

I think the thread shows that this is not true. The problem here is that your answer lacks specificity, and contains the very ambiguity that is creating problems. Are we to consider a contradiction as if it were a variable that just happens to be false (e.g. ¬p), or are we to consider a contradiction as if it is a simple (non-complex) falsity (e.g. "If a conditional is false and its antecedent is true, then its consequent is by definition false")? The problem is that we can only be pretending to consult the truth table of a contradiction, and classical logic is premised upon the consultation of truth tables.

Edit:

I would suggest reading my post here:

How is it that both (B∧¬B) and ¬(B∧¬B) can have the exact same effect on the antecedent, allowing us to draw ¬A? How is it that something and its negation can both be false? This is key to understanding my claim that two different senses of falsity are at play in these cases.

Do (A implies B) and (A implies not B) contradict each other?

Please give your reasoning, if you choose to answer, for why you think they do, or don't, contradict each other. And if you think they do contradict each other, does that mean they can't both be true at the same time? — flannel jesus

It depends upon the values given to the variables.

They can contradict one another at times. At other times, they can both be true. They cannot do both at the same time.

Some A's have a plurality of implications. If A implies both, B and C, then "A implies B" and "A implies not B" is better understood as "A implies B and C". C is not B.

A implies B and C. C is not B. A implies both, B and not B. No contradiction.

QED

I've been ignoring Tones... — Leontiskos

Your loss.

I think the thread shows that this is not true. — Leontiskos

Then the thread is in erorr. (p ^ ~p) is false in classical propositional logic.

The problem here is that your answer lacks specificity — Leontiskos

Not at all. A contradiction in first order predicate logic is an expression of the form (φ ^ ~φ). It is not an expression of the form ~φ. The lack of specificity here is your attempt to make use of a notion of contradiction that is not found in classical propositional logic.

How is it that something and its negation can both be false? — Leontiskos

Whether or not we affirm the negation of the consequent... — Leontiskos

Nowhere in that post do you affirm (B∧¬B).

A implies B and C. C is not B. A implies both, B and not B. No contradiction. — creativesoul

10 pages later...

Your loss. — Banno

I read his responses to Lionino, but many of those posts are just completely blank. He deletes what he wrote. His ready-made approach doesn't answer the questions that are being asked, and you and Tones are two peas in a methodological pod.

Then the thread is in erorr. (p ^ ~p) is false in classical propositional logic. — Banno

This answer proves that you do not understand the questions that are being asked. If one wants to understand what is being discussed here they will be required to set aside their ready-made answers. They will be required to examine the logic machine itself instead of just assuming that it is working.

Whether or not we affirm the negation of the consequent... — Leontiskos

Nowhere in that post do you affirm (B∧¬B). — Banno

I never said I did. Read again what you responded to. "Whether or not we affirm the negation of the consequent..."

You are telling me that (B∧¬B) is false, and that this is presumably the reason why we can draw ¬A in the first argument. But in the second argument we can draw ¬A because of ¬(B∧¬B). So I ask again: How is it that something and its negation can both [function as the second premise of a modus tollens]? By calling (B∧¬B) is false you are presumably thinking of the first argument (and therefore both arguments) as a modus tollens.*

* You are thinking of the first argument as a modus tollens enthymeme, which is how I was conceiving of it earlier in the thread as well.

Working again in the context of this post, consider its first argument:

• A→(B∧¬B)
• ∴ ¬A

Now consider the way that is interpreting this first argument (and I think this is the same way that many others tend to think about this precritically):

• A→FALSE
• ∴ ¬A

This relates to what I said earlier, namely that a logical sentence that contains a contradiction should perhaps not be considered "well-formed" (I preempt an objection <here>). FALSE is not a term in classical logic, and we have no clear understanding of how it is supposed to be used. Banno seems to be wanting to use FALSE as the second premise in a modus tollens argument in order to draw ¬A. Is that permissible? Does this new term FALSE really work as a substitute for the second premise of a modus tollens? I don't think there is a clear answer.

Now suppose we have:

• A→FALSE
• ¬FALSE
• ∴ ¬A

In this case can we also use ¬FALSE to draw the modus tollens? Is this new term that we have introduced into classical logic negatable? There is no clear answer.

Another way to read the first argument, and the one I prefer*, is as follows:

• A→ABSURD
• ∴ "A cannot be affirmed"

This is exactly what I said originally:

You think the two propositions logically imply ~A? It seems rather that what they imply is that A cannot be asserted. — Leontiskos

I am trying not to repeat what I have said elsewhere, but the equivocation between what is false and what is absurd or contradictory was pointed out earlier:

You could also put this a different way and say that while the propositions ((A→(B∧¬B)) and (B∧¬B) have truth tables, they have no meaning. They are not logically coherent in a way that goes beyond mere symbol manipulation. We have no idea what (B∧¬B) could ever be expected to mean. We just think of it, and reify it as, "false" - a kind of falsity incarnate.*

* A parallel equivocation occurs here on 'false' and 'absurd' or 'contradictory'. Usually when we say 'false' we mean, "It could be true but it's not." In this case it could never be true. It is the opposite of a tautology—an absurdity or a contradiction. — Leontiskos

Speaking in natural language, the opposite of false is true, and yet the opposite of a contradiction is not true. Or rather, 'false' and 'contradictory' are opposites of 'true' in entirely different ways.** Thus when Banno says that a contradiction (b∧¬b) is false, does he mean that it is false or that it is FALSE? It could be treated as false just as we treat ¬q as false, but in that case there is no possible modus tollens on the first argument (because in that case (p→¬q) would require a second premise that negates the consequent, i.e. ¬¬q or just q). In this case the opposite of false is true and everything carries on fine*** at least for the present moment, for as soon as we interact with (b∧¬b) again we are again susceptible to treating it as FALSE instead of false.

But if Banno instead treats (b∧¬b) as FALSE then the opposite of FALSE is ...? We don't know. This is really the opposite of ABSURD or the opposite of a real contradiction, and the opposite of such a thing is not simply 'true'.

Introducing ABSURD in the way I did above destroys the LEM of classical logic. Introducing FALSE significantly complicates the LEM of classical logic, and it is possible that it also destroys it.

* The one I prefer assuming we allow contradictions in our logical formulas, which I rather doubt that we should.

** I would want to say that the opposite of 'contradictory'/'absurd' is 'coherent' or 'consistent', not 'true'.

*** Fine except for the odd wrinkle that a cousin of an exclusive-or is produced where there would otherwise always be an inclusive-or (link).

This answer proves that you do not understand the questions that are being asked. If one wants to understand what is being discussed here they will be required to set aside their ready-made answers. — Leontiskos

Ah, so it's an esoteric mystery. :wink:

Nowhere in that post do you affirm (B∧¬B).
— Banno
I never said I did. Read again what you responded to. " — Leontiskos

The consequent is (B∧¬B)
The negation of the consequent is ~(B∧¬B)
Affirming the negation of the consequent is ⊢~(B∧¬B)
if you don't affirm the negation of the consequent, you affirm (B∧¬B).

Nowhere do you do this. Nowhere in these examples is it the case that "...something and its negation can both be false". That is, you do not show that somehow classical propositional logic affirms both
~(B∧¬B) and (B∧¬B).

Indeed, while your second example is a case of modus tollens, this is not clear for the first.

(The second is
1. A→(B∧¬B) assumption
2. ¬(B∧¬B) assumption
3. ¬A 1,2 modus tollens)

Modus Tollens tells us that "Given ψ→ω, together with ~ω, we can infer ~ψ". In the first example you do not have ~ω. It might as well be a Reductio, although even there it is incomplete. It should be something like:

1. A→(B∧¬B) assumption
2. A assumption
3. B∧¬B 1,2, conditional proof
4. ~A 2, 3 reductio

ans so A→(B∧¬B)⊢~A

_________________
And yes, A→FALSE is not well-formed in classic propositional logic. So if your first example is to be understood as using MTT, it is not an example from classic propositional logic. Again, that is not something I have supposed, and you misattribute it.

Which takes us back to what I pointed out earlier - you are mixing various logical systems. The equivocation here is on your part. Don't put the blame for your poor notation on to me.

(Edit: Actually, Open Logic builds propositional logic from, amongst other things, ⊥. (Definition 7.1). And v(⊥) = F - the valuation of ⊥ is "false" - in Definition 7.15. In this sort of build, φ → ⊥ could be well formed. @TonesInDeepFreeze might be able to clarify.)

Indeed, while your second example is a case of modus tollens, the first is not. — Banno

I am attributing the modus tollens to you because you are the one arguing for ¬A. If you are not using modus tollens to draw ¬A then how are you doing it? By reductio?

1. A→(B∧¬B) assumption
2. A assumption
3. B∧¬B 1,2, conditional proof
4. ~A 2, 3 reductio — Banno

As I noted earlier in response to Tones' reductio, a reductio is an indirect proof which is not valid in the same way that direct proofs are. You can see this by examining your conclusion. In your conclusion you rejected assumption (2) instead of assumption (1). Why did you do that? In fact it was mere whim on your part, and that is the weakness of a reductio.*

What if we reject (1) instead? Then A is made true, but it does not imply (B∧¬B). Your proof for ¬A depends on an arbitrary preference for rejecting (2) rather than (1).

Don't put the blame for your poor notation on to me. — Banno

What is at stake is meaning, not notation. To draw the modus tollens without ¬(B∧¬B) requires us to mean FALSE. You say that you are not using a modus tollens in the first argument. Fair enough: then you don't necessarily mean FALSE.

* A reductio requires special background conditions. In this case it would require the background condition that (1) is more plausible than (2).

a reductio is an indirect proof which is not valid in the same way that direct proofs are. — Leontiskos

A reductio is as much a proof in classical propositional logic as is modus tollens.

In your conclusion you reject (2) instead of (1). Why do you do that? — Leontiskos

Simply because I matched your example, which has

A→(B∧¬B)
∴ ¬A — Leontiskos

and not ~A⊢A→(B∧¬B).


Again, don't blame me for your problems.

edit: corrected A⊢A→(B∧¬B)/~A⊢A→(B∧¬B)

Simply because I matched your example — Banno

You think you get to arbitrarily reject (2) instead of (1) because I gave an example of the unaccountable inference that some in this thread are drawing? My whole point is that ¬A should not follow. What I gave is an example of the argument (claim?) that hypericin originally gave <here>.

So are you agreeing with me that the reductio does not prove ¬A?

What?

The reductio shows that A→(B∧¬B)⊢~A. As pointed out.

It could equally be used to show that A⊢~A→(B∧¬B); but that was not the issue you raised.

Edit: correction

The reductio shows that A→(B∧¬B)⊢~A. As ↪hypericin pointed out. — Banno

Again:

As I noted earlier in response to Tones' reductio, a reductio is an indirect proof which is not valid in the same way that direct proofs are. You can see this by examining your conclusion. In your conclusion you rejected assumption (2) instead of assumption (1). Why did you do that? In fact it was mere whim on your part, and that is the weakness of a reductio.*

* A reductio requires special background conditions. In this case it would require the background condition that (1) is more plausible than (2). — Leontiskos

Are you saying that if your logic professor asked you to justify an answer to my question you would tell him, "This guy on the internet set out an incomplete argument which he doesn't accept, in which the conclusion was ¬A. Therefore we must reject (2) instead of (1)"?

(i.e. "I matched [his] example" ())