(a→b) ↔ (¬a∨b)
¬(a→b) ↔ ¬(¬a∨b)
However (a∨b) and ¬(¬a∨b) aren't the same
So ¬(a→b) and (a∨b) aren't the same

(a→(b∧¬b)) ↔ (¬a∨(b∧¬b))
¬(a→(b∧¬b)) ↔ ¬(¬a∨(b∧¬b))
(¬a→(b∧¬b)) ↔ ¬(¬a∨(b∧¬b))
Since ¬(¬a∨(b∧¬b)) is the same as (a∨(b∧¬b))
(¬a→(b∧¬b)) ↔ (a∨(b∧¬b))

Here's another outright error.

A reductio is not truth-functional. — Leontiskos

Given a proof of B and ~B from A as assumption, we may derive ~A as conclusion — Lemmon

Or if you prefer: φ→(ψ^~ψ)⊢~φ

Or if you think it is only truth-functional if it fits in a truth-table,


At some point one has to realise that Leo has such an odd notion of logic that he is unreachable.

(It would seem that you are wrong in claiming that classical logic treats contradictions as false. — Leontiskos

Again, no.



F's all the way down.

Yep.

I gather you worked through this? Nice.

Leo does that sort of thing - claims you have said something you haven't, if it suits his purposes.

I gather you worked through this? Nice. — Banno

Yea, a→(b∧¬b) can be read as "A implies a contradiction" but ¬(a→(b∧¬b)) cannot be read as "A does not imply a contradiction", it is read instead as "not-A implies a contradiction". "A does not imply a contradiction" would rather be (a→¬(b∧¬b)). So the opposite in natural language is not the same as the opposite in logical language, in this case.

But with that in mind
¬◇(a → (b∧¬b)) entails □a
How should we read this in English? Because "{It is not possible that A implies a contradiction} entails A is necessary" is not obviously right.

Isn't it something like that "if it is not possible that A implies a contradiction, then A is necessarily true"?

Or "If in no possible world A implies a contradiction, then A is true in every possible world"?

I would guess so. But then the issue has come back, just because something can't possibly imply a contradiction, does that make it necessarily true?
Besides, ¬◇(a→(b∧¬b))↔◇(¬a→(b∧¬b)) is invalid, so the issue can't be solved like the original one was.

But I guess it can be solved in a similar way:
¬◇(a→(b∧¬b))↔□¬(a→(b∧¬b)) is valid
¬◇(a→(b∧¬b))↔□(¬a→(b∧¬b)) is also valid
Since ¬◇(a→(b∧¬b)) is the same as □(¬a→(b∧¬b)), it can be read as "It is necessary that not-A implies a contradiction". From that alone I think we can accept that it follows that necessarily A.
So, since ¬◇(a→(b∧¬b)) would be read by many as "It is not possible that A implies a contradiction", is that the same thing as "It is necessary that not-A implies a contradiction"? If not, "It is not possible that A implies a contradiction" is not a correct reading of ¬◇(a→(b∧¬b)).

So, since ¬◇(a→(b∧¬b)) would be read by many as "It is not possible that A implies a contradiction", is that the same thing as "It is necessary that not-A implies a contradiction"? — Lionino

□¬(a→(b∧¬b)) would be "It is necessarily not the case that A implies a contradiction"

□¬(a→(b∧¬b)) and □(¬a→(b∧¬b)) are the same formula
https://www.umsu.de/trees/#~8~3(a~5(b~1~3b))~4~8(~3a~5(b~1~3b))
So at least my reading is correct.
The issue with "It is necessarily not the case that A implies a contradiction" is that, if we remove the □ from □¬(a→(b∧¬b)), we end up with ¬(a→(b∧¬b)), and this can't be read as "It is not the case that a implies a contradiction".

¬◇(a → (b∧¬b)) entails □a — Lionino

Something about this is that the more general ¬◇(a ↔ (b∧¬b)) |= □a is also true. If we read ¬◇(a↔(b∧¬b)) as "It is not possible that A is False", ¬◇(a → (b∧¬b)) |= □a starts to make a bit more sense.

..we end up with ¬(a→(b∧¬b)), and this can't be read as "It is not the case that a implies a contradiction" — Lionino

Why not? I'm not seeing the issue here.

Elvis is not a man – ¬A
Elvis is a man does not imply that Elvis is both mortal and immortal – ¬(A → (B and ¬B))
Therefore Elvis is a man – A
¬A, ¬(A → (B∧ ¬B)) entails A. That doesn't make sense — Lionino

So ¬(A → (B∧ ¬B)) is the same as (¬A) → (B∧ ¬B), which may be read as "Not-A implies a contradiction", it can't read as "A does not imply a contradiction". — Lionino

Elvis is not a man – ¬A
Elvis is not a man implies that Elvis is both mortal and immortal – ¬(A → (B and ¬B))
Therefore Elvis is a man – A
¬A, ¬(A → (B∧ ¬B)) entails A, from contradiction everything follows. — Lionino

Elvis is not a man – ¬A
Elvis is a man does not imply that Elvis is both mortal and immortal – (A → ¬(B and ¬B))
These two do not entail that Elvis is a man. — Lionino

¬(a→(b∧¬b)) |= a
A not implying a contradiction does not mean that A.
So ¬(a→(b∧¬b)) can't be read as A not implying a contradiction

But (a→¬(b∧¬b)) can be read as such, and it does not entail A. Thus "A does not imply a contradiction" is (a→¬(b∧¬b)), not ¬(a→(b∧¬b))

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. — creativesoul

Same point I made earlier about alternative readings.

Hey Janus!

Evidently, normal parlance does not translate into logical notation so easily. I think there also may be a difference between "notB" and "not B". Given that no one paid much attention, I take it that my ignorance of formal logic was too obvious to mention.

:grin:

I've been reading the replies and trying to better understand, but with so little experience, and no time nor desire to practice, I'll remain an interested bystander.

It depends upon the values given to the variables. — creativesoul

Hello, creative. How are the fish hooks?

It exactly does not depend on the values given to the variables. That's kinda the point of using variables - you get to put different things in and get the same result.

So a+b = b+a regardless of what number you stick in to the formula, and a^(a→b)⊢b regardless of what statement you put in, too. Or so it is supposed to go.

Cheers. I'm similarly insufficiently tutored, so I cannot understand all the subtleties of formal logic, unless they are clearly enunciated in natural language. It seems to me, since formal logic is only an adjunct, a helpmate, to natural language, that anything that cannot be translated back into natural language such as to make intuitive logical sense, is useless (for philosophy if not tout court).

One of the problems in this thread has been that the OP was not couched in formal logical terms, and just what was meant by 'notB' was not explained.

notB is just ¬B
no ambiguity

So a+b = b+a regardless of what number you stick in to the formula, and a^(a→b)⊢b regardless of what statement you put in, too. Or so it is supposed to go. — Banno

Yeah, I get that much. As you said, that's kinda the point of using variables. I was just thinking that some statements implied a plurality of others, and hence, unless the others contradict one another, implying B and not B(C) does not imply a contradiction.

The international move has taken a year to get settled, but things are going well. Thanks for asking.

C is not B. Does that translate into notation the same way that notB does?

No because you are losing information as to what C stands for. B and notB will always be either True and False, or False and True. B and C can be any combination.

[Tones] is a pill and iinundates 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"? — Leontiskos

The hypocrisy there is astounding. The poster has written a whole lot of posts in this thread. Possibly a lot more characters I have written in this thread. Possibly more than anyone in this thread. I was away for a period while the poster had entered a lot posts, most of which are not short. When I got back, I replied to them. Only an arse would think that is not fair. And the poster, in a petty way, counts my posts, while some of them are individual for convenience. Moreover my posts have a lot of space in them due to my formatting. The poster writes stuff and I respond, often in detail. No one is a "pill' for that. In contrast with the poster's petty counting of posts, I don't begrudge the poster for filling up a lot of posting boxes. I think people should post as much as they want, explain as much as they want.

Presumably he is the only one you [Banno] believe has "explained this at length"? — Leontiskos

A mere presumption indeed. And what is the point of the remark?

I read his responses to Lionino, but many of those posts are just completely blank. He deletes what he wrote. — Leontiskos

What in the world?! The poster takes issue with the fact that it happens sometimes that one needs to delete! I deleted some posts that were only started, because those posting boxes were out of sequence when I came back to finish them. Then l finished the posts in posting boxes that were in better sequence. Wow, how petty he is!

Tones gave a translation of the latter as:
"It is not the case that if A then B & ~B
implies
A"
I still can't make sense of it. — Lionino

That might be because, for ease of clarity, the sentence needs parentheses.

"¬(A→(B∧¬B)) entails A"

In order not to conflate with 'entails' to stand for semantic entailment, I'll simplify:

~(A -> (B & ~B)) -> A

I merely used the ordinary interrelations of the symbols.

(it is not the case that (A implies (B and it is not the case that B))) implies A

or

(not (A implies (B and not B))) implies A

or

if (not (A implies (B and not B))) then A

But you still have not told me what your point is in asking me this!

What is the definition 'analogical equivocity'?
— TonesInDeepFreeze

It is the kind of equivocity present in analogical predication, where a middle term is not univocal (i.e. it is strictly speaking equivocal) but there is an analogical relation between the different senses. This is the basis for the most straightforward kind of metabasis eis allo genos. The two different senses of falsity alluded to above are an example of two senses with an analogical relation. — Leontiskos

I am still looking at references to get a grasp of those terminologies. I think I have at least a picture of the notion of analogical equivocal and analogical univocal, mostly as I find the notions in certain philosophy of religion, but I guess found more generally in philosophy also. If I am not mistaken, the main idea, in greatest generality (not specifically regarding theological concerns) is that we have different kinds of subject to apply predicates to. When the predicate applied to the subjects means the same among the subjects, then that is univocal. But when the predicate means differently among the subjects then that is equivocal. That's the best come up with so far. But there is more terminology* and I don't follow what it means as applied to logic in this thread. *Especially it would help to know the translation of "metabasis eis allo genos".

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

— javra

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? — javra

I asked because I wanted to know whether you think they are equivalent, and if not, then knowing in what ways they are different would shed light on the differences in how the law of non-contradiction is taken. And, if they are equivalent then I could use my formulation in also your context, as I prefer my formation with which we don't have to give the caveat each time we talk about the kind of statements we're studying, including the law of non-contradiction and others.

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

A and notA do not occur — javra

Is A a statement?
— TonesInDeepFreeze

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

Then my formulation does not accord with your notion, since my formulation takes the law of non-contradiction to regard statements.

if not [a statement], 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. — javra

I get you. A broad sense.

get the sense you might now ask further trivial questions devoid of any context regarding why they might be asked. — javra

They're not trivial. And I asked them to better understand your view.

Thank you for suggesting the SEP article. I am familiar with it even if I have not carefully studied all of it.

one can’t pretend to represent a contradiction in the form of a proposition and then apply the LEM — Leontiskos

Does "apply" there refer to proofs of "(A -> (B & ~B)) -> ~A" or "~(B & ~B)"? LEM is not needed to prove those.

"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. — Janus

I don't think that way, except possibly as elliptic. Especially if the context demands analysis of the logical connections, I would says something like this instead:

'If water is present then oxygen is not present' contradicts 'if water is present then oxygen is present'.

As to to whether the above is true, of course, in many everyday contexts, it is regarded as true. In other contexts it is not regarded as true.

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. — Janus

If I understand you correctly, I agree, and I touched on a similar example a while back.