'tis a think of beauty, but cannot be used to show truth tables within posts in TPF.

Other methods are clumsy.

- Yes - I wasn't sure, but it fortuitously solved my conundrum as well.

- See

Then give your proof. — Leontiskos

Are you serious? You don't know how to prove it yourself?

Proof:

(1) (A -> B) ... premise

(2) (A -> ~B) ... premise

(3) A ... toward a contradiction

(4) B ... from (1), (3) modus ponies

(5) ~B ... from (2), (3) modus ponens

(6) ~A ... from (4), (5) and discharge (3) by contradiction

Truth table:

A B......A->B......A->~B......~A
T T..........T............ F..............F
T F..........F.............T..............F
F T..........T.............T.............T
F F..........T.............T.............T

All rows in which both A->B and A->~B are true are rows in which ~A is true.

/

Do I get anything for doing your homework for you? Cash? Philosophy Forums poker chips? Internet Brownie points?

A -> B. But that's not imply. that's "Necessarily leads to."
— Philosophim

Wrong. Material implication does not require necessity. — TonesInDeepFreeze

You misunderstand. A -> B is not implication as we may use it in English. In other words, "It being cloudy implies that it will rain soon," is not the same as "It being cloudy necessarily means that it will rain soon. Yes, in logic, its called an "implication", but in standard English its equivalent to "Necessitates". That notation is If A, then B. Or if A, necessarily B. Its not, If A, maybe B.

Thus, if one plugs the word 'necessitate' in, its clear that A necessitates B and A necessitates not B contradict each other.

What I see some people doing is declaring A -> A or B, which is perfectly fine. But that is not the same as stating A -> B, (which means it will only lead to B) then in the next statement A -> !B (A will lead only to not B).

On the other hand, Tones says that both propositions imply ¬A. That is true if "both props" is understood as (A → B) ^ (A → ¬B) and "imply ¬A" as the proposition being True means A is False. That much is accurate. But taking (A → B) and (A → ¬B) individually, neither need A to be False for them to be True (see: my logic tables in first page).

Edit: we posted at the same time.

Again, a tool not unlike the tree proof generator, that produced a png or other image that could be inserted into a post, would be most helpful.

but cannot be used to show truth tables within posts in TPF. — Banno

Well I screenshot, press control v on imgur.com, then copy image url address and use it on "image" button

and "imply ¬A" as the proposition being True means A is False — Lionino

Yes, this was my concern. Tones requires the assumption, as I thought he must.

Yep. A workable solution, but a bit convolute for my taste.

You misunderstand. — Philosophim

I replied exactly to what you wrote. What you wrote is wrong.

A -> B

or

material implication

is not "A necessarily implies B".

Tones says that both propositions imply ¬A. — Lionino

Both propositions together imply ~A. The conjunction of the propositions implies ~A. The premise set {A->B, A->~B} implies ~A.

That is true if "both props" is understood as (A → B) ^ (A → ¬B — Lionino

Right.

https://people.cs.rutgers.edu/~elgammal/classes/cs205/implication.pdf

"Different forms for implications
p implies q is equivalent to
If p then q
q if p
q whenever p
q when p
q follows from p
p is sufficient for q
a sufficient condition for q is p
q is necessary for p
a necessary condition for p is q"

So what?

'->' is ordinarily regarded as standing for material implication that is not "P necessarily implies Q" nor "P necessarily leads to Q".

So what?

'->' is ordinarily regarded as standing for material implication that does not require necessity. — TonesInDeepFreeze

Please demonstrate your claim.

Look in any textbook on symbolic logic.

↪Philosophim

Look in any textbook on symbolic logic. — TonesInDeepFreeze

If you're not going to bother answering like I did, I'm going to hand wave you away. Don't be a troll. Show it.

Are you serious? You don't know how to prove it yourself? — TonesInDeepFreeze

Rather, I'm interested in you doing something more than making curt pronouncements from on high. This is a philosophy forum, after all.

Here is the alternative notion of contradiction that you are overlooking:

“opposite assertions cannot be true at the same time” (Metaph IV 6 1011b13–20) — Aristotle on Non-contradiction | SEP

Wiki might suffice to show you the difference between material implication and strict implication. That might be what you have in mind.

Tones is correct.

It is not trolling to point out an incorrect statement, and it not trolling nor handwaving to suggest that one can look in textbooks to see that the statement is incorrect.

Or look in any resource you like to see that '->' in logic is ordinarily understood as the material conditional and any textbook in symbolic logic will explicitly state that '->' is the sentential connective such that 'P -> Q' is interpreted as true if and only 'P' is interpreted as false or 'Q' is interpreted as true. And that is material implication.

Philosophim must be talking about conditions.

If A (being True) is necessary for B (to be True), B→A. A is a necessary condition of B.
If A is sufficient for B, A→B. A is a sufficient condition of B.
Necessary and sufficient: A←→B. A is a necessary and sufficient condition of B, so A and B are in constant conjunction (queue: Hume).

Another confusion that stems from 'p'/'a' meaning "[given proposition] is True" or meaning a variable that may take values 0/False or 1/True.

curt — Leontiskos

Actually, your imperative "Then give your proof" was curt, especially as spoken to someone giving you correct information.

The proof is so simple that one would think that someone posting claims about the matter would know the proof or would just look it up, or even run it in their mind for five seconds, rather than a challenging "Then give your proof". People don't need to give such utterly basic proofs. It's like saying, "Then give your proof that 1+1=2". Moreover, as good as proofs had already been given in this thread.

Run it in your mind for five seconds: If A implies both B and ~B, then A implies a contradiction, so we infer that A is not the case. You really need to challenge people to prove that for you?

opposite assertions cannot be true at the same time — Aristotle on Non-contradiction | SEP

That is the law of non-contradiction. What I said is a more formal way of saying the same.

Yes, we say 'necessary' and 'sufficient' conditions. But that is not "necessarily implies' or 'necessarily leads to'.

If P -> Q, then P is a sufficient condition for Q and Q is a necessary condition for P, and that is not to suggest any modal operator. But we don't say that P necessarily implies Q.

But that is not "necessarily implies' or 'necessarily leads to'. — TonesInDeepFreeze

Yes, the first phrase means □(p→q), the second means nothing to my knowledge.

That would be my leaning too.

That is the law of non-contradiction. What I said is a more formal way of saying the same. — TonesInDeepFreeze

The first page of the thread contains two basic ways of defining contradictions. You gave a third: mere negation and the attendant inverted truth table. That's a legitimate extrapolation, but still different from a non-formal assessment of contradiction.

and "imply ¬A" as the proposition being True means A is False
— Lionino

Yes, this was my concern. Tones requires the assumption, as I thought he must. — Leontiskos

I don't require such an assumption. "imply ~A" is not even a proposition.

I didn't say it is.

I don't claim that one may not discuss all kinds of non-formal, formal, alternative formal, or philsophically formal or informal, or mystical New Age incantational informal senses.

"imply ¬A" as the proposition — Leontiskos

'imply ~A' is not a proposition, and I didn't say that it is, so I don't require it as an assumption.