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Is it true that these two conditionals if A then B and if not-A then B cannot be both true?

Example :

  • "If I stay then I will eat fish"
  • "If I didn't stay then I will eat fish"

The reason I think it maybe the case, is because one can decide to eat fish whether she stays or didn't stay home (maybe she decides to eat fish outside at a restaurant), so one can say that : "if I stay then i will eat fish, and if I didn't stay then I will eat fish".

The problem I think this may lead to a contradiction is through Modus tollens, the conditionals are equivalent to the following respectively :

  • "If I didn't eat fish then I didn't stay"
  • "If I didn't eat fish then I stayed".

So, it must be that something is wrong. Because if "I didn't eat fish is true", then it follows that both "I stayed" and "I didn't stay" would be true from logical necessity .

So, it must be that one cannot say : "Whether A or ~A, it's always B" ... since that would lead to a contradiction.

Is there something I missed?

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    They can both be true, and it doesn't lead to a contradiction. It leads to B being true. Indeed, this is a standard form of proving B by cases. It is intuitive: if I ate fish no matter what then I did eat it, and it can't be true that I didn't. – Conifold Feb 27 at 22:26
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    If not B then ⊥, as they denote falsehood/contradiction these days. – Conifold Feb 27 at 22:36
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    Are you using "if - then" to refer to material conditional or some other type of conditional statement? Just looking at the truth table for A -> B, it's true when A is true and B is true, but it's also true when A is false and B is true. – Hypnosifl Feb 27 at 22:37
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    There is an informal alternative definition of "if" meaning "if and only if" (sometimes written as "iff"), where these two statements would indeed be contradictory. But in formal logic, "if" is only a one-directional implication. – Darrel Hoffman Feb 28 at 15:55
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    If I ride my bike then water is wet. If I don't ride my bike then water is wet. If water is not wet then I didn't ride my bike. If water is not wet then I rode my bike. <- all true statements – user253751 Feb 28 at 16:43
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You said in a comment that you were referring to the material conditional, not other notions of if/then like the antecedent being a cause of the consequent, or the antecedent logically implying the consequent. So let's get rid of the if/then structure and write them explicitly as material conditionals:

A) "I stay" -> "I eat fish"

B) "I didn't stay" -> "I eat fish"

You are worried about the possibility that both A) and B) are true simultaneously, and we apply modus tollens to each one in turn to get two more true statements:

C) "I didn't eat fish" -> "I didn't stay"

D) "I didn't eat fish" -> "I stay"

It is indeed possible that A) "I stay" -> "I eat fish" and B) "I didn't stay" -> "I eat fish" are simultaneously true--the material conditional is defined by its truth table, and the truth table indicates that as long as the consequent is true, the material conditional is true regardless of whether the antecedent is true. So in the case where "I eat fish" is true, both your first two material conditionals A) and B) are automatically true. And if "I eat fish" is true, there's no problem with modus tollens, since both C) "I didn't eat fish" -> "I didn't stay" and D) "I didn't eat fish" -> "I stay" involve a false antecedent in this case, and the truth table for the material conditional shows that it's always true when the antecedent is false, regardless of whether the consequent is true or false.

To put it another way, if you learn that C) "I didn't eat fish" -> "I didn't stay" is true but the antecedent "I didn't eat fish" is false, then you can't deduce anything about whether "I didn't stay" is true or false. Likewise if you learn D) "I didn't eat fish" -> "I stay" is true but "I didn't eat fish" is false, again you can deduce nothing about whether "I stay" is true or false. So as long as "I didn't eat fish" is false, learning that both of your second two statements C) and D) are true doesn't lead to any contradictory deductions about whether you stayed or didn't stay.

On the other hand, let's consider the other case where "I eat fish" is false/"I didn't eat fish" is true. In the case that the consequent is false, the truth table shows it is not possible that both A) "I stay" -> "I eat fish" and B) "I didn't stay" -> "I eat fish" are simultaneously true. When the consequent is false, the material conditional is true if the antecedent is false as well, but the material conditional is false if the antecedent is true.

So, if "I eat fish" is false, there are now two different possibilities:

1) If "I stay" is false as well, then A) "I stay" -> "I eat fish" is true (both antecedent and consequent are false) but B) "I didn't stay" -> "I eat fish" is false (antecedent is true but consequent is false).

2) If "I stay" is true, then A) "I stay" -> "I eat fish" is false (antecedent is true but consequent is false) while B) "I didn't stay" -> "I eat fish" is true (antecedent and consequent both false).

The main point to note here is that under any circumstances where "I eat fish" is false (or equivalently 'I didn't eat fish' is true), only one of your original two statements A) and B) can be true while the other must be false, which implies only one of the two derived statements C) and D) can be true while other must be false as well. Whereas your paradox arose from assuming both of the first pair A) and B) were true and then applying modus tollens to both to get two more true statements C) and D) that both had the antecedent "I didn't eat fish", and finding that these two statements led to inconsistent conclusions about whether you stayed. But what I've shown above is that C) and D) can only both be true if their antecedent "I didn't eat fish" is false, in which case nothing can be deduced from them about whether you stayed, whereas if "I didn't eat fish" is true than only one of C) or D) can be true. Either way, there's no contradiction.

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  • Ah, I see now, so in general one can say that I assumed false what I already presumed to be true. It's just like saying : If x is a number then x/0 doesn't exist. Where the inverse conditional presumes that x/0 is true in : if x/0 then x is not a number. And to get the contradiction I have assume x/0 exists, which I already decided is false. Thank you so much ! – SmootQ Feb 28 at 8:30
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The problem is that you've created your logical statements in the form of a causal relationship when no causal relationship exists. Normally we would expect the statement:

  • If I stay, I will eat fish

to mean that 'eating fish' is a logical consequence of 'staying', which carries the implication that if you have not eaten fish, you must not have stayed. But your second statement means that 'eating fish' is a logical consequence of 'leaving' as well, and since 'staying' and 'leaving' are logical complements (covering all possible cases), these two statements, combined, amount to:

  • Whatever I do, I will eat fish

So the logical contradiction is not between your original two statements, but rather between your implicit assertion that you will eat fish in every possible case, and your explicit assertion that you did not eat fish. You're doing the logical equivalent of dividing by zero; asserting that something which must happen in all cases did not happen.

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  • Thank you for your reply, "I eat fish" is always true, so in this case, not being true leads to a contradiction. Thanks again – SmootQ Feb 28 at 8:32
  • Wow! How did you learn to understand and argue in such a manner!? – Ardent Coder Feb 28 at 11:32
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    @ArdentCoder: By doing it wrong, lots and lots and lots of times. 😀 One of my favorite quotes from Wittgenstein: "“Don't for heaven's sake be afraid of talking nonsense! But you must pay attention to your nonsense." – Ted Wrigley Feb 28 at 17:16
  • @TedWrigley Great! You are amazing :) – Ardent Coder Feb 28 at 17:23
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There is no problem here. Assuming material conditionals, if it is the case that

If A then B, and if not-A then B,

then B is simply a tautology, as it is true in every possible case. You are right the the above conditionals entail

If not-B then A, and if not-B then not-A,

but this shows that not-B cannot be true. Given that B is a tautology, this is to be expected.

Of course "I will eat fish" is not a tautology. So one might wonder what goes on when people say things like "Whether I stay or not, I will eat fish". Well, there are two options. Either they are not using material conditionals (such that the modus tollens reasoning doesn't go through), or they are making false (perhaps idealized) statements. Neither of these options should seem surprising.

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    B doesn't need to be a tautology though, it can be any true statement including a contingent one, that's enough to guarantee that both "A -> B" and "not-A -> B" are true, regardless of whether A is true or false. Just look at the truth table for the material conditional, it is always true when the consequent is true. – Hypnosifl Feb 28 at 0:07
  • @Eliran, thanks for your reply. "shows that not-B cannot be true", I think this sums it up. – SmootQ Feb 28 at 8:40
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In standard propositional logic, where “if…then…” is the material conditional, “if A then B” and “if not-A then B” can certainly both be true together. Indeed, “(if A then B) and (if not-A then B)” is equivalent to simply “B”.

Your everyday example with staying vs. eating fish is a bit misleading, because it pushes us towards thinking of the subtler natural-language connotations of “if…then…”, as connoting for instance a causal or temporal sense of implication. The question becomes clearer with a more abstract example; for instance:

  • If it is Tuesday, then 2+2=4.
  • If it is not Tuesday, then 2+2=4.

These are clearly both true — on any day of the week, 2+2=4. So yes, two such propositions can hold together.

Generally, “(if X1 then Y) and (if X2 then Y)” is equivalent to “if (X1 or X2) then Y”. So “(if A then B) and (if not-A then B)” is equivalent to “if (A or not-A) then B”; but “A or not-A” is always true, so this is in turn equivalent to “if true then B”, and hence to just “B”.

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    I see, 2+2=4 is a nice example, thank you so much : It's kind of a tautology, and assuming it to be false, makes it look like a contradiction. +1 – SmootQ Feb 28 at 16:46
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The issue you bring up seems to be how human beings today view conditional propositions. That is the famous IF . . . Then. . . type of sentence which also is called hypothetical propositions.

Many people tend to believe that conditional propositions MUST BE SUFFICIENT OR NECESSARY which turns out false in reality. "If I stay then I will eat fish" is not necessary one bit. I hate fish personally so the consequent is false. I am still alive without eating fish. If I stay the night can also be false. I stayed somewhere else than where your proposition refers. Me staying somewhere that the proposition refers is NOT FORCED upon me. Do you get it? Both the antecedent and the consequent of the conditional propositions you originally stated are false (or better yet COULD BE FALSE) in reality. Nothing is sufficient nor necessary that if both the antecedent and consequent of both of your propositions are false the world ends or I cease to exist. So this means some conditional are not either necessary or sufficient. Take for example, "If boxer Tyson fury beats Deontay Wilder, then I am a monkey's uncle." Surely this condition is in the same category as your two conditionals propositions: nine of them have any necessary aspect nor sufficient aspect for the consequent to be true. I would think some conditional statements are for pure Rhetorical purposes. I would hope you see this and agree.

Language is strange in that way. Sometimes what we mean is not explicitly expressed when we originally make a statement. Somewhere we see we left out details or we could have said the statement in a better way. This brings me to my next point. How you stated the propositions could be interpreted differently. On top of that the question along with the statements could be interpreted differently.

Let me explain, you wrote two propositions. Did you mean to use them conjunctively as one long formula: [ (a-->b) & (~a-->b) ] ? Or did you mean to look at each separately and try to decide if one is true must the other be false? That is, given a -->b is TRUE must ~a-->b FALSE? Well it turns out both can be true and both can be false based on the CONTENT of what the propositions are about -- not just form alone. Also we have to be CLEAR when you say TRUE what do you mean. Do you mean this conditional is TRUE by truth table, is it true by us using or senses like touch or sight, is this objectively true under all circumstances, etc. You will always get the guy that says well if the antecedent is false the statement is TRUE! Well specifically it is true via truth table and could very well be FALSE in the real world. TRUE doesn't necessarily mean real world true.

Finally, I took the QUESTION you asked not the propositions you wrote as a question comparing propositional values. That is you are given two propositions and you know the value of only one, what can you do to determine the truth value of the other proposition?
That is, for example "if Fido is a dog, then Fido is a mammal" and we are told this is True. What is the truth value of the second proposition "If Fido is not dog, then Fido is a mammal." I would think this is a different interpretation than the other answers so far that none have taken.

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The examples you give are unnecessarily complicated. Consider a statement B that is a tautology, i.e. trivially true, e.g. "a yellow car is a yellow car".

Then, 'A implies B' and 'not-A implies B' are both true for any choice of A. The reason for this is that, if B is simply True, then

A implies True = not-A or True = True

for any A.

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