My Math Forum Implications and the "inverse error" fallacy

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 May 12th, 2009, 10:55 AM #1 Senior Member   Joined: Apr 2009 Posts: 201 Thanks: 0 Implications and the "inverse error" fallacy Hello, there's something I don't quite understand: The condition "->" is valid if: T -> T F -> F F -> T But isn't F->F the same as "denying the antecedent"? Thanks
 May 12th, 2009, 11:05 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Implications and the "inverse error" fallacy No. Denying the antecedent is a -> b a = F ----- b = F (wrong) But both F -> T and F -> F are true, so you can't conclude anything about a -> b given that a is F.
May 12th, 2009, 11:13 AM   #3
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Re: Implications and the "inverse error" fallacy

Quote:
 Originally Posted by CRGreathouse No. Denying the antecedent is a -> b a = F ----- b = F (wrong) But both F -> T and F -> F are true, so you can't conclude anything about a -> b given that a is F.

Isn't saying if a is false, then b is false ( a - > b, not a :. not b) the same as F-> F? (if the antecedent is false, then the consequent is false??)

May 12th, 2009, 11:40 AM   #4
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Re: Implications and the "inverse error" fallacy

Quote:
 Originally Posted by CRGreathouse No. Denying the antecedent is a -> b a = F ----- b = F (wrong) But both F -> T and F -> F are true, so you can't conclude anything about a -> b given that a is F.
Hmm.. I think I understand: a->b , not a :. not b isn't valid because given "not a", it could either be F->T or F->F..
So does this mean that in certain situations, if it is clearly F->F, denying the antecedent is valid?

EDIT: Nevermind, I took the "->" implication as an act of deducing; I see that the determinant of its validity is non-contradiction

 May 12th, 2009, 03:45 PM #5 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Implications and the "inverse error" fallacy In case there's any doubt: (a = 2, b = 3) -> a + b = 5 If a = 1 and b = 4 then you have F -> T If a = 1 and b = 1 then you have F -> F If a = 2 and b = 3 then you have T -> T But you can't have T -> F.
 May 12th, 2009, 06:55 PM #6 Senior Member   Joined: Apr 2009 Posts: 201 Thanks: 0 Re: Implications and the "inverse error" fallacy thanks again, could you actually show me "F->F" (or if you don't mind, the conditionals) in other deductions, sentences, propositions..etc? thanks like.. If a then b, not b then not a
 May 12th, 2009, 09:47 PM #7 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Implications and the "inverse error" fallacy I'm not entirely sure what you're asking. (Knowing a -> b doesn't let you conclude any of a, not-a, b, or not-b, it just tells you relationships between them.) If you just want examples like my last, though, I can do that. (x is an integer, y is an integer) -> x + y is an integer for (x, y) = (pi, 3/2) x is positive -> (2^x > 1) for x = -1 x is the President of the United States -> x is the Commander-in-Chief of the United States for x = CRGreathouse x is a human -> x is a featherless biped for x = an octopus
 May 13th, 2009, 07:58 AM #8 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: Implications and the "inverse error" fallacy I'm not sure if this has been answered... The problem is that you are confusing evaluating an implication with known values and concluding the values based on the implication. In other words: Given a->b, we know something about the relation of the truth value of a and the truth value of b: We know if a is true, b must be true, and if b is false, then a must not be true (because then T->F). Here, we're using the assumption that the implication is valid to infer the value of a or the value of b On the other hand, if "a = false" is given, The implication a->b is valid, because it is only invalid when a is True and b is False. Here we're using the assumption that a is false to infer the validity of the implication Putting these ideas together, the statements you gave infers the truth value (validity) of the implication based on the truth values of the terms (a and b), while denying the antecedent infers the truth value of term b based on the validity of the implication and term a. I hope that wasn't too dense...
 May 13th, 2009, 09:16 AM #9 Senior Member   Joined: Apr 2009 Posts: 201 Thanks: 0 Re: Implications and the "inverse error" fallacy Thanks guys, I think that you both answered my problem to its core I have one more thing that needs to be cleared up though, and it may sound strange.. Anyway, the nature of the implication "->" seems to have different attitudes.. For example, if we said, True: Mr.Jones is a Canadian Citizen (a) -> True: Mr. Jones is a human being (b), the truth or the implication in this sentence is necessarily true - there is no way around its truth - you cannot be a citizen if you are not a human being. However, if you said, False: Mr.Jones is not a Canadian Citizen -> False: Mr. Jones is not a human being or False: Mr. Jones is not a Canadian Citizen -> True: Mr.Jones is a human being, the truths or the implications of these sentences are less self-evident and not "necessarily true"; even though it MAY be true, if you choose one, you could always hypothesize the other. It seems to me that while the validity of "->" is at times logically undeniable, it is something else at other times. Thanks again
May 13th, 2009, 10:20 AM   #10
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Re: Implications and the "inverse error" fallacy

Quote:
 Originally Posted by ElMarsh I have one more thing that needs to be cleared up though, and it may sound strange.. Anyway, the nature of the implication "->" seems to have different attitudes.. For example, if we said, True: Mr.Jones is a Canadian Citizen (a) -> True: Mr. Jones is a human being (b), the truth or the implication in this sentence is necessarily true - there is no way around its truth - you cannot be a citizen if you are not a human being. However, if you said, False: Mr.Jones is not a Canadian Citizen -> False: Mr. Jones is not a human being or False: Mr. Jones is not a Canadian Citizen -> True: Mr.Jones is a human being, the truths or the implications of these sentences are less self-evident and not "necessarily true"; even though it MAY be true, if you choose one, you could always hypothesize the other. It seems to me that while the validity of "->" is at times logically undeniable, it is something else at other times.
x is a Canadian citizen -> x is a human being

This statement is correct because if the second part is false, the first part cannot be true. As it happens in example the other three cases are possible:
F -> F: x = a rock
F -> T: x = Tony Blair
T -> T: x = Stephen Harper

But let's suppose that for x = Madison the Canadian goose, x is a Canadian citizen but x is not a human being. All this shows is that the implication
x is a Canadian citizen -> x is a human being
is wrong.

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