5 ms·

I believe you are talking about the inverse, as opposed to the contradiction. Consider a * b Contradiction: a * -b Inverse: -a * -b In the theorem that he pr

I believe you are talking about the inverse, as opposed to the contradiction. Consider a * b Contradiction: a * -b Inverse: -a * -b In the theorem that he provides us, H(X) = H(-X), it could be argued either way as to whether he meant contradiction or inverse. However, his example of "we should work harder on correct proofs; we should work harder on incorrect proofs" very clearly uses two variables, and the form of the second is of a contradiction. EDIT: Wow did I screw my terms up. Changed Negation to Contradiction, and Contrapositive to Inverse. Thank you barrkel for pointing this out.

barrkel 16y ago
Contrapositive is the rule that takes "if A, then B" to "not B, therefore not A". The contrapositive is true if the initial proposition is true. But the grandparent posited "we shouldn't work hard on getting our proofs correct" as the correct negative to "we should work hard on getting our proofs correct". If it were a contrapositive, then it would be true when the article's proposition is true; but that's not the case (one can't simultaneously work and not work hard), so it's not a contrapositive. A contrapositive to "we should work hard on getting our proofs correct" could be "if we're not working hard on it, it's not a proof" - though it depends on how you interpret the statement as a proposition.
vog 16y ago
> I screw my terms up. Changed Negation to Contradiction, and Contrapositive to Inverse. Even with those changes, your post doesn't make any sense to me. The negation of logical statements has a pretty well defined meaning [1]. It seems to be neither what you call "inverse" not what you call "contradiction". Also, I don't see what the " * " operator means in your example. It appears to be some operator to combine two logical statements "a" and "b". However, the original statement isn't a combination of smaller statements at all. (In particular, the original statement is not an implication, i.e. a term of the form "if a then b" or "from a follows b" or similar.) Anyway, for whatever you mean by " * ", the negation is simply "- (a * b)". The expressions you are proposing ("-a * b" and "a * -b") seem to be different from that. In summary, it is totally unclear what you mean by "a", "b", " * ", "inverse" and "contradiction". [1] The negation is true for exactly those instances for which the original statement is false.
- trobertson 16y ago
  Fair points. I was basing my terminology off of this table, after realizing that I had used incorrect terms. http://en.wikipedia.org/wiki/Contraposition#Comparisons http://en.wikipedia.org/wiki/Contraposition#Comparisons > If it is meant to be a synonym of what I'm calling negation [1], it would be "- (a * b)" for whatever you mean with " * ". However, this is by no means equal to "(-a) * (-b)". This is absolutely correct, and was what originally prompted my comment. In your original comment, you say that > The correct negation is: "we shouldn't work hard on getting our proofs correct." which, if we let a = "should work" and b = "getting proofs correct", would mean that you said (-a) * (-b) is the negation, as opposed to -(a * b). By the quoted text, the negation would be either (-a) * b, or a * (-b). This distinction is what I was trying to communicate, though it seems I did a poor job of that.
  - vog 16y ago
    Okay, so your " * " operator means "implication" [1], because the terms "inverse", "contradiction" and "contrapositive" in your mentioned Wikipedia article refer only to implications. However, the original statement isn't an implication. Also, neither your proposed part "should work" nor the other part "getting proofs correct" are logical statements. Those are just parts of a sentence. In other words, your decomposition doesn't make any sense. [1] That is, "a * b" = "if a then b" = "from a follows b" = "a implies b"

barrkel 16y ago

Contrapositive is the rule that takes "if A, then B" to "not B, therefore not A". The contrapositive is true if the initial proposition is true. But the grandparent posited "we shouldn't work hard on getting our proofs correct" as the correct negative to "we should work hard on getting our proofs correct". If it were a contrapositive, then it would be true when the article's proposition is true; but that's not the case (one can't simultaneously work and not work hard), so it's not a contrapositive. A contrapositive to "we should work hard on getting our proofs correct" could be "if we're not working hard on it, it's not a proof" - though it depends on how you interpret the statement as a proposition.

vog 16y ago

> I screw my terms up. Changed Negation to Contradiction, and Contrapositive to Inverse. Even with those changes, your post doesn't make any sense to me. The negation of logical statements has a pretty well defined meaning [1]. It seems to be neither what you call "inverse" not what you call "contradiction". Also, I don't see what the " * " operator means in your example. It appears to be some operator to combine two logical statements "a" and "b". However, the original statement isn't a combination of smaller statements at all. (In particular, the original statement is not an implication, i.e. a term of the form "if a then b" or "from a follows b" or similar.) Anyway, for whatever you mean by " * ", the negation is simply "- (a * b)". The expressions you are proposing ("-a * b" and "a * -b") seem to be different from that. In summary, it is totally unclear what you mean by "a", "b", " * ", "inverse" and "contradiction". [1] The negation is true for exactly those instances for which the original statement is false.

trobertson 16y ago
Fair points. I was basing my terminology off of this table, after realizing that I had used incorrect terms. http://en.wikipedia.org/wiki/Contraposition#Comparisons http://en.wikipedia.org/wiki/Contraposition#Comparisons > If it is meant to be a synonym of what I'm calling negation [1], it would be "- (a * b)" for whatever you mean with " * ". However, this is by no means equal to "(-a) * (-b)". This is absolutely correct, and was what originally prompted my comment. In your original comment, you say that > The correct negation is: "we shouldn't work hard on getting our proofs correct." which, if we let a = "should work" and b = "getting proofs correct", would mean that you said (-a) * (-b) is the negation, as opposed to -(a * b). By the quoted text, the negation would be either (-a) * b, or a * (-b). This distinction is what I was trying to communicate, though it seems I did a poor job of that.
- vog 16y ago
  Okay, so your " * " operator means "implication" [1], because the terms "inverse", "contradiction" and "contrapositive" in your mentioned Wikipedia article refer only to implications. However, the original statement isn't an implication. Also, neither your proposed part "should work" nor the other part "getting proofs correct" are logical statements. Those are just parts of a sentence. In other words, your decomposition doesn't make any sense. [1] That is, "a * b" = "if a then b" = "from a follows b" = "a implies b"