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  Posted byu/[deleted]2 years ago
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  ## Logic: If 1 equals 2 then 4 equals 5. Is this a true statement?
  
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  level 1
  Theory of Computing14 points · 2 years ago
  
  The statement "if p then q" is true if p is false.
  
  This is known as a vacuous truth.
  
  level 1
  24 points · 2 years ago
  
  If 1 equals 2 then 4 equals 5
  
  This is vacuously true.
  
  level 2
  3 points · 2 years ago
  
  Ah. Trivial.
  
  Thanks.
  
  level 1
  5 points · 2 years ago · edited 2 years ago
  
  Logic: If 1 equals 2 then 4 equals 5. Is this a true statement?
  
  Yes. This is how material implication works---if the antecedent is false, then the conditional is true, regardless of the truth of the consequent.
  
  Since our example has no relations between different elements we assumed it must be transitive because the condition doesn't need to be satisfied (or is satisfied already?)
  
  Your R is transitive, but your reasoning is not quite correct. It can be that a R b and b R c. But if this happens then a = b = c so a = c so a R c. We don't require that a, b, and c be distinct just because we name them with different variables.
  
  level 1
  Undergraduate1 point · 2 years ago
  
  Yep! That's true! It's called "vacuous truth" when the premise of an implication is false.
  
  level 1
  1 point · 2 years ago
  
  As others have mentioned, your statement is vacuously true, but there's also a direct, arithmetical proof:
  
  We have 1 = 2 and 3 = 3. Adding the equations, we get 3 + 1 = 3 + 2, that is, 4 = 5.
  
  The fact that you can do this turns out to be important in the foundations of mathematics.
  
  level 2
  1 point · 2 years ago
  
  Isn't it also true that if 1=2, then 4=27 ?
  
  level 3
  2 points · 2 years ago
  
  It's not hard to prove that if 0=1 in any ring, then all elements of that ring are equal to 0 (0 being the general additive identity, and 1 the multiplicative). And since if we assume that 1=2 (in the integers), we must also have 0=1, so all the integers are 0. Most mathematicians I know think this is rather boring, and doesn't result in a particularly useful ring.
  
  level 4
  2 points · 2 years ago
  
  The trivial ring is not particularly useful, but the fact that you can prove any equality starting from 0=1, without invoking the concept of vacuous truth, is actually useful in foundations.
  
  For example, you can base arithmetic on intuitionistic logic instead of classical logic: the resulting theory, Heyting arithmetic, is strictly weaker than usual arithmetic.
  
  This yields an obvious question: what do we get if we base arithmetic on minimal logic instead? Minimal logic does not have the principle of explosion, so arguments by "vacuous truth" are no longer available. But the argument above demonstrates that we can prove any equality starting from 0=1, without invoking explosion! Therefore, the resulting theory is just as strong as Heyting arithmetic.
  
  level 3
  1 point · 2 years ago
  
  Of course it is. It's not this particular equality that turns out to be important, but the general principle used in the proof above: you can prove any equality starting from 0=1, without invoking the concept of vacuous truth.
  
  level 1
  1 point · 2 years ago
  
  It's vacuously true, as would be something like "If whales are space shuttles, then I have seventeen eyes and three noses" When you have a statement "if A then B" and A is false, then the conditional as a whole is vacuously true, so B can be whatever you want it to be.
  
  level 1
  0 points · 2 years ago · edited 2 years ago
  
  If you assume a false statement to be true, you can conclude anything you want. This is the principle of explosion.
  
  It also calls to mind the logic of "if then".
  
  "If the sky is green, pigs can fly" is a true statement because the sky isn't green, so it doesn't matter what the predicate is. Likewise, "if the sky is green, I will eat a slice of pie" is also true, whether or not I decide to eat pie regardless as long as the first portion is false.
  
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