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Using two forms of provability:

  • Identity Elimination/Transitivity
  • AnaCon: Analytical Consequence

Below, "Larger(x,y)" means "x is larger than y", "Smaller(x,y)" means "x is smaller than y", and "SameSize(x,y)" means "x is the same size as y".

What would be the best way (makes more logical sense) to prove the following:

  1. Larger(b, c)
  2. Smaller(b, d)
  3. SameSize(d, e)
    --> Larger(e, c) <-- this is what we are trying to prove

I have the following proofs:

Proof # 1

4) Smaller(b, e) AnaCon: 2, 3
5) Larger(e, b) AnaCon: 4
6) Larger(e, c) AnaCon: 5, 1

Proof # 2

4) Smaller(b, e) AnaCon: 2, 3
5) Larger(e, c) AnaCon: 4, 1

Proof # 3

4) Smaller(c, d) AnaCon: 1, 2
5) Larger(e, c) AnaCon: 4, 3

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    I don't have enough points here, could someone please add the following tags: provability and fitch-proofs. Thanks2011-03-07
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    Does "Larger(x,y)" mean "x is larger than y" and "Smaller(x,y)" mean "x is smaller than y"? Or the other way around? In any case, Proof #1 must be incorrect; you have Smaller(e,b) in 2, and you are deducing Smaller(b,e); or was 2 supposed to be with d instead of e?2011-03-07
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    This is confusing: if you are saying (1) b is *larger* (or perhaps *smaller*) than c and (2) e is *smaller* (or perhaps *larger* than b), then I don't see how you hope to conclude anything about the relationship between e and c. Either might be *larger* than the other, or they might be the *same size*: it is easy enough to produce examples of each. If this is a partial order then there might even be no relationship at all.2011-03-07
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    @KerxPhilo, @Henry: Given (4) in both "Proof #1" and "Proof #2", I suspect that 2 was supposed to be "Smaller(b,e)", rather than "Smaller(e,b)". If so, perhaps KerxPhilo can clean this up and explain the notation being used?2011-03-07
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    Sorry, there was a error in step 2, it has now been corrected.2011-03-07
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    @Arturo, Yes, Larger(x, y) means "x is larger than y" and Smaller(x, y) means "x is smaller than y"2011-03-07
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    @KerxPhilo: All three proofs seem to me to be about the same; neither is better ("makes more logical sense") than any other. "makes more logical sense" is, alas, a partial order, not a total one.2011-03-07
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    Maybe I should have not shown the three proofs and seen what others came up with :) Anyone can see another proof available?2011-03-07
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    I like the following proof: $c < b < d \approx e$.2011-03-07
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    @KerxPhilo: There are likely *many* different ways at arriving at the same thing, some more circuitous than others. In some sense, you are being more explicit in #1 than in the others (e.g., in #4, you are first going from Smaller(c,d) to Larger(d,c), then using this and SameSize(e,d) to get (5); you skipped the first step, but you were explicit about in in #1). Every proof will touch the following "bases": b is larger than c and smaller than d, so d is larger than c, and therefore e is smaller than c.2011-03-07

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As edited they now all look equivalent. Proof #1 is slightly more explicit when using the fact that Smaller(y,x) implies Larger(x,y) , which seems to be implicit in the others. You could also have another 3 step version

4) Larger(d, b) from 2 - reverse

5) Larger(d, c) from 4, 1 - transitive

6) Larger(e, c) from 5, 3 - substitution

and I suspect there are other variants