# Which proof makes more logical sense?

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. Thanks –  KerxPhilo Mar 7 '11 at 0:49
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? –  Arturo Magidin Mar 7 '11 at 0:58
@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. –  Arturo Magidin Mar 7 '11 at 1:23
I like the following proof: $c < b < d \approx e$. –  Yuval Filmus Mar 7 '11 at 1:28
@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. –  Arturo Magidin Mar 7 '11 at 1:29