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Consider the following argument.

1) Arthur is taller than Brian 2) Brian is taller than Chris Conclusion: 3) Chris is shorter than Arthur.

Here is how I would formalise this reasoning using predicate logic:

1) Tab 2) Tbc Conclusion: 3) Tac

T= taller than a= Arthur b= Brian c= Chris

Is my attempt correct? More specifically, my question is: am I right in thinking that there is no need to introduce a new predicate letter (let's say the letter S) in order to say "shorter than"?

Many thanks

Fisher

user405159
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  • In general (i.e., without any axioms abut the predicates involved), deducing $Tac$ from $Tab$ and $Tbc$ is not any more valid than deducing $Sca$ ... – Hagen von Eitzen Feb 12 '17 at 18:01
  • Ok. So I can't automatically read Tac as «Chris is shorter then Arthur"? – user405159 Feb 12 '17 at 18:14
  • Could someone give the correct formalization of my argument (in such a way that the argument is valid)? – user405159 Feb 12 '17 at 18:20
  • Two more pemises : 1) $\forall x,y,z [(Txy \land Tyz) \to Txz]$, we need tarnsitivity, and 2) $Sxy \leftrightarrow Tyx$.. – Mauro ALLEGRANZA Feb 12 '17 at 18:27
  • Ok, it's clear now! – user405159 Feb 12 '17 at 18:33
  • Without 1) the argument is not valid; if we interpret $Txy$ as "$x$ is Father of $y$", from "John is Father of Tom" and "Tom is Father of Paul" does not follow that "Paul is Son of John". – Mauro ALLEGRANZA Feb 12 '17 at 18:38
  • Ok. But in fact the worry behind my initial question was a bit different and didn't really concern the validity of the argument. My worry is: why if Lab (L = loves, a = Arthur b= Betty) means both Arthur loves Betty and Betty is loved by Arthur, why then Tac doesn't mean both Arthur is taller than Chris and Chris is shorter than Arthur? – user405159 Feb 12 '17 at 18:50
  • @Fishermansfriend Tac can be understood as Chris is shorter than Arthur, but if you have a separate predicate for Taller and Shorter, then you need to add the axiom $\forall x \forall y (Txy \leftrightarrow Syx)$ in order to infer that using formal logic. Remember, to the logic system, T and S are just arbitrary symbols .. And kind of meaningful connection will have to be made explicit. – Bram28 Feb 12 '17 at 19:01
  • Ok! So the same applies to "loves" and "is loved by", if you have a separate predicate for "loves" and "is loved by". Thanks a lot! – user405159 Feb 12 '17 at 19:18
  • Maybe useful Logical Constants and Logical Form. – Mauro ALLEGRANZA Feb 12 '17 at 19:28

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