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The question I am trying to answer is:

Transcribe the following to English:

Adam is the tallest one in the class.

Transcription guide:

$Cx$: $x$ is in the class

$Txy$: $x$ is taller than $y$

$a$: Adam

The question also says I have to explicitly state that Adam is not taller than himself. The answer given by the book is,

$\forall x [(Cx \land x\neq a) \rightarrow Tax]$,

however I thought this transcribed to 'Adam is taller than everyone in the class that is not himself' which does not say whether he is taller than himself or not. So I said the sentence could be transcribed as,

$\forall x[Cx \rightarrow (Tax \equiv x \neq a)]$

or

$\forall x [(Cx \land x\neq a) \rightarrow Tax] \land \neg Taa$.

Could you please tell me what the correct answer is?

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  • $\begingroup$ The first one. $a$ denotes Adam; thsu $Cx \to x \ne a$ means "$x$ is in the Class and he is not Adam". $\endgroup$ – Mauro ALLEGRANZA Sep 11 '19 at 6:52
  • $\begingroup$ By Exportation, this is the same as : $∀x[Cx → (x≠a \to Tax)]$ that is near to the second one. $\endgroup$ – Mauro ALLEGRANZA Sep 11 '19 at 6:54
  • $\begingroup$ Don't forget to specify that Adam is in the class. $\endgroup$ – Graham Kemp Sep 11 '19 at 7:02
  • $\begingroup$ @MauroALLEGRANZA Are you saying that the book's answer is correct or that the first one of my answers is correct? $\endgroup$ – Lachie Sep 11 '19 at 7:25
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If the question explicitly asks you to say that Adam is not taller than himself, than the answer in the book did indeed forget to add the $\neg Taa$ statement. So, your second answer is better than the book's. And, your first answer also implies that Adam cannot be taller than himself. So, good for you.

But, as Graham Kemp points out, none of the answers make it clear that Adam is the class ... which is clearly true if Adam is to be 'the tallest one in the class'. So, you should add this to either of your answers, meaning you can do:

$\forall x[Cx \rightarrow (Tax \equiv x \neq a)] \land Ca$

or

$\forall x [(Cx \land x\neq a) \rightarrow Tax] \land \neg Taa \land Ca$

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