Hypothetical syllogism fallacy

Question

Earlier in the textbook I'm reading we proved the sequent: $$\{ (\phi \rightarrow \psi),(\psi \rightarrow \chi) \} \vdash (\phi \rightarrow \chi)$$

We are later given an alleged counterexample:
$\phi$ is the statement 'I imply you are a donkey'
$\psi$ is the statement 'I imply you are an animal'
$\chi$ is the statement 'I imply the truth'

I can kind of 'sense' what is going wrong, but I'm having difficulty expressing the exact issue. I can see that 'I imply the truth' is tied to the statement 'I imply you are an animal', but that's as far as I get. Could someone explain what the exact issue preventing this from being a counter example is?

Edit: corrected sequent

Answer 1

Hypotetical syllogism is the valid argument form :

$\{ (ϕ → ψ), (ψ → χ) \} ⊢ (ϕ → χ)$

and Chiswell & Hodges' example (page 62) regards it.

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Where is the source of the alleged counterexample ?

In the use of the truth predicate of natural languages that changes reference from the first occurrence : $(ψ → χ)$ to the second one : $(ϕ → χ)$.

In the first case, the expression means "It is true that you are an animal", while the second expression means "It is true that you are a donkey".

Thus, the two equal expressions have different meanings according to the context, and the result we get substituting them for the same variable in the formula "breaks" the validity of the argument.

If we rewrite the statements as follows:

$ϕ$ is the statement 'You are a donkey'

$ψ$ is the statement 'You are an animal'

$χ$ is the statement 'It is true that you are an animal'

the counterexample vanishes, because the conclusion will be :

"if (You are a donkey), then (It is true that you are an animal)".

The antecedent if FALSE, and thus the conditional is TRUE (regardeless of the truth-value of the consequent).