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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

Cdizzle
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1 Answers1

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Hypotetical syllogism is the valid argument form :

$\{ (ϕ → ψ), (ψ → χ) \} ⊢ (ϕ → χ)$

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

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).

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Mauro ALLEGRANZA
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  • Sorry if I wasn't clear, I can see that the sequent is valid. It follows that the substitutions for the metavariables won't form a counterexample, my problem was it seems like a counterexample and I had difficulty explicitly pointing out why it is not.

    The trouble with the substitutions given in the book is they begin with 'I imply', which means the statement 'I imply you are a donkey' is true even if the statement 'you are a donkey' is not. So the antecedent of the conclusion would be true.

     – Cdizzle Apr 08 '18 at 04:40
  • @Cdizzle - yhe ansewr is above, and C&H comments are clear: the use of natural language can cause troubles. In the substitution $(ϕ→ψ)$ the natural language sentence is "I say that you are a animal and what I assert is true' while the the second one is 'I asert that you are a donkey and what I assert is true'. What is said "to be true" is not the same in the two cases. – Mauro ALLEGRANZA Apr 08 '18 at 07:22
  • So there isn't any deeper reason for the issue, it's simply a case of natural language not being fully compatible with formal languages? – Cdizzle Apr 08 '18 at 07:51
  • I suppose any deeper issue would be more a problem of linguistics than mathematics, so this does answer my question. Thank you. – Cdizzle Apr 08 '18 at 07:58