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LINGUISTIX&LOGIK, Tony Marmo's blog
Friday, 18 March 2005


The (Non-)Transitivity of Knowledge reports

If some propositional attitudes are interpretable as negation of propositions, should veridical reports containing verbs like to know interpreted as the simple statement of the propositions? This question seems to make sense, since it has been observed that to know something entails that that thing is true. However, a claim like (1a) might mean that the propositional attitude contributes nothing to the meaning of the whole expression. Moreover, if (1a) is true, then (1b) should be the case:
(1) Hypothesis A
a. Γ (π)|=π
b. (Γ (π)|=π)&(π|=β) → Γ (π)|=β

Let us call one of the ideas that seems to underlie the intuition formalised in (1a) Hypothesis A? and re-write it as (2):
(2) Hypothesis A?
a. Γ (π)=π
b. ∴ Adding to know to a proposition σ adds nothing to its meaning.

Hypothesis A? would lead to think that the negation of Γ (π) is the negation of π itself:
(2?) ¬Γ (π)=¬π

Should (2?) hold, then statements of the type not know π and those of the type to know that π is the not case would mean the same. But the evident contrast between sentences (3a) and (b) below does not confirm such prediction:
(3) a. We know that Giselle is not a singer. ≠
b. We do not know that Giselle is a singer.

This contrast suggests that Γ (π) is not equal to π but rather that it means one epistemic agent has access to a truth π , while ¬Γ (π) does not mean that ¬π , but rather that an agent has not access to a truth π .
Still this finding only excludes hypothesis A?, while it would be possible to maintain hypothesis A. Additional evidences, on the contrary, suggest that to know is perhaps the most opaque of the attitudinal verbs, in the sense that sentences with to know somehow block transitivity. Consider this example:
(4) a. People can skate on the lake. |= Its water has frozen.
b. John knows that people can skate on the lake. |≠ Its water has frozen.

Being the entailment in (4a) valid, and if (1b) applies, then the addition of John knows? should not affect the entailment. But (4b) disconfirms such expectation: the mere fact that John knows that people can skate on the lake does not mean right at the same moment that the water of the lake is covered by a thick layer of ice.
Now consider this other hypothesis:
(5) Hypothesis B
(π |=β) → (Γ (π)|= Γ (β))

This second hypothesis is not true either, as shown by (6)
(6) a. Oedipus killed the man he met at the crossroads.
|= The oracle has been fulfilled.
b. Oedipus knows he killed the man he met at the crossroads.
|≠ Oedipus knows the oracle has been fulfilled.

So evidences point to the contrary conclusion, although an expression of the type to know π entails the truth of π , it somehow unmakes π |=β :
(7) Non-transitivity
(Γ (π)|=π)&(π |= β) → (Γ (π)|≠β)& (Γ (π) |≠ Γ (β))

The non-transitivity of veridical reports requires closer examination and more attention. For the sake of economy, such topic cannot and will not be herein investigated in more detail. Here it will suffice to say that the apparent non-transitivity of veridical reports is also an anti-trivialisation mechanism.

Posted by Tony Marmo at 12:29 GMT
Updated: Friday, 18 March 2005 12:36 GMT

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