Paul Horwich’s minimal theory of truth consists of all propositions of the form:
<p> is true iff p.
As was pointed out by Anil Gupta (Philosophical Perspectives 1993), Horwich’s minimal theory does not logically imply certain general facts about truth. Horwich (Truth, 2nd ed.) acknowledged Gupta’s point, but he thought that those general facts nevertheless are non-logically implied by the theory. What is involved, he thinks, is the following non-logical but truth-preserving rule:
If S is a set of premises all which attribute a certain property P to a proposition, such that every proposition is attributed that property by one of the premises, then S non-logically entails that every proposition has P.
Panu Raatikainen (Analysis 2005) assumed that what Horwich had in mind is the omega-rule, but he correctly points out that the omega-rule would not be of help to Horwich, because it is only applicable to a denumerable infinity of premises. But there are far more propositions than natural numbers.
Panu is right about the usefulness of the omega-rule. But doesn’t his point provide clear evidence that the rule Horwich alludes to just isn’t the omega-rule? Instead, it is the rule stated above; that rule requires for its application more premises than there are natural numbers (and as many premises as there are propositions). So, it is not the omega-rule, and the rest of Panu’s criticism somehow misses its target.
To see whether the generalization-problem can be solved by Horwich, one should forget about the omega-rule and instead focus directly on the rule above. The crucial questions are: Is the rule truth-preserving? And can it explain how finite minds arrive at the required generalizations starting from Horwich’s theory?
Posted by Benjamin.