Significant Sententialism in Transparent Intensional Logic and Martin - Löf ’ s Type Theory ∗

Let A be the mathematical proposition that 3 is a prime number. The challenge for the constructivist is to define the semantic conditions, fulfilment of which allows an attributer to issue a claim to knowing that Lulu knows that A is true. Knowledge claims are called judgements. Hence the task becomes to specify when one may make a judgement that attributes to Lulu the judgement that A is true. Judgmental attitudes are indirectly sentential for the following reason. One justifies in part one’s judgement that (1) is true by invoking appropriate linguistic acts that Lulu has performed. It is partly an open issue exactly what those acts must be but Lulu’s inscribing a proof that 3 is a prime number seems essential. Once the assertion conditions of attitude ascription are in place, CTT puts forward an inference (-schema) with those conditions as premises and (1) as conclusion. The typing then serves to make explicit the types of the proof-objects involved in making the judgement that (1) is true. The task the TIL realist faces is to identify the structured hyperintension which the attributer claims to yield True relative to the world and time at which the claim is made just in case Lulu knows that 3 is a prima number. Structured hyperintensions are rigorously defined as constructions, which are no constructivist mathematical proofs but procedures, of one or more steps, which given a certain typed input yield a certain typed output (or, in some clearly circumscribed cases, none). The procedures are platonic objects which should not be confused with their being processed by someone in some situation at some world during some interval. In the case at hand we must identify the construction which constructs a possible-worlds proposition that returns True only at such world-time couples at which it is true that Lulu is related via the knowing-relation to the construction that constructs True by applying the property of being a prime number to 3. Lulu’s attitude is constructional by being a relation to a construction, and is indirectly sententialist for the following reason. Her knowledge that A is true necessarily goes via a proof which is encoded in mathematical language (‘arithmetese’). The motivation for making the proof part of the account of the attribution of knowledge de dicto as in (1) is that the account would be incomplete without an adjacent mode of presentation of the knowledge in question. TIL doesn’t couch linguistic meaning in terms of assertability conditions but in terms of objectual procedures, or constructions. Hence TIL is silent on what you must have judged to be