Formal games and forms for games

Interestingly, it is not just the lexical ambiguity of "duck" which enters here. There is also structural ambiguity. Precisely what structures are involved will be a matter for theoretical discussion, but it is clear that the difference is that between John's seeing her doing something, and his seeing something of hers. The assignment of senses to sentences is not one-to-one. Con sideration of active and passive sentences, and (1) above show us that the assignment is many-many. Every student learning how to translate English sentences into logical notation discovers this early on. We also learn how limited are the resources of expression in standard first order logic by comparison with English. There are clear, unam biguous English sentences that cannot be translated into the language of first order logic. We recognise this in our talk of 'fragments' of English which enjoy the metamodality of logical parsability. Moreover, it is not just a case of lacking structurally similar logical translations, with inverted iotas instead of Russellian quantified complexes doing duty for definite articles. The limitations of first order logic are more serious than that. The problem is that, even allowing for such excesses as a million occurrences of an existential quantifier to cope with one occurrence of the word "million", there are sentences that cannot be translated into first order logical notation at all. This is so even with a minimal constraint upon such translation: that the logical translation and the English sentence have the same truth conditions (where for simplicity I am assuming the English sentence to be unambiguous). There are unambiguous English sentences whose truth-conditions cannot be captured by any sentence of (existing) formal languages.