Tutorial 5: Connectives PLIN0009 Semantic Theory 1 Sentential Conjunction: Semantic Types Recall how semantic types work. In Lecture 4, we looked at sentences like the following. t John e VP xe, ty loves xe, xe, tyy Mary e Let us now add the connective and to our inventory. We will irst consider its sentential use, andS, which conjoins two (declarative) sentences, as in: S S John e smokes xe, ty andS S John e VP xe, ty loves xe, xe, tyy Mary e Fill in the empty boxes with appropriate semantic types. 2 Sentential Conjunction: Denotation Let us analyze the connective and based on the logical connective ^ from Propositional Logic. Recall from PLIN0001 that (ϕ ^ ) is true (= denotes 1) if both ϕ and are true (= denote 1), and is false (= denotes 0) otherwise. We can represent this using a truth-table as follows. ϕ (ϕ ^ ) TT T TF F FT F FF F The denotation of and then should apply to two truth-values (one at a time) and return 1 iff both of the truth-values are 1. Write down the entry: (1) For any model M, M vandSw = (Note that this meaning will not depend on M) 1 3 VP Conjunction: Semantic Types The same word and can be used to conjoin VPs as in: S John e VP swims xe, ty andIV VP xe, ty loves xe, xe, tyy Mary e Fill in the empty boxes with appropriate semantic types. 4 VP Conjunction: Denotation (dificult) Let us now analyse the denotation of and for VP conjunction. The idea is that ‘DP VP1 and VP2’ is true iff both ‘DP VP1’ and ‘DP VP2’ are true. That is, and combines with two type xe, ty functions (one at a time, as before) and returns another type xe, ty function that takes an entity x and returns 1 iff both of the input functions map x to 1. (2) For any model M, M vandIVw = 5 Nominal Conjunction (open discussion) The same word and can apply to two proper names, as in John and Mary. Discuss how the above analysis could or could not be extended to such cases. 2.