My semantics professor uses functions to teach her class. She wrote the following sentence and three examples in one of her slides.
[[smile]] is a function that takes something, let’s call it x, and gives back T iff x smiles.
(note that c@ denotes 'any given context'; the [[ ]] aren't really important, but they denote the truth condition of the utterance written between them. So [[mi casa está en llamas]] is true, iff my house is infact on fire)
- F :{x: x is an entity in c@} ---> {T, F}
- [[smiles]]:{x:x is an entity in c@} ---> {T,F}
- [[smiles]]^c@ (x)=1 if x smiles in c@
The first two examples seemed questionable to me: She calls them functions, but makes a relation from the element x to the elements T and F, but functions are 1:1 relations. But, I'd bet on her having a better understanding of what she teaches than I do.
The last example just seems odd: I've never seen a function notated that way; I've never seen a function that explicitly depend on the state of affairs in the real world. However, it doesn't seem that much different than writing (∃x)(Ex ∧ Sx), where E denotes an entity in a given context, and S denotes [[smiles]].
Are any of these expressions properly notated functions?
Can you, and if so, how would you, notate a function whose output depends on the state of affairs in the real world?
Thank you,