🚀 go-pugleaf

In article <rkaivola.718902434@mits> rkaivola@mits.mdata.fi (Risto Kaivola) writes:

     Exactly how should one formalize the English notion of 'only if'
   in the language of the propositional calculus?
     That is, given the sentence
   "Tom will visit us only if we invite him.",
   is the correct formalization

   A = "Tom will visit us."

   B = "We invite him."

   A  -->  B

Yes.

Think about it this way: When is "Tom will visit us only if we invite
him" false?  It's certainly false if Tom visits without being invited,
i.e., if A is true and B is false.  If the sentence is true, any of the
other cases -- Tom is invited and visits, Tom is invited but doesn't
visit, and Tom is not invited and doesn't visit -- is possible.  Well,
those are precisely the truth conditions for A -> B.  So if you're going
to treat "Tom will visit us only if we invite him" as a truth-functional
compound of "Tom will visit us" and "We invite him", it's simply a
conditional.  (Of course some, including myself, would argue that
attempting to treat this as a truth-functional compound is not good
enough for some purposes.  For example,

    Tom will visit us only if we invite him.

might be thought of as indicating a general disinclination of Tom to
visit uninvited, making this something like a universal generalization
of a conditional rather than a conditional.  But this is beside the
point, which is that "only if" corresponds to -> in the same way that
"if" corresponds to <-.)

                                                        -- rar