Chapter Seven, Tutorial One
The semantics for SL is given by truth tables. The idea, as presented in chapter two, is that meaning can be understood from truth conditions. For example, a conjunction's meaning is understood by knowing that it is true if and only if both its conjuncts are true. Similar definitions can be given for PL.
For example, suppose we are interested only in 18th century US presidents. (Remember? There were only two. John Adams was the second, lest someone forget.) So, we may take the following interpretation:
universe of discourse: George Washington and John Adams
Similarly for all the connectives, we take over the truth conditions from SL. The only new part of the semantics is the definition of truth conditions for the two quantifiers.
Fortunately, it's almost as easy to think about semantics for the quantifiers. For example, "all US presidents of the 18th century are male", or '(^y)My', is true just in case all members of the universe of discourse satisfy the predicate 'M', that is, just in case everything in the group of two is male.
It's worth stressing that this definition of truth condition for '(^y)My' is just restating the obvious: the upside down-'A' just means "all" or "every". We can stress the obvious a bit more:
For the given interpretation, then, '(^y)My' comes to the same thing as 'Mg&Mj'.
Similarly, to say that someone (among our universe of discourse) was the second US president, '(%z)Sz', is simply to say that there is at least one member of the universe of discourse which is an S.
For the given interpretation, then, '(%z)Sz' comes to the same thing as 'Sg v Sj'.
Why, then, do we have the quantifiers in addition to to the ampersand and wedge? The reason is that we often want to describe all members of the universe of discourse even when there are many of them, or few names. We have to be able to give truth conditions for a sentence like "all positive numbers are greater than zero". (You don't want to write out a long conjunction for this one!)
Now, before moving on to the next page, which of the following are true on the given interpretation? (universe of discourse: George Washington and John Adams, g: George Washington, j: John Adams, Mx: x is Male, Fx: x is the first US president, Sx: x is the second US president, Bxy: x was president before y.)