In an expression without quantifiers
we work with two-letter expressions in which a capital letter
represents some characteristic (or predicate) while after it is a
small letter from anywhere in the alphabet up to the last three letters
(x,y, and z are variables, and in this course we use them only
following a quantifier)
Alice
is a student. Sa Alice is a good student. Sa & Ga Aice and Bob are students. Sa & Sb
Quantifiers tell us whether we are talking about everyone is a
particular set of individuals or only some of them.
Everyone is anxious. (x)Ax
Everyone is an anxious student. (x)(Ax & Sx)
All the students are anxious . (x)(Sx -> Ax)
Only the students are anxious. (x)(Ax -> Sx)
No one is worried. (x)~Wx
Not everyone is worried. ~(x)Wx
No teachers are worried.
(x)(Tx -> ~Wx)
Some teachers are worried. (Ex)(Tx & Wx)
other ways of talking about specific individuals that are
"translated" in the same way:
There are worred teachers.
There is a teacher who is worried.
There are teachers who are worried.
Many teachers are worried.
A few teachers are worried.
Any student who is not anxious will be relaxed. (x)[(Sx & ~Ax) -> Rx]
There are students who are not anxious but are relaxed. (Ex)[(Sx & ~Ax) & Rx]
A key thing to remember is that the letter used as the variable in a
quantifier "binds" that same variable inside the following parenthesis.
If every student is anxious then a few teachers are worried. (x)(Sx -> Ax) ->(Ex)(Tx & Wx)
or (x)(Sx -> Ax) ->
(Ey)(Ty & Wy)
Be careful to identify the main connective in an expression. (x)(Sx -> Ax)
the quantifier is the main connective ~(x)(Tx -> Wx)
the curl is the main connective (x)(Sx -> Ax) -> (Ex)(Tx & Wx)
the second arrow is the main connective
