At least one of these things is true:
Ted is going to get a job or he won't have money for college.
Jt
v ~Mt
Only one of these things is true: Ted
is going to get a job or he won't have money for college.
(Jt
v ~Mt) & ~ (Jt & ~Mt)
Ordinarily we suppose that "or" is inclusive
(the sense we have when we say that you may take either cream or sugar
in your coffee), but when it is important to emphasize that "or" is meant
to be exclusive (the sense we have when we
say that you may have either coffee or tea) we have to express the thought
in a more complex way.
Do we always have to think about how "or" is meant to be used?
Not at all. You'll realize, for instance, that P
v ~P by itself rules out the possibility of both disjuncts
being true at the same time, so we have no need of a more complicated expression.
Logic is easy and it is fun. E
& F or F & E
Logic is easy if it is fun. F
-> E (not E -> F)
Logic is easy only if it is fun. E
-> F (not F-> E)
Conjunction (and) and disjunction (or) are commutative and associative. That means that the connectives & and v behave just like + or x in arithmetic: you can change their order or their grouping without changing the results.
Implication is not commutative: the order does matter. It is
not associative, since the wff p->(q->r)
does not give us the same truth table as (p->q)->r.
Look at the tables below (and see why with three variables we have to double
the number of lines from four to eight).
| p | q | r | p->(q->r) | (p->q)->r |
| T | T | T | T | T |
| T | T | F | F | F |
| T | F | T | T | T |
| T | F | F | T | T |
| F | T | T | T | T |
| F | T | F | T | F |
| F | F | T | T | T |
| F | F | F | T | F |
In working with conditionals it is crucial that we symbolize correctly. A sufficient condition (what we hear with "if") becomes the antecedent and is to the left of the connective, regardless of the word order in an English sentence. A necessary condition (what we hear with "only if") becomes the consequent and is symbolized to the right of the connective.
It's false that if logic is fun then it is
not easy. ~(F -> ~E)
If it is false that logic is fun then it is
not easy. ~F -> ~E
Negation (not) involves a unary connective with the tilde (or curl).
This means that we have to be careful about the scope
of the negation. The first sentence involves denying the entire conditional
(and so it is equivalent to saying we can have logic be both fun and easy).
The second sentence involves only a false antecedent.