1 1 9B a s i c P r o p o s i t i o n a l C o n n e c t i v e sNow we can look at an argument involving conjunction. Here is one that
is ridiculously simple:
Harry is short and John is tall.
∴ Harry is short.
This argument is obviously valid. But why is it valid? Why does the conclusion
follow from the premise? The answer in this case seems obvious, but we will
spell it out in detail as a guide for more difficult cases. Suppose we replace
these particular propositions with propositional forms, using a different
variable for each distinct proposition throughout the argument. This yields
what we will call an argument form. For example:
p & q
∴ p
This is a pattern for endlessly many arguments, each of which is called a sub-
stitution instance of this argument form. Every argument that has this general
form will also be valid. It really does not matter which propositions we put into
this schema; the resulting argument will be valid—so long as we are careful to
substitute the same proposition for the same variable throughout.
Let’s pursue this matter further. If an argument has true premises and a
false conclusion, then we know at once that it is invalid. But in saying that an
argument is valid, we are not only saying that it does not have true premises
and a false conclusion; we are also saying that the argument cannot have a
false conclusion when the premises are true. Sometimes this is true because
the argument has a structure or form that rules out the very possibility of
true premises and a false conclusion. We can appeal to the notion of an argu-
ment form to make sense of this idea. A somewhat more complicated truth
table will make this clear:
Premise Conclusion
p q p & q p
T T T T
T F F T
F T F F
F F F F- Susan got married and had a child.
- Jane speaks both French and English.
- Someone who speaks both French and English is bilingual.
- Ken and Naomi are two of my best friends.
- Miranda and Nick cooked dinner.
- I doubt that John is poor and happy.
97364_ch06_ptg01_111-150.indd 119 15/11/13 10:15 AM
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