Let's look at the classic example from Aristotle:
All men are mortal, and Socrates is a man,
so Socrates is mortal.
What makes the argument work is the way in
which we make use of a common idea (being a man) that is present in the
two premises to arrive at the implied connection between Socrates and what
we can predicate of him. (To predicate
something of a person is to say that the individual has this or that characteristic.)
We are going to take this in two stages. First, we will look at the individual variables--naming someone and saying something about him--and then in the next lesson we will look at what we have to do to talk about groups of individuals. We do this by setting up a code that will use capital letters for the predicates and small letters for the individuals themselves. Let's start with these examples.
Fx: x is friendly
Gx: x greets people
Hx: x is happy
a: Al
b: Bob
Please note that Fx by itself is not a wff
FaHbA would represent
the idea that Al is friendly and Bob is happy.
FaHaA would have
Al be both friendly and happy.
FaFbA would tell
us that both Al and Bob are both friendly.
FNa tells us
that Al is not friendly (we never use "N" after the small letter)
We are using all the same signals as before,
and we will use the same set of rules for derivations with one important
addition: PQCN=PQNA
CN subs
(We could get this result with several steps this way:
PQCN
PNQON MI
PQNA deM
We use the CN subs rule as a shortcut.)
If Al is friendly then he greets people.
If he is friendly then he is happy. Therefore, Al is someone who
greets people and is happy.
FaGaC, FaHaC, Fa |- GaHaA
1. FaGaC
Ass
2. FaHaC
Ass
3. Fa
Ass
4. Ga
1,3 C elim
5. Ha
2,3 C elim
6. GaHaA
4,5 A int
Sometimes we are going to find that we cannot use a direct derivation, even though it will be obvious that we have a valid argument. An alternative is the use of an indirect derviation, which allows us to imagine (hypothesize) that the intended conclusion is false. In a valid argument, by definition, it would be inconsistent to have a false conclusion together with all true premises. Working out what follows from the original premises and the imagined false conclusion will lead us to a contradiction. When we have shown that, we are able to say that the opposite of the false hypothesis must in fact follow. To do this we work with what is called a subordinate derivation, indicated by indenting the lines (and being careful not to think any line in the indented group would be true otherwise).
If Al is friendly then he greets people.
If he greets people then Bob is happy. Therefore, if Al is friendly
then Bob is happy.
FaGaC, GaHbC |- FaHbC
1. FaGaC Ass
2. GaHbC Ass
3.
FaHbCN Hyp
4.
FaHNbA 3 CN subs
5.
Fa
4 A elim
6.
Ga
1,5 C elim
7.
Hb
2,6 C elim
8.
HNb 4
A elim
9.
HbHNbA 7,8 A int
10. FaHbC
3-9 hyp elim
Please note how we show the results of
testing the hypothesis and then "eliminating" it.
Indirect derivations will work for any valid argument. Often, though, they are longer than a direct derivation when that is possible, and often too they use the same steps.
Finally, let's look at something that may seem peculiar (you've already seen it in an assignment) but follows from our definition of a valid argument (a pattern that does not allow a set of true premises to be consistent with a false conclusion). When we have inconsistent premises, by definition we cannot have an invalid argument!
P,PN |- Q (using a direct derivation)
1. P Ass
2. PN Ass
3. PQO 1 O int
4. Q
2,3 O elim
P,PN |- Q (using an indirect derivation)
1. P Ass
2. PN Ass
3. QN
Hyp
4. PPNA
1,2 A int
5. Q 3-4 hyp
elim