note
that the following section occasionally makes reference to a text often
used at Pierce, Hurley's A Concise
Introduction to Logic
One convention that I ignore is to
express general patterns with small letters (as you see in most texts)
and actual argument forms with capital letters. In the examples
below I use the capital letters P,Q,R,S with the understanding
they can be replaced with any other letter. Also, when we talk
about rules, we think of how more complex expressions can be used to
take the place of a single letter (we call these substitution
instances). For instance, for the rule of modus ponens we have the basic
pattern: P
-> Q, P |- Q MP Then we have unlimited substitution
instances, such as (P v Q) -> R, P v Q |-
R MP
or P -> (Q & R) , P |- Q &
R MP
Just as different textbooks use
different symbols to express relationships they also use different sets
of rules. The most common is what is called an intelim system
(short for "introduction and elimination") with these basic patterns:
Introducing a
conjunction: P,
Q |- P &Q (Conj)
Eliminating a conjunction: P & Q |- P
(Simp) Introducing a
disjunction: P |- P
v Q (Add)
Eliminating a disjunction: P v Q, ~P |- Q (DS) Introducing a conditional:
none
Eliminating a conditional: P -> Q, P |-
Q (MP)
Most texts also allow for eliminating
a conditional by having the MT pattern: P -> Q, ~Q |- ~P Do pay attention to
the fact that MP works only left to right and MT works only right to
left. We do not have a rule
for introducing
a conditional, but often we have the same result through using a
conditional derivation. Also, we can use the addition rule and
then make use of material implication to change the disjunction to a
conditional.
The additional rules in the Hurley
text: (P -> Q)
& (R -> S) , P v R |- Q v S (CD) note that although the destructive
dilemma is explained in Chapter 6, it does not make it into the list of
rules P -> Q, Q -> R
|- P -> R (HS)
Substitution or equivalence rules
(called axioms in our text) allow even just parts of an expression to
be changed (always keep in mind that inference rules apply only
to the main connective or operator).
Exchanging conditionals and
disjunctions: (P
-> Q) :: (~P v Q) Impl Changing the position of the parts of
a conditional: (P
-> Q) :: (~Q -> ~P) Trans Changing the order in a conjunction
or a disjunction: (P
& Q) :: (Q & P) Com (P v Q) :: (Q v
P) Com note that ordinarily
we allow this axiom to be part of how we apply the inference rules of
Conj, Simp, Add, and DS
For example: (A v B)
-> C, B |- C; we can use the Add rule to go from B to A v B
without having to first say B v A and then using the axiom Com Changing the grouping in a
conjunction or disjunction: [P & (Q & R)] ::
[(P & Q) & R] Assoc [P v (Q v R)] ::
[(P v Q) v R] Assoc Negating a conjunction or a
disjunction: ~(P
& Q) :: ~P v ~Q DM ~(P v Q) :: ~P
& ~Q DM Double negation: P ::
~~P DN note that ordinarily
we allow this axiom to be part of how we apply the inference rule of DS
For example: A v ~B ,
B |- A ; we can use DS right away without first having to say ~~B
with DN Reworking expressions
involving both
conjunction and disjunction: [P & (Q v R)] ::
[(P & Q) v (P & R)] Dist
[P v (Q & R)] :: [(P v Q) & (P v
R)] Dist Changing to or from a
biconditional:
(P <-> Q) :: [(P -> Q) & (Q ->
P)] Equiv
(P <-> Q) :: [(P & Q) v (~P &
~Q)] Equiv
These are
rules not included in my online text but do appear in the recommended
exercise below. A rule that is useful in
changing the
appearance of a more complex conditional: [(P & Q) -> R]
:: [P -> (Q -> R)] Exp And a final rule that is occasionally
very useful: P
:: (P v P) Taut
P :: (P & P) Taut
