Logic: Natural Deduction Rules (Handout)
Here A, B and C can be any formula you like. X and Y represents lists of formulas
(assumptions). We present all the rules in sequent form.
1. Rule of Assumption
A
WA
2. &-Introduction (&-I)
3. &-Elimination (&-E)
WA
YWB
X, Y W A & B
WA&B
XWA
X
X
4. Modus Ponendo Ponens (MPP)
WA→B
YWA
X, Y W B
X
WA
XWA∨B
X
X
WA
XWB∨A
X
WA
X W ¬¬A
WA→B
Y W ¬B
X, Y W ¬A
9. ¬¬-Elimination (DNE)
11. Disjunctive Syllogism (DS)
WA∨B
Y W ¬A
X, Y W B
X
X
12. Reductio Ad Absurdum (RAA)
W ¬A
WA∨B
Y, A W C
Z, B W C
X, Y, Z W C
X
X
10. Modus Tollendo Tollens (MTT)
X
WA→B
W ¬¬A
XWA
X
W B & ¬B
WB
7. ∨-Elimination
8. ¬¬-Introduction (DNI)
X, A
5. Conditional Proof (CP)
X, A
6. ∨-Introduction (∨
∨-I)
WA&B
XWB
X
.
WA∨B
Y W ¬B
X, Y W A
X
.
A Strategy for Constructing Natural Deduction Derivations
Unlike the method of truth trees, there is no (simple) method for constructing a derivation
of a valid sequent. Constructing derivations requires a certain amount of imagination
and creativity! (There is, in fact, a method, but it requires listing all possible
derivations in a mechanical way, and then “selecting the one which works”. This is very,
very inefficient, but possible.)
However, Paul Tomassi in his textbook Logic presents (pp. 107-8) the following strategy,
which he calls The Golden Rule:
You are presented with a sequent of the form:
A1, …., An : B
where A1, …, An are the premises and B is the conclusion.
Golden Rule:
1. Is the main connective of the conclusion the conditional →? If so, apply the
strategy for CP. I.e., assume the antecedent and try to derive the consequent.
If not, ask:
2. Is the main connective of one of the premises the disjunction ∨? If so, apply
the strategy for ∨-E. I.e., assume each disjunct separately, and try to derive the
conclusion B.
If not:
3. Try RAA. I.e., assume the negation of the conclusion and attempt to derive a
contradiction.
Tomassi adds,
Never lose sight of the fact that each and all of the above strategies can work
together in a single proof. I.e., the pursuit of an overall strategy may necessitate a
sub-proof which itself requires a different strategy. Hence, apply the Golden Rule
at the outset to identify an overall strategy, and then reapply as necessary
throughout the process of proof construction.
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