Material implication

Symbolic Logic: How to
Use Material Implication
Michael Potts, Ph.D.
Professor of Philosophy
Methodist University
Fayetteville, North Carolina

 In material implication, there must either
be a disjunction as a part or whole of a
line, or a conditional statement as a whole
or part of a line. The rule is set up as
follows:
 Material Implication (Impl.)
p  q
p  q

 Material implication is a rule of
replacement, meaning that the
statements on each side in the rule are
logically equivalent—you can replace one
with the other. If you did a truth table
with the formula (p  q)  (p  q), the
result would be a tautology. This implies
that:

◦ 1. Material implication works both directions;
that is, if you have on a line in a proof p  q,
you can derive p  q, and if you have on a line
in a proof p  q, you can derive p  q.
◦ 2. As with all rules of replacement, material
implication can be applied to either the entire
line at the level of the main connective or to
part of a line. For example:

Material implication applied to a whole line:
1. A  B
2. A  B 1, Impl.
Material implication applied to part of a
line:
1. (A  B)  (C  D)
2. (A  B)  (C  D) 1, Impl.

In all cases of material implication, the left
side of the line or part of a line to which
you apply it is reversed when it comes to
the  sign, and either the  is changed to a
 or the  is changed to an . Various
examples are on the next slide:

A  B A  B A  B A  B
A  B A  B A  B A  B
A  B A  B A  B A  B
A  B A  B A  B A  B

Note more complex structures; for
example:
1. (A  B)  (C  D)
2. (A  B)  (C  D) 1, Impl.
1. [(A  D)  (R  S)]  [P  (Q  R)]
2. [(A  D)  (R  S)]  [P  (Q  R)]
1, Impl.

◦ To set up hypothetical syllogism (answer is on
the next slide):

Answer:
1. A  B
2. B  C /A  C
3. A  B 1, Impl.
4. B  C 2, Impl.
5. A  C 3, 4, HS

To set up modus ponens (answer is on the next
slide):

Answer:
1. (A  B)  (C  D)
2. A /C
3. A  B 2. Add.
4. A  B 3. Impl.
5. C  D 1, 4, MP
6. C 5, Simp.

Answer:
1. A  B
2. B  A /A
3.  A   B 1, Impl.
4. B  A 2, Impl.
5. A  A 3, 4, HS
6. A  A 5, Impl.
7. A 6, Taut.

Answer:
1. A  B
2. A  B /A  B
3. A  B 1, Impl.
4. B  A 2, Com.
5. B  A 4, Impl.
6. (A  B)  (B  A ) 3, 5, Conj.
7. A  B 6, Equiv.

Answer:
1. (A  B) /B
2. (A  B) 1, Impl.
3. A  B 2, DM
4. B 3, Simp.

◦ Answer:
1. (A  B)  (C  D)
2. (A  B) /D  C
3. A  B 2, DM
4. A  B 3, Impl.
5. C  D 1, 4, MP
6. D  C 5, Com
7. D  C 6, Impl.

Material implication

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Material implication