Formal Proofs, Subproofs, and Assumptions
PHIL 012 February 21, 2001
Overview
 Subproofs
 Disjunction Elimination
 Negation Introduction
 Rules for Assumptions
 Sample proofs
Subproofs
There are two circumstances (so far) when we will use a
subproof:
 Disjunction Elimination (Proof by cases)
 Negation Introduction (reductio ad absurdum)
Disjunction Elimination ( Elim.)
 You would use Disjunction Elimination when
1. You have a statement whose major connective is a
disjunction.
2. You are reasonably sure that what you are trying to
prove (either as a final or an intermediate step)
follows from either side of the disjunction.
 In order to use Disjunction elimination, you will
introduce two subproofs.
 Each subproof begins with the assumption that one of the
sides of the disjunction is true.
 Each subproof then utilizes that assumption together
with any steps from the parent proof to establish the truth
of the same statement.
 In order to make sure we use subproofs properly, we use
fitch notion (vertical and horizontal lines) to indicate the
scope of each subproof and to distinguish assumptions
from other steps.
AB
A



P
B



P
P
Assumption
Steps in subproof 1.
Assumption
Steps in subproof 2
Example
Given (P  Q)  ( R  S ), prove (P  R) using  Elim.
1. (P  Q)  ( R  S )
2. P  Q
3. P
 Elim 2
4. P  R
 Intro 3
5. R  S
6. R
7. R  P
8. P  R
9. P  R
 Elim 5
 Intro 6
Com 
 Elim 1, 2-4, 5-8
Negation Introduction ( Intro.)
 You would use Negation Introduction when you want to
prove a claim by showing that the assumption of its
negation leads to a contradiction.
 Proof by Negation Introduction involves the use of one
subproof.
 Your first step is to introduce as an assumption the
opposite of what you are trying to prove.
 Then, using steps derived from the assumption and
possibly the parent proof, you generate a contradiction.



P
previous steps in proof (if any)
P



Q  Q
Assumption
Steps in subproof
A contradiction
 Intro
 Note that there is a more limited form of contradiction
and a more general form.
 The limited form requires a contradiction of the form, Q  Q, which
is the opposite of a tautology
 The more general involves a set of unsatisfiable statements, such as
Cube(a)  Tet(a).
 The limited form always will be false in every language as it is logically
contradictory independent of meaning.
 The more general form is a function of the meanings of the terms in the
language.
Example
Given V  W and V, prove W  X.
1. V  W
2. V
/WX
3. (W  X)
4. W  X
5. W
6. V  W
7. (V  W)
8. (V  W)  (V  W)
9. W  X
 Intro 3-8
DeM 3
 Elim
 Intro 2, 5
DeM 6
 Intro 1, 7
Rules for Subproofs
 You may not use a step of a subproof outside of its scope.
 Once a subproof is complete no steps apart from its conclusion
(including its assumption) are usable.
 You may use steps from a parent proof at any time or the parent
of a parent, provided that the parent proofs are in the same
scope.
Sample Proofs
1. G
2. H
/ (G  H)  I
3. G  H
 Intro 1, 2
4. (G  H)  I
 Intro 3
1. (A  B)  C /  C  B
2. (A  B)
3. B
2  Elim
4. B  C
3  Intro
5. C  B
4 Com 
6. C
7. C  B
6  Intro
8. C  B
 Elim 1, 2-5, 6-7