Subsection 3.2.4 Contradictory Premises
What happens if we choose premises that aren’t just wrong (i.e., they don’t correctly describe the world)? What happens if they actually contradict each other? It turns out that, if we do that, in even one case, we’ll be able to prove any conclusion we can come up with.
To see why this is so, let’s abstract away from any particular premises that we might choose. Let’s consider:
an arbitrary premise we’ll simply call p , and
some arbitrary conclusion we’ll call q .
Now suppose that we add the premise ¬ p . We know that it’s not possible for both p and ¬ p to be true. (Recall Aristotle’s Principle of Non-Contradiction.) But suppose that we claim that they are. What happens? Let’s now try to prove q from our two (contradictory) premises. To do this we must show that this is a tautology:
( p ∧ ¬ p ) → q
Here’s the truth table:
| p | q | ¬ p | p ∧ ¬ p | ( p ∧ ¬ p ) → q |
|---|---|---|---|---|
| T | T | F | F | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
We’ve proved q . And we’ve done it without any appeal to what p and q actually are and without any premises that connect p and q in any way. Whatever p is, p ∧ ¬ p is false. F → q is always true (from the definition of implies). So q is true.
For example if we assert both of the following claims:
The moon is made of green cheese.
The moon is not made of green cheese.
Then we can prove any of the following:
Elephants can fly.
Elephants cannot fly.
The king of France is a unicorn.
Notice why this is. Look at the next to the last column of the truth table. It’s all F ’s. Recall the definition of the expression p → q . It’s true whenever p is false (as well as whenever q is true). So if the first operand (the part before the →) is false, the entire expression will always be true.
Big IdeaWith contradictory premises we can prove anything. Beware.
Of course, in toy examples like this one, it’s easy to see that we’ve added contradictory premises and so our logic engine is of no use in attempting to determine truth. There’s a much more serious problem, however, if we have thousands, hundreds of thousands, or even millions of premises. This can happen when Boolean logic is used to solve real, practical problems like the design of computer circuits. In those cases, engineers must be very careful and they must exploit powerful design tools to guarantee the premises are not contradictory.
Exercises Exercises
Exercise Group.
1. Let’s give names to the following statements:
S: I should study Spanish today.
G: I should study Government today.
H: I will stay at home today.
Assert the following premises:
[1] H → ( S ∨ G ) If I stay home today I should study Spanish or Government.
[2] ¬ G I am not going to study Government today.
[3] H I will stay at home today.
We wish to prove that I should study Spanish today.
1.
(Part 1) We need to show that the claim that the premises imply the conclusion is a tautology. Which of the following statements is that claim:
( H → ( S ∨ G )) ∧ ¬ G ∧ H ∧ S
(( H → ( S ∨ G )) ∧ ¬ G ∧ H ) → S
(( H → ( S ∨ G )) ∨ ¬ G ∨ H ) → S
(( H → ( S ∨ G )) ∧ ¬ G ) → S
2.
(Part 2) We will use a truth table to prove this claim. As one builds a truth table, there are sometimes choices about what intermediate expressions to make explicit columns for. But some expressions would be useless. Which of these would get us nowhere in building the truth table that we need?
H → ( S ∨ G )
¬ G ∨ H
S ∨ G
¬ G ∧ H → S
3.
(Part 3) Use the Truth Table app to build the table that proves the claim.
Exercises Exercises
Exercise Group.
2. Let’s give names to the following statements:
C: I will serve cake at my party.
P: I will serve pie at my party.
B: I will buy a cake.
I: I will buy a pie.
M : I have money.
Assert the following premises:
[1] C ∨ P I will serve cake or pie at my party.
[2] C → B If I serve cake, I will buy a cake.
[3] B → M If I buy a cake, I have money.
[4] ¬ M I have no money.
We wish to prove that I will serve pie at my party.
1.
(Part 1) We need to show that the claim that the premises imply the conclusion is a tautology. Which of the following statements is that claim:
(( C ∨ P ) ∧ B P ∧ ¬ M )→ P
(( C ∨ P ) ∧ ( C → B ) ∧ ( B → M ) ∧ ¬ M ∧ ¬ B ) → P
P → ¬ C
(( C ∨ P ) ∧ ( C → B ) ∧ ( B → M ) ∧ ¬ M ) → P
2.
(Part 2) We will use a truth table to prove this claim. We know that the number of rows in our truth table grows as the number of propositional variables grows. We’ve defined five variables in this problem. But we don’t have to enter into the truth table any that aren’t involved in the proof. How many of the variables that we’ve defined do we actually need to use to do this proof?
2
3
4
5
3.
(Part 3) How many rows will the truth table have?
4
8
12
16
20
4.
(Part 4) Show the truth table that proves our claim.
Exercises Exercises
Exercise Group.
3. Let’s give names to the following statements:
N: It’s raining.
C: It’s clear.
U : Unicorns are purple.
Assert the following premises:
[1] N It’s raining.
[2] C → ¬ N If it’s clear, it’s not raining.
[3] C It’s clear.
1.
(Part 1) Using these premises, we wish to prove that unicorns are purple. Which of the following statements, if it’s a tautology, proves the claim:
( N ∧ ( C → ¬ N ) ∧ C ) → U
( N ∨ C ) → U
( N ∧ ( C → ¬ N ) ∧ C ) → ( U ∨ ¬ U )
( C → ¬ N ) ∨ N ) → U
2.
(Part 2) Write out the truth table. Can you prove the claim that unicorns are purple?
( N ∨ C ) → ¬ U
( N ∧ ( C → ¬ N ) ∧ C ) → ( U ∨ ¬ U )
( N ∧ ( C → ¬ N ) ∧ C ) → ¬ U
( ( C → ¬ N ) ∨ N ) → ¬ U
3.
(Part 4) Write out the truth table. Can you prove the claim that unicorns are not purple?
4.
(Part 5) Suppose that we want to delete premises until it’s no longer possible to prove that unicorns are purple. (After all, we don’t actually have any premises that say anything about unicorns.) Which of the following deletions will do the job:
Deleting N is the only thing that will accomplish the task.
Deleting C is the only thing that will accomplish the task.
Deleting C → ¬ N is the only thing that will accomplish the task.
Deleting any one of the premises will accomplish the task.