33.3: Multiple Claims

When we have more than one claim, we can use truth tables to compare them, and in doing so determine more complicated relationships.

Consistency

The first relationship we will test for is consistency. Consistency is a pretty simple relationship. It means the claims can be true at the same time. As with tautology, contradiction and contingency sometimes it will be obvious that two claims are consistent. Unrelated claims are always going to be consistent for instance. Troy likes apples and Godzilla is the king of monsters are consistent because they have nothing to do with each other, and we don’t need a table to test that. Still, there will be times where claims are complex (or when you want to compare a large number of claims) and tables can be very useful in those cases. For simplicity’s sake, we will focus on pairs of complex sentences, but there is no limit to the number of claims you could test for consistency at the same time.

In terms of what the tables should look like, two claims are consistent if there is any row in which both the claims are true at the same time. Let’s look at the table comparing ‘P v Q’ and ‘P & Q’.

Both claims are true on row 1, so these claims are consistent.

Now let’s look at an example where the claims aren’t consistent. Below is the table comparting ‘P & Q’ and ‘~P & ~Q’.

Notice there is no line where both claims are true, so the claims are not consistent. When claims are not consistent, we say they are inconsistent, and to reiterate, this means that they can’t both be true at the same time.

Equivalence

We can also use tables to compare statements to determine if they are logically equivalent. When claims are logically equivalent, they both contain the same information. You can think of it as the two statements saying the same thing (even if they don’t look like they do). To test for equivalence, we make a truth table and again put both statements on it. If the statements have the same truth value on all lines, then they are logically equivalent. The following truth table shows that ‘~(P v Q)’ and ‘~P & ~Q’ are logically equivalent:

The following truth table shows that ‘P v Q’ and ‘P & Q’ are not logically equivalent, because the results on the table differ on rows 2 and 3.

Exercises

Construct truth tables for the following sets of statements to test for consistency and logical equivalence.

1. P v ~Q ; P → Q
2. ~(P v ~P) ; P & ~P
3. ~(Q v P) → P ; ~Q
4. P v [P & (Q → R)] ; P v (Q v R)
5. (P → Q) & (Q → P) ; P ← → Q
6. P v ~P ; P v Q
7. P ← →(Q ← → R) ; P ← → R
8. P← →Q ; Q← → P
9. ~(P v ~Q) ; P → Q
10. P v ~(P ← → Q) ; (P & ~Q) v (~P & Q)
11. P → Q ; Q → P
12. P → Q ; ~(Q → P)
13. ~ (P ← → Q) ; (~P & Q) v (P & ~Q)
14. ~ (~P v ~~Q) ; P & ~Q
15. P ← → (Q → ~R) ; ~P v ~Q
16. P v (Q v R) ; (P v Q) v R
17. P & (Q & R) ; (P & R) & Q
18. P & (Q v R) ; (P & Q) v R
19. P & (Q v R) ; (Q v R) & P
20. ~(P → Q) v [P v (~Q → P)] ; ( P ← → Q) v [(P v ~P) & (Q v ~Q)]

Selected Answers

1. P v ~Q ; P → Q

These claims are consistent as they agree on rows 1 and 4, but they are not equivalent as they disagree on lines 2 and 3.

8. P← →Q ; Q← → P

These claims are equivalent as they agree on all rows. All equivalent claims are consistent so they are also consistent.

14. ~ (~P v ~~Q) ; P & ~Q

Before even doing a table in this case we can use the negation manipulation rules to make things simpler for us. ‘~ (~P v ~~Q)’ becomes ‘P & ~Q’. Now we could do a table, but we turned the first claim into exactly the same thing as the second one, so we know they are equivalent.