From $A⇒B$ and $B⇒C$ can we conclude $C$?

Question

Is the following valid:

Premises: If Bob eats breakfast, he won't eat lunch.
If Bob doesn't eat lunch he will have an early dinner
Conclusion: Bob will have an early dinner

I think this is valid. The premises are in the form $A ⇒ B, B ⇒ C$. They are telling us that $C$ is true. Then it follows that we will never have a true premise and false conclusion. Is that accurate?

score 6 · Answer 1 · answered Nov 07 '21 at 20:30

6

It's invalid because we aren't given that Bob actually eats breakfast. A valid conclusion would be $A\implies C$. That is, if Bob eats breakfast then he will have an early dinner.

answered Nov 07 '21 at 20:30

Karl

11,446

score 5 · Answer 2 · answered Nov 08 '21 at 00:10

Then it follows that we will never have a true premise and false conclusion.

It does not follow.

A conditional statement is valued as true when either the antecedent is false or the consequent is true.

So it is possible to value $B\to C$ and $A\to B$ as true, when $C$ is false .

score 2 · Accepted Answer · answered Nov 08 '21 at 02:52

Here's an indirect analysis:

If your original argument is indeed valid, then adding the third premise "Bob won't have an early dinner"—noting that we now have a consistent set of three premises—will not introduce a contradiction.

Therefore, by contrapositive, you original argument must have been invalid.

From $A⇒B$ and $B⇒C$ can we conclude $C$?

3 Answers3