Does the order matter with and statements?

Question

I’m trying to evaluate the validity of the statement p implies q, r implies p, q implies r, therefore p.

When I did the truth table, I got all true values, and the statement also seemed valid by the law of syllogism. However, the answer key states that the statement is invalid.

The text I’m using rephrased the propositions as a long “and” relation. I figured that since it should be commutative, I could use the law of syllogism. I’m not sure whether the order matters though.

What am I missing in this picture? Why is this statement invalid?

Answer 1

Your truth table could look like

p  q  r  p⟹q q⟹r r⟹p all3⟹ 
T  T  T    T    T    T      T
T  T  F    T    F    T      F
T  F  T    F    T    T      F
T  F  F    F    T    T      F 
F  T  T    T    T    F      F
F  T  F    T    F    T      F
F  F  T    T    T    F      F
F  F  F    T    T    T      T

Reduce this to the cases where $p \implies q$ and $q \implies r$ and $r \implies p$ to get

p  q  r  p⟹q q⟹r r⟹p all3⟹ 
T  T  T    T    T    T      T
F  F  F    T    T    T      T

to see that this does not always lead to $p$ being true, and so the three do not imply $p$

Answer 2

Sure, you can use the law of syllogism, but all that that gets you is further conditionals like $p \to r$, or $r \to q$. And using all three you can get $p \to p$ .. but that is not the same as $p$ either. Indeed, using the Law of Syllogism you cannot get $p$, $q$, or $r$ by themselves.

And that's for good reason: the argument is indeed invalid. If $p=q=r=False$, then all premises are $True$, but the conclusion is $False$.

And finally, no, the order of the and statements does not matter. As you say, the conjunction is commutative. So, swapping those around will not suddenly make this into a valid argument. No matter how you order the premises, the conclusion does not logically follow.

Answer 3

Your truth table missed a last column of evaluating $P$ under all conditions.

Of course when $P$ is false it is false regardless of what you have in the previous colums.