Thread: critical rows - what if there are none?

Why is an argument valid if there are no critical rows?

I know that if all premesis' are true (this forms a critical row), then the conclusion must be true to make the argument valid.
I understand that if all premesis' are true, and a conclusion is false, then the argument is invalid.

but why is an argument valid if there are no critical rows?

Originally Posted by dunsta

Why is an argument valid if there are no critical rows?

I know that if all premesis' are true (this forms a critical row), then the conclusion must be true to make the argument valid.
I understand that if all premesis' are true, and a conclusion is false, then the argument is invalid.

but why is an argument valid if there are no critical rows?

I believe it's the same reason that: suppose p is a contradiction; then "p implies q" is true, regardless of the truth of q. I find the reasoning is a little easier to see/accept when we write "if p then q" as opposed to "p implies q." The reasoning is: since there is no case in which p is true, thus there is no case in which "if p then q" is false, therefore "if p then q" is true no matter what q is.

Does this seem to answer your question? If I've missed something, I might not be the right one to provide a full answer.

thanks undefined
I understand the reasoning you outlined, but as for seeing how it relates to there being no critical rows, then the arg is valid. I guess this is something I will have to remember rather than understand why.

Thanks for the help.

Undefined gave the reason why. Look at it this way: Suppose there are no critical rows. Then the premises are always false, since a critical row corresponds to a situation in which the premises are all true. But if the premises are always false, then conditional is always true. When your premises are just one proposition, it's of the form B --> A. When your premises are many, it's of the form (B & C & ... & Z) --> A for however many premises you have. You know from the basics of the truth table that, whenever B is false, the whole conditional is true. Likewise, if (B & C & ... & Z) is always false, then the conditional is always true. But if there are no critical rows, then (B & C & ... & Z) is always false, thus the conditional is always true.

Think of a valid argument as being one in which

there are no possible circumstances in which all the premises are true and the conclusion is false.

Then reason this way:

If there are no possible circumstances in which all the premises are true, then there are no possible circumstances in which all the premises are true and the conclusion is false. So any argument that has premises in which there are no possible circumstances in which all those premises are true is a valid argument.