Logical Validity v. Soundness

The word logic is derived from the Greek word logos. Logos has several nuanced meanings, but the one with which I am presently concerned pertains to deductive argument. Logic is simply the rules by which one must abide in order to present a valid, deductive argument and nothing more. In the political environment in which we live, one in which the vicious polemic is king, this is sometimes forgotten. However, logic is of paramount importance to anyone either looking to hire a blogger or hoping to be hired as a blogger. Deductive reasoning is what distinguishes a thoughtful post from a derelict jeremiad.

Deductive Validity

Arguments are made up of premises and conclusions. By using the basic rules of logic, one follows the premises of an argument in order to arrive at a conclusion. When the rules are followed, the argument is valid. Put in the overly formal language of my old textbook, “The Logic Book”:

An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false. An argument is deductively invalid if and only if it is not deductively valid. (p. 11)

For example, if Max is taller than Sandy, and Sandy is taller than Bill, then Max is taller than Bill. It can just as easily be thought of in numbers. Think of Max as 3, Sandy as 2 and Bill as 1.

3 is greater than 2.

2 is less than 3 and greater than 1.

3 is, therefore, greater than 1.

If this sounds a lot like geometry, then you’re getting the message loud and clear. From the days of Euclid and Pythagoras up until today, logic, geometry and mathematics have been known to be inalienable from one another.

Deductive Soundness

There’s one catch, though. A deductively valid argument is not necessarily true. By definition, validity simply means that, if an argument is constructed without false premises, it will always yield a true conclusion. That “if” in there is key.

Take the above example. Just because the premises state that Max is the tallest of the three doesn’t make it so. Perhaps I mistook Max for Bill. Perhaps there I had something vindictive in mind. Whatever the case may be, it’s not important. What is important in assessing the validity of the argument is its structure, and, in this case, the structure is impeccable.

When assessing the soundness of an argument, however, the truth does come into play. If it turns out that Max is not the tallest of the three, then the above example is what is known as an unsound argument. This is because the relations between the three people are not accurate. To once again refer to the stuffy language of the textbook:

An argument is deductively sound if and only if it is deductively valid and all its premises are true. An argument is deductively unsound if and only if it is not deductively sound. (p. 12)

The easiest way to remember the difference between deductive validity and soundness is to think of validity as the structure of the argument takes—If A, then B; A; therefore, B. Soundness, on the other hand, is the substance of the argument. If one cannot present the former, then the latter won’t be taken seriously.