Tag Archives: Modus ponens

Deductive Soundness and Validity (How to win arguments part 2)

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validity and soundness

In my last post, I explained what a deductive argument is and looked at conditional statements, which play a central role in deductive reasoning. Now I’d like to talk about soundness and validity. Being able to establish quickly whether a deductive argument is valid or not allows you to work out if you need to devote more time and energy to unpacking it. If it’s invalid, you don’t have to examine it any further; you can put it aside and focus your attention on other arguments. But if it’s valid, it’s time to start the arduous process of working out whether it’s sound (and, thus, whether you have to accept its conclusion).

But what do these terms mean? We use them in various ways in our every day life, but in the realm of critical thinking they have specific definitions. Validity refers to an argument’s form; a deductively valid argument is one that has one of two forms. Basically, the premises need to guarantee the conclusion. You may remember from my previous post that this is the definition of a deductive argument. So, when we’re saying that an argument is deductively valid, all we’re really saying is that it actually is successfully deductive. To determine whether it is or isn’t, we have to look at the conditionals.

A conditional deductive argument will always contain a conditional statement, some kind of follow-up statement, and a conclusion. An example of this would be: “If something is a green plant, it photosynthesizes. My cactus is a green plant. Therefore, it photosynthesizes.” The follow-up statement in this case is affirming the sufficient condition. This is one valid form of deductive argument, also known as a modus ponens argument. The other form of valid deductive argument is known as a modus tollens argument – this is an argument in which the second premise denies the necessary condition. An example of this might be “If something is a green plant, it photosynthesizes. My cat does not photosynthesize, therefore it’s not a green plant.” In each of these arguments, the premises guarantee the conclusion. If a conditional deductive argument denies the sufficient condition or affirms the necessary condition, the result is an invalid argument: “If something is a green plant, it photosynthesizes. My cat is not a green plant, therefore it doesn’t photosynthesize.” The way I was taught to remember this is to look for matching letters – affirming the sufficient and denying the necessary are the valid forms. It’s possible for an invalid argument to have only true premises and a true conclusion, as in the example just given – validity refers only to whether or not the conclusion is guaranteed by the truth of the premises.

What about soundness? A sound deductive argument is a valid deductive argument with only true premises (and, thus, has a true conclusion – if a deductive argument is shown to be sound, you can’t disagree with its conclusion). This is what people want to achieve when they construct a deductive argument. But it’s hard to establish whether an argument really is sound, because determining whether it’s premises are true can be difficult and may involve subjective reasoning. Note that validity and soundness are two different things; an argument can be valid without being sound (though it can’t be sound without being valid). Consider my first example in the paragraph above. Cacti are indeed green plants. And they do photosynthesize. But it’s not true that all green plants photosynthesize; dodder, a kind of parasitic plant, can have a green colour but does not photosynthesize. So the first premise and the conclusion are true, but the second premise is false. The argument is valid, but it isn’t sound.

Knowing how to identify valid deductive arguments can allow you to have far more fruitful discussions with people; you can get straight to the work of examining the premises of their valid arguments to determine whether they’re true, without wasting time on arguments that are structurally flawed. Hopefully this post will help you learn how to do so.