Tag Archives: Deductive reasoning

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


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.


Conditionals (How to win arguments part 1)



Being able to argue properly, with yourself and others, is an incredibly valuable skill. Being able to critically evaluate arguments is not only extremely useful in the workplace – it can help you make better life choices for yourself and the people around you. This skill is generally known as “critical thinking”, and it is a philosopher’s bread and butter. Now, there’s no point in denying that we all use our brains differently, and that some of us are naturally good at this kind of thinking. But anyone can get better at thinking critically by studying arguments, including those to whom it comes naturally. So, for your entertainment and edification, I’ve decided to put together a few posts on the topic.

Let’s start with deduction. A deductive argument, simply speaking, is one in which the premises (if true) guarantee the conclusion. In other words, if the premises of a valid deductive argument are true, the conclusion can’t possibly be false. An example of this could be; “If I don’t study for my philosophy exam on wednesday, I will not get a good mark on it (first premise). I’m not studying for my philosophy exam on wednesday (second premise), therefore I will not get a good mark on it (conclusion)”. As you can see, if the premises of this argument are true, the conclusion must also be true. Note that this doesn’t mean that the conclusion actually is true, because the premises might be false. Compare this to; “If I don’t study for my philosophy exam on wednesday, I will not get a good mark on it. I will study for my philosophy exam on wednesday, therefore I will get a good mark on it”. This is a bad argument, deductively speaking, because the premises don’t guarantee the conclusion. This is because the form of the argument is invalid. I’ll get to validity later, but for now I want to focus on conditionals.

Conditional statements (often shortened to “conditionals”) are statements like the first premise in the example arguments above. “If then” statements, essentially. Conditional statements contain two different conditions (hence the name): a necessary condition and a sufficient condition. To put it very simply, the sufficient condition is the one that comes after the “if”, and the necessary condition is the one that comes after the “then”. In the examples above, “not studying for my philosophy exam” is the sufficient condition and “not getting a good mark on it” is the necessary condition. The “if” and “then” are not part of the conditions. Knowing the difference between these different kinds of condition, and being able to identify them, is an essential skill when it comes to evaluating deductive arguments.

However, identifying them can be difficult, because conditional statements are often not given in the standard “if then” form. In such sentences, it’s possible to work out which is which by looking at the meaning of the statement (what is necessary and what is sufficient), but it’s also possible to translate it into “if then” form. Consider this statement: “only people who don’t study for their philosophy exam will get a bad mark on it.” This translates into: “if you get a bad mark on your philosophy exam, then you didn’t study for it.” The original sentence does not say that everyone who doesn’t study for their philosophy exam will get a bad mark on it; rather, it says that those who do get a bad mark will not have studied. This becomes a lot clearer when the sentence has been translated, because it’s easier to identify the conditionals.

There are various different ways in which you can go about translating conditional statements, such as memorizing certain key words and phrases (“only if”, for instance, usually precedes a necessary condition), but I’ve found that the easiest way is to use a conditional statement that you know is true (preferably one that’s very obviously true), and seeing how it would be rearranged in the form of the sentence you’re translating. I tend to use “if you’re a father, then you’re a parent”, which was the example given to me when I was learning how to do this. Consider the example sentence I gave you earlier. If I translate my “if then” sentence into the form it has, I get “only parents are fathers” (since the “if then” sentence is true, it must still be true when it’s rearranged – that’s how I know where to put the “father” and where to put the “parent” – “only fathers are parents” would be a false statement). I can now see that the necessary condition and sufficient condition have changed place, with the necessary condition coming first. All I have to do now is identify what the original conditions were (“not studying for your philosophy exam” and “getting a bad mark on it”) and lable them as necessary and sufficient.

Translating conditional statements like this is a skill, and as such you have to practice doing it. There’s a great little game on Khan Academy that let’s you do just this (don’t worry, it’s not too mathsy): https://www.khanacademy.org/math/geometry/logical-reasoning/e/conditional_statements_2

You can also find more information on conditional statements and deductive arguments on that page – I’ll be following up on this post with one on deductive validity in the near future. Until then, I hope you are well, and as always if you have any questions or suggestions please leave a comment below!