Valid and Sound Arguments

This is the first post I am making in what will hopefully become a long and detailed series of posts about how to think more clearly about difficult questions. Since we ought to try to think clearly in every domain of life, we must begin the discussion at the broadest level, with the most important things to understand about clear and logical thinking. The obvious place to start here is the idea of a philosophical argument.

Defining ‘Argument’ and Related Terminology

First things first, understand that this is not an argument in the popular definition of two people bickering back-and-forth about some topic. This is nothing like arguments you see on the internet, or arguing with your parents or siblings. There need not be any anger at all behind a philosophical argument, just as there doesn’t have to be (and effectively never is) any anger behind a mathematical argument. When I use the word argument in the context of a philosophical argument, all that I mean is an effort to systematically put together a collection of information to draw some sort of meaningful conclusion.

A philosophical argument can come in a variety of levels of detail, sophistication, and length. However, since at its heart all we mean by an argument is an effort to combine some information to draw a meaningful conclusion, any philosophical argument worth its salt can be boiled down into a format called a syllogism – which is more or less just a fancy way of saying a bullet-by-bullet presentation of the pieces of information that are relevant to the argument. In a syllogism, the initial pieces of information are called premises, and the culminating new information drawn from these facts is called the conclusion. Most often, a philosophical argument will consist of a syllogism along extended discussions of why each premise should be considered to be true, and these discussions are called defenses of the premises. Here is a basic example of a philosophical argument that illustrates these definitions.

  1. If it is raining right now, the ground is wet.
  2. It is raining right now.
  3. Therefore, the ground is wet.

The collection of statements 1-3 is a syllogism. Statements 1 and 2 are premises, and statement 3 is the conclusion. To make the argument more detailed, you could make a defense of Premise 1 by convincing your listener that water makes things wet, and you could defend Premise 2 by looking out a window, walking outside, or checking a weather app.

Also important in discussion of arguments are objections to an argument. An objection to an argument is basically just anything that is meant to show that the conclusion in the argument is actually invalid. For instance, in the above argument, you could object to Premise 1 by noting that ground inside of a building or underneath a shelter won’t become wet when it is raining, or you can object to Premise 2 by showing your friend that there are no clouds in the sky. These considerations would, if they are accepted as true, make the conclusion “Therefore, the ground is wet” not necessarily true.

Objections can come in many forms, and methods of objection are a topic for later discussion. But before we can embark on that discussion, we first must lay out clearly what a philosophical argument actually has to accomplish in order to be considered as success. There are two basic criteria that arguments aim to satisfy – an argument must be valid and sound before it should be trusted as correct.

What is a Valid Argument?

Of the two conditions mentioned, validity is weaker (in the sense that being sound requires validity). An argument is valid if it is the case that, if we take as absolute truth every premise, then we cannot help but affirm the truth of the conclusion. Sometimes, the term valid is instead replaced with logically valid, because what we mean is that we can use the rules laid down by the formal theory of logic to move us from premises to conclusions. The argument about rain I presented earlier is logically valid in this sense – it has the format

  1. If A, then B
  2. A
  3. Therefore, B

In the philosophy of logic, this form is given the title modus ponens – Latin for the mode of affirming. I’m not going to try to explain why this is accepted as logically valid – I quite frankly don’t know how to boil this down any more than it already is. If anyone reading this believes 1 and 2 but not 3, please do try to explain to me why. So long as A and B genuinely mean the same thing in all three statements, this is unavoidable.

Why does validity matter? Aren’t there other ways to know something than pure logic? Well, yes, sort of. It is true that very, very little can be known using pure logic alone. But the “impure” aspects of a philosophical argument should never, ever be found in the logical validity of an argument. Rather, uncertainty ought to arise out of the truth of a premise – which is where the idea of a sound argument comes in.

What is a Sound Argument?

In order to define a sound argument, I find it helpful to first show examples of arguments that satisfy validity but not soundness. This is an “argument” that Steve went to jail – but upon reading the argument it should be instantly clear that something is awry (which is the whole point):

  1. Everyone who goes to jail commits a crime.
  2. Steve was sent to jail.
  3. Therefore, Steve committed a crime.

Anyone who has studied logic will affirm that this argument is logically valid – if you accept as true both Premise 1 and Premise 2, then the conclusion that Steve committed a crime cannot be avoided. And yet, we all know there is something wrong here. As much as we would like Premise 1 to be true, it is not true. Sometimes, people who have not committed any crimes go to jail. So, Premise 1 is false.

This example shows what we mean by an unsound argument – which we ought to reject. Now, informally, a philosophical argument should be called sound if it is logically valid, is based upon truth and which genuinely leads a thinking person to truth. To further clarify this, here is a definition of a sound argument that I find quite reasonable:

An argument is sound if and only if it is logically valid, all its premises are in fact true, and its premises are more likely to be true than false in light of the evidence.

Now, I will provide some examples to show why I include all the points in this definition that I do. If the argument is not logically valid, then by definition I am perfectly within my intellectual rights to accept as true every premise and to still reject the conclusion – so logical validity is required. If there is any confusion here, I will provide examples later in this series of logically invalid arguments that sometimes pass as valid (google the genetic fallacy, the ad hominem fallacy, or the fallacy of composition for some examples). I hope that I do not need to defend here how important it is that our beliefs line up with the basic laws of logic (which is really what logical validity means).

The argument about jail already shows how an argument can be logically valid but that it might be based on a false premise. Interestingly, we might here note that if an argument is logically invalid, we cannot therefore deny the conclusion. In the jail argument, it is quite possible that Steve is actually in jail. It is quite possible that Premise 2 and the conclusion are both correct. The important point here is that Premise 1 is not correct, so the argument is not valid because you cannot deduce the conclusion from the premises – since one of them is false, you can’t use it to meaningfully learn anything by this method.

So, this example shows why in my definition of a sound argument I require that all the premises be actually true. However, there is yet another way that an argument can be unsound, which is more complicated. To see what I mean, suppose that we go back in time 1000 years – to the time when everyone believed that the Sun revolved around the earth. People believed this, among other reasons, because no evidence had yet been produced that the earth actually revolves around the Sun. Now, in this situation, suppose I propose the following argument:

  1. If the Earth revolves around the Sun, the Sun would appear to rise and set in the sky.
  2. The Earth revolves around the Sun.
  3. Therefore, the Sun appears to rise and set in the sky.

To a modern reader, this is a sound explanation of why the Sun appears to rise and set (albeit massively simplified – of course you’d want to give a visual or mathematical demonstration of how Premise 1 actually works). However, to a person living 1000 years ago, Premise 2 will seem very highly unlikely in light of the evidence that they had available to them. So, the man living 1000 years ago need not believe that the conclusion is true because of this argument – they can instead believe that the Sun appears to rotate around the Earth because it actually does – because to this man, that is where all his evidence points.

It is for this reason that in discussing whether an argument is sound – roughly speaking, whether it ought to convince intelligent and informed people of its conclusions – we have to take into account what sorts of evidence are available to us. This is why in the definition of soundness that I give, I specify that every premise must be more likely than not given the evidence. To be fair here, you could require a standard other than “more likely than not” – for instance, in a court of law very often this standard is increased to “beyond a reasonable doubt” because of the horrible consequences of a false conviction. But despite the potential for changing the “degree of likelihood” we demand of our evidence, I find the “more likely than not” approach most reasonable.

Conclusion

From this post, we have defined the minimal criteria that an argument should meet in order to be convincing. In summary, in order to convince your listener of a point, your line of reasoning – your argument – should line up with the laws of logic, it shouldn’t rely on any false premise, and all premises in your argument should be supported by evidence. In the future, I will spend more time elaborating on how to properly approach the premises of an argument and evidence for each premise carefully and even mathematically.

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