Logic 104: Evaluating Arguments

Hey Everybody, Marcus here.

This is going to be the final video in my series on logic. In this video, we will explore how arguments are to be evaluated; namely, we will explore the concepts of validity and soundness. However, before we get into the subject of this video, let us once more go over sentences, propositions, and arguments so these concepts are fresh in our mind. Let us begin with sentences and propositions.

There is a difference between sentences and propositions. Sentences are the grammatically correct utterances we issue in the English language. A proposition or sometimes referred to as a statement, on the other hand, is what an unambiguous declarative sentence asserts. We also learned that no exclamation like Hi! can be a proposition. We learned that questions cannot be propositions. And finally, we learned that ambiguous sentences, such as sentences with indexicals cannot be propositions.

Now, let us review the structure of an argument.

An argument is composed of at least one or more premises. An argument has exactly one conclusion though you can construct many arguments with the same premises to derive multiple conclusions from them. The premises and the conclusion of an argument are always propositions. Pretty much no arguments you will encounter in the real world will be presented in the formal argument structure. As such, you will be tasked to extract the arguments from the various passages you encounter in the real world.

To help you extract argument you can use premise indicator words as well as conclusion indicator words. However, you will also encounter a great number of passages that in themselves do not contain an argument. There are broadly 5 types of passages which do not contain arguments. These are simple non-inferential passages which can further be broken down to warnings, pieces of advice, statements about beliefs or opinions, loosely associates statements, and report. The other types of non-argumentative passages are expository passages, illustration passages, explanations, and conditional statements.

Okay. You now know what propositions are and what arguments are. In logic we investigate, among other things, the so-called logical properties of propositions and arguments. In ordinary language, we do not distinguish sharply between those entities but it is crucial that you learn to distinguish them.

It is logical nonsense to say that an argument is true or false. The premises of an argument, which are always propositions, may be true. The conclusion of an argument, which is also a proposition, may be true. But an argument cannot be true.

It is likewise logical nonsense to say that a proposition is valid or invalid.
One way to understand why this distinction is needed is to remember that an argument involves an inference, a logical movement of sorts, from the premises to the conclusion. When we evaluate an argument, we evaluate how good the inference is. This evaluation is quite different from the evaluation of the premises. It is also important to remember that one can reason well, namely, validly, given false premises. Consider this argument:

P1: All stars emit light.
P2: Venus is a star.
C: Therefore, Venus emits light.

This is a logically valid argument. The argument, however, is not sound as someone reaches a false conclusion while reasoning validly from a false premise. After all, Venus is a planet, and not a star.
Now, we will learn a little bit more about validity and introduce another feature of arguments, namely, soundness, but before we can do this we need to understand one incredible feature of reasoning; its formal nature.

Consider the following innocently looking example:

P1: All borkalorks transmit AIDS.
P2: But no nerkakirks transmit AIDS
C: Therefore, no nerkakirks are borkalorks.

The conclusion that you have surely reached is that no nerkakirks are borkalorks. After all, it stands to reason that If all borkalorks transmit AIDS, and nerkakirks don’t, they can’t be borkalorks. You could reason out the right conclusion even though we pulled borkalorks and nerkakirks right out of our ass. And yet, and this is quite incredible, if you think about it, we could reason about borkalorks and nerkakirks!

This is all because reasoning is a formal affair. What matters to an argument is not so much the content of a proposition as its logical structure which is also called logical form. Arguments are valid, or invalid, in virtue of exhibiting a certain logical form. This is why you can reason correctly about things you have just made up.

Let us get an intuitive grasp of what logical form is. Let’s start by filling in the conclusions to these arguments:

P1: John will turn right or left.
P2: John did not turn left.
C: Therefore, John turned right

P1: Kay will have fruit or ice-cream.
P2: Kay did not have ice-cream.
C: Therefore, Kay had fruit.

P1: Tim will get a rabbit or a hamster.
P2: Tim did not get a hamster.
C: Therefore, Tim got a rabbit.

As before, you surely did not have any problems in drawing the right inferences. But in this case, you may also have noted that despite the fact that the arguments here are all different, they nonetheless share something. They have the same logical form. Let us write one of the examples in a more detailed fashion, so that we can explicitly see all the propositions involved.

First, we can look at the original argument as it has been presented so far. Next, we can present the expanded version of the argument. The expanded version converts the premises into explicit propositions. Now, let us use some color to mark the relevant segments of the premises and conclusion to highlight the structure more clearly.
As we can see in the form outlined version, there appears to be some color repeating itself. Let us continue this exercise and see where we get.

Look at the “Form Only” version of the argument. The different boxes stand for different propositions. Note that the same proposition must always go into the same colored box. What you see outside the boxes – the phrases ‘or’, ‘it is not the case that’ are so-called logical constants. It is the study of their behavior that is the proper task of a logical theory.

Because it would be hard for logicians to use differently-colored or differently-shaped boxes, they have adopted the convention of using so-called propositional variables, which simply name such boxes. It is accepted as a convention that propositional variables are written by means of the small letters of the alphabet starting with p, q, r, etc. The logical form of the above arguments can thus be written as it appears in the formal notation version of the argument.

This logical argument form that we have been using so far actually has a Latin name. It is called modus tollendo ponens, and it is better known as the disjunctive syllogism. It is important that you see how every version of argument fits this form. When an argument fits a certain logical form, we say that the arguments instantiates or exhibits this logical form. All of the versions of the argument we have shown so far instantiate the logical form called disjunctive syllogism. We can also say that the original 3 arguments we showed earlier, though containing different premises and conclusions, are all instances of disjunctive syllogism.

Now, there are many different logical forms out there that you can work with. Disjunctive syllogisms are just one example. However, what is important to note is that when we say that an argument is valid, we are saying that the argument follows some logical form. Even if the argument is complete nonsense, yet follows a valid logical form, the argument as a whole must be considered valid.

Argument instances are valid or invalid in virtue of their logical form. This means that an argument instance is valid if and only if its argument form is valid. This in turn means that all instances of a logical form will be valid. Now, this is quite an incredible fact given that there are an infinite number of instantiations of any given logical form one can make; in theory at least.

Now, I have said many times in many of my videos that if an argument is valid, someone who accepts the premises must also accept the conclusion. In other words, in a valid argument, the conclusion logically follows from the premises. It will pay to pause a little to think about what this ‘must’ means. You might think to yourself: “The hell with you Marcus, you can’t tell me how to think, I do what I want, whatever!”

Now, logic does recognize your right to refuse accepting the conclusion of a valid argument. However, this can only be done if you also refuse to accept one of the premises of that argument. If, on the other hand, you do accept the premises of a valid argument as true then you indeed must accept the conclusion of that argument; otherwise, you are retarded. And if you see the argument as valid, you will in fact have no trouble at all in seeing that someone who accepts the premises cannot but accept the conclusion.

Let us close our discussion of argument validity with a formal definition.
An argument is logically valid if and only if it is impossible for the conclusion of that argument to be false while the premises of that argument are true.

Let us now turn to argument soundness and unsoundness. Logically valid arguments need not have true premises. In fact, the relationship between truth and validity is a very complex one and I will not cover it in this video series. However, logicians have introduced a special term to cover those valid arguments that also have true premises. They have called such arguments “sound.” That is pretty much it. On the other hand, if one or more of the premises is false, then the argument is unsound though it may be valid.
Let us get a formal definition in place for soundness.

An argument is sound if and only if it is logically valid and all of its premises are true.
As far as argument construction and evaluation is concerned, one obviously ought always to strive for both validity and soundness. Validity is usually a lot easier to achieve though many people even fail at that. Defeating an argument follows the same general principle. If you can demonstrate that a given argument is invalid, namely, it does not follow a logical form, then you win. The content of the premises is no longer relevant if an argument is found to be invalid. However, if you find that an argument is valid, you are forced to demonstrate that one of its premises is false.

The premises of any argument are usually the conclusion to different arguments. In this way, you can look to demonstrating that the argument used to support a premise of another argument is valid or invalid, sound or unsound. This pattern repeats itself until you usually end up arguing over metaphysical issues.

Now, I will close this video and video series by providing an explanation of why logic works at all. We all understand the concept of gravity to one degree or another. We like to think that gravity is somehow just built into reality. Gravity is considered to exist external to the mind and not as a part of the human existential condition. In contrast, color may be argued to not exist external to the human existential experience even though we can say the light frequencies that cause our brains to display to us color are external. Now, Aristotle speculated that logic works and exists in the a similar manner as gravity does and not like color does.

Aristotle thought that logic, and specifically, his three laws of logic; namely, the law of non-contradiction, the law of excluded middle, and the law of identity, are built into reality itself. Reality, simply put, is logical as its foundation. If we take up this explanation, then it makes perfect sense why logic works. It works because reality exists and could not exist if it was not logical. When we reason, we are not actually performing some humanly invented activity. What we are doing is simply correctly looking at reality as it is. To not look at reality logically, in turn, is to not look at reality at all but onto some make believe humanly bastardized version of reality.

Though the Aristotelian view on logic’s relationship to reality has been challenged, I still believe it is correct.

Now, this video series is in no way an exhaustive look at logic. This series just scratches the surface. However, if you understand at least what I presented in this series and adopt it into your thinking, writing, and speaking, you will be way ahead of the crowd.

So, I hope you enjoyed this series, and thanks for watching!

Go team!

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