Argumentation and Logic, part 1

We thought it would be good to do a brief primer on logic before moving forward with any other posts. This will be a series consisting of two posts, the first dealing with some of the basics of argument and propositional logic, and the second with informal fallacies. Obviously, these two posts will hardly even scratch the surface of the field of logic, but it is nonetheless an important foundation to have for any reasoned debate. In this age of internet trolls and drive-by commenting, these two posts alone will take the reader well past the average person’s understanding of logic.

In seeking truth, one of our most important tools is argument because it allows us to bolster our position and to convince others. If flaws are found, then we are forced to rethink our position, consequently strengthening or abandoning it. Defining the word "argument" is actually fairly difficult, and strangely, one of the best definitions comes from - of all places - Monty Python: "an argument is a collected series of statements to establish a definite proposition." More precisely, an argument is a series of statements that serve as the premises leading to a conclusion.

We use arguments every day, whether we are conscious of it or not. For example, consider the following conversation: "I'm going to Best Buy to get a new TV." "Don't go to Best Buy." "Why not?" "They have terrible prices." In essence, what is being argued is:

1. If a store has terrible prices, one shouldn't shop there.
2. Best Buy has terrible prices.
3. Therefore, one shouldn't shop at Best Buy.

There are two basic types of argument: deductive argument (as in the case above), in which the premises guarantee or entail the conclusion, and inductive argument, in which the premises make the conclusion more probable than other competing conclusions.

Deductive Arguments

The arguments that follow the rules of logic, as one might follow the rules of mathematics, are deductive arguments. Deductive arguments are those that are studied in college courses on logic, with all the letters and strange symbols (again, sort of like math). Contrary to the claims of postmodernism that have had so much influence, these rules of logic are rather unyielding. In fact, they are completely unyielding, but a brief defense of logic against the onslaught of postmodernism will need to be left for another day.

A good deductive argument must achieve two requirements: it must be valid, and it must be sound. A valid argument is one whose conclusion follows from the premises based on the rules of logic (like the one about Best Buy). Here is a good example of an invalid argument:

1. If the Cleveland Indians are the best team in baseball, they will score over 12 runs some games.
2. The Cleveland Indians scored over 12 runs in a game.
3. Therefore, the Cleveland Indians are the best team in baseball.

It could be the case that the Indians are the best team in baseball, but it is far more likely that they had one really good outing in yet another disappointing season.

For an argument to be sound, it must be both valid and the premises must be true. An argument may be formally valid and nonetheless completely ridiculous.

1. If the sky is blue, the moon is made of cheese.
2.The sky is blue.
3.Therefore, the moon is made of cheese.

Strange as it may seem, that is a valid argument. The conclusion does follow from the premises. The problem is that the first premise is ridiculous; it is quite obviously false.

Laws of Logic

Now, I've mentioned the rules of logic a few times in the post, but haven't yet said what they are. That is intentional. There are a fair number of them, and in explaining all of them, this would look a whole lot less like a blog post and a lot more like a textbook. The interested reader is advised to check out a book on logic or read more online. That being said, there are three classic laws known as the "laws of thought" that govern ... well pretty much anything related to thinking, including the rules of logic.

The first is the Law of Identity: every thing is the same with itself and different from another. That probably comes off as so incredibly obvious that it's not even worth mentioning. Nonetheless, it's pretty important for the development of philosophical argumentation.

The second is known as the Law of Non-contradiction (LNC): two contradictory statements cannot both be true in the same sense at the same time. It is this, among other things, that more glib flavors of postmodernism (and perhaps some of the more serious versions) rail against. "It might be true for you but not for me." Now this statement could be correct in a sense. I may have a broken thumb, but you may not. In that case, it is true for me that I have a broken thumb, but not true for you that you have a broken thumb. But it is nonetheless the case that I have a broken thumb; it cannot be that for you I do not have broken thumb. The facts of the matter do not depend on one's disposition. And in any case, that is not the sense in which the statement is usually given. It is a denial of the LNC.

I once heard it claimed that the law of non-contradiction was simply a "western" idea, and that the history of thought in the East was much too rich and complex to include something so restrictive. There are two responses to this. The first is that it is simply false. There are several ancient and medieval Indian sources that state the law of non-contradiction. And secondly, even if it were true that eastern thought did not give credence to the LNC, as Ravi Zacharias has said, "Even in India we look both ways before we cross the street because it is either me or the bus, not both of us!" Again, the claims of postmodernism on this subject will have to be dealt with in another post.

And finally, the Law of Excluded Middle: a single proposition is either true, or its negation is true; it cannot be in any intermediate state between the two. This, in conjunction with the law of non-contradiction, leads to what is known as the Principle of Bivalence: propositions are either true or false. Now it might sound like the law of excluded middle and the principle of bivalence are exactly the same, but they are distinct, and the distinctions are rather difficult to explain without getting into some pretty heady stuff. If you're feeling brave you can look them up on Wikipedia for a start.

The application of these laws is indispensable for the correct and effective use of a deductive argument. Now, onto inductive arguments.

Inductive Arguments

Inductive arguments are "bottom-up", in that it derives a probable explanation from a collection of examples. "Every time I let go of this ball, it falls to the ground. So if I drop it again, it will fall to the ground." Inductive arguments are probabilistic. In other words, an inductive argument does not guarantee the conclusion, but only makes the case that the conclusion is probable. With an inductive argument, it is possible that the premises be true and no invalid inferences are made, and yet the conclusion nonetheless be false.

There is a common misconception that deductive reasoning is much stronger than inductive reasoning. This is not necessarily the case. If the premises of a deductive argument are true and the argument is also valid, then the argument is essentially airtight. But say the status of the premises is uncertain. They may be debatable, or perhaps only possible, but not necessarily true. In that case, the deductive argument only makes the conclusion epistemically possible; it may be false, but if the evidence points towards the premises being true then it is probably also true.  Contrast this with an argument that is inductive where evidence in favor of the conclusion is overwhelming. Gravity, for example. It has been so consistent in its action throughout the eons that it is known as a LAW, despite the fact that we can't actually prove in the next moment that the ball will indeed fall to the ground.

Abductive Arguments

In closing, I should mention that technically there is a third form of argument, known as abductive argument. It can be defined as "inference to the best explanation." Let's say Greg goes down to his mailbox only to find that it's been smashed, and there are bits of wood strewn around.  How did the mailbox get smashed?  There are literally an infinite number of possible explanations; perhaps it was aliens who also accidentally dropped a few splinters of wood they were doing tests on; or maybe a neighbor's dog has been genetically modified so that is so strong it can crush a mailbox with a tree branch. Or, it could be that those teenage miscreants down the street destroyed the mailbox with a baseball bat. The third option seems like the most reasonable explanation, prima facie, but an actual application of abduction, using all relevant facts, could help determine if that is actually the case; the best explanation is then inferred as being true. It is interesting to note that much of science relies on abductive reasoning.

In my next post I'll cover some of the informal fallacies that are found in conversations every day.  Some of them can be pretty entertaining, as the penguin has so ably demonstrated.
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  1. To add one more bit of info to this post, I would like to point out a further distinction between deductive and inductive arguments:

    One cannot discover any new information by deductive argument, while an inductive argument actually does (probabilistically) discover new information. That is, if a deductive argument is sound, the conclusion will always be found implicitly in the premises. On the other hand, the conclusion of an inductive argument is extrapolated from the premises, thereby "discovering" new information. A premise may say "Y has followed from X every time in the past," and the conclusion extrapolates this to "Y will always follow from X."

  2. So is the "Law of the excluded middle" and/or the "principle of bivalence" like the old saying "a half-truth is a full lie" (i.e. if any part of a statement is untrue, the whole statement is considered false?)

  3. Not quite; the Law of Excluded Middle and principle of bivalence deal only with claims and arguments with a single possible truth value.

    For example, the statement "I went to the grocery store," makes a single claim, so it has only one possible truth value. The principle of bivalence claims that it is either true or false; it cannot be on some continuum in between. But the statement "I went to the grocery store and punched the cashier in the face," makes two claims. For the principle of bivalence (and the Law of Excluded Middle) to apply, one needs to split up the statement into separate components. Each component can only have one truth value- either true or false.

    "Half-truths" are a bit more difficult to pin down. They can be statements with multiple truth claims with at least one of those being false, they can be a single claim that doesn't give all the relevant data, or they can be deceptive in some other way (e.g. having a double meaning).

    When a witness swears to "tell the truth, the WHOLE truth, and NOTHING BUT the truth," she is essentially swearing to give no half-truths. So it's certainly a worthwhile proverb, but not directly analogous to the Law of Excluded Middle or the principle or bivalence.

    That was probably a much longer answer than you wanted, but that will probably become par for the course on this blog.