These lessons introduce the logic of arguments. They build on the logic of statements.

  1. Arguments & truth
  2. Logical fallacies

1. Arguments & Truth

Building Arguments with Statements

The Logic & Language series introduced logic as a way of representing and analyzing sentences, but I skirted around questions of truth and establishing good arguments. I'd like to take a stab at those topics in this video.

We've payed attention to the structure of sentences and used symbolic logic to represent that structure. On that front, logic provides a handy way to sort out and clarify statements. Please make sure to go through those videos - if you understand that material, you can see that the logic of a statement like 'water is liquid' may be represented as (∀x)Wx⊃Lx (for all x, if x is water then x is a liquid). Now that you have that kind of grasp on logical statements, you can extend your logic to whole arguments with multiple statements.

When I say argument, I'm afraid you may be hearing 'heated exchange of words'. I mean instead a series of statements made in support of an assertion, together with the assertion drawn from those statements. The supporting statements are premises of the argument and the assertion that follows from them is the conclusion.

Right now I want to convince you that coffee is a food, not a beverage. Yeah, you heard that right. I'll start with some sentences we both accept. (1) Coffee is a bean (C⊃B). Remember your predicates here - I'm stating that (∀x)Cx⊃Bx. (2) A bean is a food, or (B⊃F). So it follows that (3) coffee is a food (C⊃F)!

I brought you to a conclusion using two supporting premises. You might think I'm pulling something over on you here - maybe you can't even put your finger on it. Hold your criticism for now and consider the structure of my argument first.

The Form of an Argument (formal matters)

Is the form of my "Coffee is food" argument any good? Let's abstract the argument to A is B, B is C, therefore A is C. You'll find that, however you fill in this structure, the argument seems to work. I bet you're even smart enough to see why, you really are, so let's move along.

If, on the other hand, I tried to fill out an argument with a structure like (premise 1) A is B, (premise 2) A is C, (conclusion) therefore B is C, it turns out that the argument doesn't work. Apply this form to a few cases, and you'll figure out pretty quickly that it fails.

Arguments with a working structure are valid. Arguments that are broken get the negative label invalid (please read that as in--lid). It would be premature of us to stop there and declare that we have a watertight argument. We've checked our structure, but we haven't thought about whether we've filled our structure with correct statements.

The Content of an Argument (informal matters)

The premises of an argument can be true or false, and the conclusion true or false as a result. I could propose the argument that all soups are cold (A⊃B), anything that's cold can't burn your mouth (B⊃C), so soups can't burn your mouth (A⊃C). The "Soup can't burn" argument has a valid structure, but the first of its supporting premises is false, which impacts the conclusion I'm drawing.

If our argument is valid and has true premises, it's a sound argument. This may strike you as straightforward, but it's easy to get lazy (or manipulative) with your logic and end up producing or being convinced by unsound arguments. (Or, should I say, it's easy for the other guy). Notice that either truth or validity can impact the soundness of your argument - unsound arguments include an invalid argument with true premises, a valid argument with one or more untrue premises, and an invalid argument with untrue premises; only a valid argument with true premises is sound.

You may be aware of this, but our argument structure introduced above - A is B, B is C, so A is C (dogs have brains, brains have neurons, dogs have neurons) - is called a syllogism. So my "Coffee is food" is a valid syllogism with, I will assert, true premises. If that's all correct, the argument is sound!

Not all arguments are syllogisms. Notice that the syllogism has two conditional premises: if A then B, if B then C. The conclusion follows from these conditions - therefore if A then C. We could just use one conditional premise - if A then B. If we assert in a second premise that A, we conclude that B.

Deductive Reasoning & Inductive Reasoning

I haven't looked outside these arguments to support them. I just relied on the structure and persuasiveness of the premises to make my point. The meaning and words are all there for you to evaluate, and if these are good, the conclusion's good, too. That's deductive reasoning (deduction), and the resulting argument is called a deductive argument.

Once I start checking my facts outside my bare words, things get trickier. If I want to support my assertion that coffee is beans, I have to use inductive reasoning (induction) to look at specific information in the real world. I then make abstract generalizations about 'all coffee' even though I could never hope to check if what I'm saying is true even of a fraction of the world's coffee. Such claims are probabilistic and open (until I can demonstrate that they are false), and arguments that take this line of reasoning are inductive.

For instance, I can't demonstrate that my first premise is true of all coffee ever, but I can state things that probably hold, which requires inductive reasoning. You might instinctively perform inductive reasoning by checking against your own experience ("Have I seen coffee beans?"; "Have I ever seen coffee that's not beans?"), but There are even more methodical approaches (operationalize (clearly define for observation) "coffee", "beans" and "food", statistical sampling).

We called valid and true deductive arguments sound. When we come across inductive arguments, it's preferable to talk about cogency. A cogent argument is "valid" (strong) and has probable premises.

Truth

I've thrown around the words true and truth. If you're like most of us, you intuitively feel you have some understanding of what those words mean. The difference between deductive and inductive thinking may have raised a few issues, but let's tackle the issue of truth straight on. Digest each one of the following questions, and use them to get a broad view on different theories of truth in logic & philosophy:

  1. Is a statement true when it aligns with, is consistent with and doesn't contradict other true statements? (Coherence)
  2. Is a statement true when it corresponds to something in the real world? (Correspondence)
  3. Can certain statements be asserted as true in and of themselves, self-evidently? Will truth then deductively follow from these assumptions? (Foundationalism)
  4. Is a statement true when it proves to be useful or practical? (Pragmatic accounts)
  5. Is a statement true when enough people argue or believe that it is true? (Consensus)
  6. Is truth not an actual property of a statement at all, but something else? Can 'true' or 'the truth' ever be predicated in a meaningful, non-redundant way? (Deflationism)

Whatever your intuitions, what do you make of my "coffee is a food" argument? Do you think it's sound? In the next video, I'll share a perspective on logical fallacies and take another look at that argument.

2. Logical Fallacies

When you're arguing a point using logic, you establish premises to support a conclusion. Many times, your conclusion doesn't actually follow from your line of reasoning, no matter how much you or your opponent are convinced by your argument. In that case, you may be relying on a fallacy to persuade yourself and your opponent. This word "fallacy" sounds charged and can certainly get tossed from side to side in heated debates, but in logic a fallacy is merely a descriptive term, and says nothing about how dumb you are nor about whether or not there exists some real way to support the same conclusion.

Informal Fallacies

Let me show you how it works. I made an argument in the previous section demonstrating that coffee isn't a beverage, but a food. In this argument, I relied on two distinct meanings of the word 'coffee', and two different meanings of the word 'beans' in my premises. I was intentionally vague, and used compact language to hide my choices. This simple language makes the argument much more convincing than if I disclosed the intended meaning of these words. (I relied on equivocation).

The result: the conclusion looks like it follows from the premises, but it's really not supported by them. The conclusion may or may not be true, but it is uncalled for here. This reliance on the psychological persuasiveness of a statement is a hallmark feature of informal fallacies, which are informal because the truthfulness of the conclusion's content can't be derived from the truthfulness of the content of the premises. Still, the argument's form may be perfectly valid, like my 'food is coffee' syllogism.

Here are some examples of informal fallacies:

Informal fallacy description example
argumentum ad baculum appeal to force; threatening into submission I'll hit you if you say coffee is a drink. So coffee is a food!
argumentum ad hominem focus on character flaws & bias rather than argument at hand John's a loudmouth, and he says coffee is a drink. So coffee must be a food.
argumentum ad verecundiam focus on authority, expertise or credentials rather than the argument at hand Dr. John has two PhDs, and he says coffee is a food. So coffee is a food.
argumentum ad ignorantiam asserts a conclusion based on something unknown We don't know everything about food or coffee yet. For all we know, maybe coffee is a food. So let's accept that coffee is a food.
petitio principii begging the question; asserting the conclusion to demonstrate the conclusion Since coffee is a food retail product, coffee is a food.
etc.

And the list goes on and on. In all cases, informal fallacies rely on content that does something other than support the conclusion in order to get you to accept the conclusion. You can get a feel for how the logic of the supporting line of reasoning actually runs if you just remove the fallacious statement, like this: I'll hit you if you say coffee is a drink, so coffee is a drink. Not so reasonable now, is it?

Formal Fallacies

Recall that arguments don't just have content, they also have form. We called an argument with well-structured premises and a conclusion valid. It may not surprise you to hear of formal fallacies, which arise when we shape the argument in a way that the conclusion doesn't follow from the premises, but people can still be persuaded by the conlusion because it seems to follow. Let's play around with a few arguments to see if you can spot weaknesses in their structure.

First, a refresher: a basic syllogism. Fill it in as you like; we've seen that this form works:

Premise 1: A ⊃ B
Premise 2: B ⊃ C
Conclusion: ∴ A ⊃ C

Next, a second point of reference, and an even simpler argument. If P then Q. P, so Q. This is a basic reasoning strategy, and it works!

P1: P ⊃ Q
P2: P
C: ∴ Q

What about this one? A is not B, B is not C, therefore A is C. Think about it for a moment, and check against other examples if you need to. The first premise only supports that A is NOT B, so anything we say about B in the second premise can't apply to A, positively or negatively. The conclusion doesn't follow.

P1: A ⊃ ~B
P2: B ⊃ ~C
C: ∴ A ⊃ C

This next argument runs aground on the same issue as the last. A is not B, so no matter what we say about B, it doesn't apply to A.

A ⊃ ~B
B ⊃ C
∴ A ⊃ ~C

What's wrong below? I've negated P in the conditional statement, then assumed that Q must also be negated. Perhaps Q is conditioned on something else. (If I flip the switch the light turns on; the light's on, but maybe someone else flipped some other switch!)

P ⊃ Q
~P
∴ ~Q

Spot the problem in the next argument? No? Look closer. Keep looking... because you won't find it. This one's good, and I leave it to you and your clever ways to figure out why.

P ⊃ Q
~Q
∴ ~P

One more for you. Can I argue that what follows from a conditional implies what precedes? Consider this form carefully. A implies B only states that if A holds, B also holds. It does NOT mean that B always & only follows from A. Come on, you remember your logical biconditionals, right?

A ⊃ B
B
∴ A

That's all for fallacies (well, for this introduction to fallacies - I'm sure you'll encounter plenty more!) These last two topics introduced arguments and fallacies. I'll add them to my playlist for the intro to logic if you're visiting the Youtube channel and to the lessons on nativlang.com. I hope you've enjoyed a bit more logic, and thanks for learning with me.