Ada Lovelace, the first person to write a computer program.



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<![CDATA[o describe logic as the study of how truth moves from sentence to sentence. It is not a common way to describe logic, but it's mine, and I like it. Other people describe logic as the science of good reasoning, the study of valid inference, the science of argumentation, etc. The more cynical types think of it as merely a bunch of symbols and rules for manipulating those symbols. These are good descriptions, too. Logic is a very big thing. Most people think of logic as intricately tied to human reasoning, and I'll treat it this way at first.

The notion of truth moving from sentence to sentence informs what I said earlier, about logic telling us what brain states we can be in, given the brain states we were just in. An example of being in a brain state is to believe that Sasha is a dog and that all dogs are mammals. Here, you are in a state of belief about a particular sentence. Logic tells you what other brain states you can transition to, based on that brain state. In this case, based on logic, you can transition into the brain state of believing that Sasha is a mammal. I think this transitioning from one brain state to another, based on the rules of logic, is what it means to make an inference.

You might wonder what computer programming has to do with logic. You would not be alone in wondering that. If you would just chill out and be patient, I’ll tell you! The idea of transitioning from one state into another is the way computer programs work. They tell the computer what sequence of states to transition to and from. That’s why I think of logic as a sort of programming language for human brains. You can think of an inference as a transition from one brain state to another.

To illustrate this idea, imagine a hypothetical space of beliefs one might have; call it BeliefSpace. As a thinking human, you are always occupying some spot in BeliefSpace. Usually, you are not just stationary, but are wandering around, probably lost. As someone who strives to be rational, you want to make sure that you never jump from a true spot in BeliefSpace onto a false one. I don’t know, maybe if you jump from a true spot to a false spot you fall into a pit with spikes in it or something. BeliefSpace is full of peril.

It would be neat to have a guide, or a map, that tells you what sort of jumping patterns you can make through BeliefSpace. This way, you could be sure that you never jump from a true spot onto a false spot, impaling yourself in the spike pit.

Well, it turns out we do have just such a guide. It’s called logic. Note well, logic does NOT make sure you are always standing on a true spot. Science figures out whether the spot is true or not. Logic says, “If you’re on a true spot, and you follow my jumping rules, you’ll never land on a false one.”

Ruth Barcan Marcus, expert hop-scotcher and modal logician.



Logic tells you how to jump through BeliefSpace like a happy little hopscotch expert. I once had a friend who played hopscotch in Florida. She hopscotched onto a sinkhole with spikes at the bottom. She died. If you don’t want that to happen to you, and instead you want to hopscotch joyfully and safely through BeliefSpace, you should know more about logic than How Not To Hop. You should know How To Hop Right. The Fallacy Files is a good primer on How Not To Hop. I hope this upcoming series of blogs sparks your interest in learning How To Hop Right.

People who study How To Hop Right are called logicians. To be a logician is to be constantly tortured by symbols, so that other humans can Hop Better through BeliefSpace. You will get a taste of this torture in this series of upcoming blogs on logic.

Things weren’t always torturously symbolic. The use of symbols in logic is a relatively recent advancement (mistake?). I don’t want to go into too much history. I’ll just say that for most of its history logic has been intimately tied up with the way humans reason, and humans don’t reason about symbols very often. Humans reason about ideas, objects, other people, the weather, and of course spiked pits. The goal has been to figure out the rules for saying true things, based on other true things.

For humans, it is easier to get a grip on logic when the examples deal with these relevant ideas and objects, and spiked pits. But, for a logician, this is the least interesting part of logic. Logicians don’t care about the things that are being reasoned about. We care about the relationships between the sentences (the patterns between the spots in BeliefSpace), no matter what the content of each sentence (spot). The subject of logic is a level of abstraction beyond any particular topic. Two arguments may be about totally different things, but to a logician, be exactly the same. Let me give an example.

Argument 1:

1. If Sasha is a dog, then Sasha is a mammal.

2. Sasha is a dog.

3. Therefore, Sasha is a mammal.

Me (top left), and Sasha (center, with tongue sticking out)



Argument 2:

1. If Mercury has a mass sufficient to clear its orbit of space debris, then Mercury is a planet.

2. Mercury has a mass sufficient to clear its orbit of space debris.

3. Therefore, Mercury is a planet.

Mercury (gray thing in center)

At first glance, these arguments have nothing in common. Argument 1 is about my dog Sasha, while Argument 2 is about the conditions under which a celestial body is a planet. But to a logician, these are instances of the same argument. There is no logically relevant difference between them. That’s why we use symbols. The symbols allow us to focus on only the part of the argument that we care about—its form. These two arguments map out the exact same pattern through BeliefSpace.

If you want to study these arguments as a biologist, or an astronomer, then the actual sentences matter. But if you want to study them as a logician, then you’ve got to formalize them: kick them up a level in abstraction.

We use sentence letters and logical symbols to do this. We use sentence letters to stand in for statements in the sentences, or for the sentences themselves. I’ll explain this in a moment. Here are the logical symbols:

→ : This takes the place of an “if… then…” or “…implies …” (these mean the same thing) where the ellipses are where statements go.

& : This takes the place of an “… and …”.

\/ : This takes the place of an “… or …”.

The ~, which I pronounce “squiggle,” is a negation. It reverses the truth value of the statement that it’s in front of. If you put a ~ in front of p, then you get a not-p. It takes the place of a “not…”.

So, using those symbols, let’s formalize Argument 1. We start with the first premise, which says,

1. If Sasha is a dog, then Sasha is a mammal.

It is an “if… then…”, so we take out those words, and stick the arrow between the two statements:

1. Sasha is a dog → Sasha is a mammal.

Next, we replace the statements with letters. Because the statements are different, we have to give each a different letter. I’ll use ‘d’ for “Sasha is a dog”, and ‘m’ for “Sasha is a mammal.”

1. d → m

Okay, now we’ll do premise two:

2. Sasha is a dog.

This doesn’t have an “if…then…”, “and”, “or”, or “not”. What do we do? We call this statement atomic. It is just a single sentence letter. Because we already assigned that statement a letter, where it appeared in premise one, we’ll use the same letter in premise two: ‘d’.

1. d → m

2. d

The conclus

ion is:

3. Sasha is a mammal.

This is also atomic, and we already gave it a letter: ‘m’. So, we’ll just plug the ‘m’ in:

Argument 1, formalized:

1. d → m

2. d

3. Therefore, m

Okay, that’s the formalized version of the argument. Notice that I could have used any sentence letter to represent each statement, as long as I was consistent in assigning them. Once you give a statement a sentence letter, you’ve got to give it the same sentence letter when it pops up again.

Now let’s do Argument 2. I’ll let ‘s’ stand in for the statement, “Mercury has a mass sufficient to clear its orbit of space debris,” and ‘p’ stand in for the statement, “Mercury is a planet.” Plugging those in accordingly, we get:

Argument 2, formalized:

1. s → p

2. s

3. Therefore, p

You should notice that the two formalized arguments look exactly the same, other than the particular sentence letters. But remember, I could have used any sentence letters for the statements; they were completely arbitrary. So, as far as form goes, the arguments are exactly the same: You have an “if…then…” as premise one, then the “if…” on its own as premise two, and you conclude the “then…” statement from the first premise. Same form; same pattern.

Now, to a biologist or an astronomer, these arguments are really boring. In fact, they might even be really boring to you. But, to a logician, the fun has only just begun. Once you get the hang of stripping arguments down to their bare and vulnerable form, you gain the ability to assess them for validity. Assessing an argument’s validity is like making sure the pattern through BeliefSpace won’t take you from true spots to spiked pits.

Once you reduce arguments to letters and symbols, you can manipulate them in crazy intricate ways. If you were trying to remember what the arguments were actually saying, you’d go crazy. Most logicians go crazy anyway, but they are able to accomplish a lot beforehand, since they care only about the symbols.

Saul Kripke, the craziest logician of all, demonstrating how many fucks he can hold in one hand. None!

Logicians claim that their goal is to find all of the valid argument forms (all the safe patterns through BeliefSpace). This is sort of true, as a Big Picture goal. This is what the logician wants to happen, but the immediate goal is always to solve a puzzle. The letters and symbols become little puzzles, representing a game between the logician and the universe. If you can follow the rules, and after a series of BeliefSpace safe steps land where you want, then you win. It is like chess, but for people without friends!

That’s the next thing I want to show you about logic. I think most of you really like logic, or at least the idea of being logical. However, I fear that for many of you, your familiarity with logic extends only to the taxonomy of fallacies compiled by the Fallacy Files. So, I’ll show you the puzzle-solving side of logic that lures young logicians into its bedroom, forever damning them to a life agonizing over meaningless symbols.

Playing with logical symbols serves two purposes. First, it actually makes you better at reasoning, because you internalize the rules. Second, it helps you understand what validity is, which in turn helps you understand what it is that makes something a fallacy. You can usually read an intuitive explanation of a fallacy and understand that something is clearly wrong with the reasoning. A formal understanding of logic allows you to pinpoint exactly what’s wrong, and to prove that shit is wrong.

I had to get this basic stuff out of the way, to make sure we are all on the same page. Next week, we’ll start playing my favorite game: Proving Shit.