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4.1 And Here Is Conjunction: &

The human brain is a pattern-recognition machine. It has helped us make amazing discoveries.

The philosopher Thales predicted the first eclipse in 585 BCE by recognizing patterns in the sky. Barry Marshall discovered that bacteria can cause ulcers by recognizing patterns in his gut.

Sometimes our brains see patterns and intentions even where there aren’t any, which is why the Greeks believed that floods and storms were caused by angering Poseidon.

BOOL: a logical system. The name is short for “Boolean logic”.

In this chapter we’ll use pattern recognition to our advantage. It’s time to start building our first logical system.

A symbolic system deserves a symbolic name, so we’ll call it BOOL.

We’ll give you a few examples of how BOOL works, and you have to pick up on the patterns and figure out the rest. For example, here’s how we say Pia is guilty and Quinn is guilty.


Now you try.

As you figured out, capital letters stand for sentences, like P means Pia is guilty.

Pick the capital letter that makes the most sense, and follow the English as closely as you can.

Here’s the basic rule of BOOL: pick the capital letter that makes the most sense, like you did for R.

Let’s try another one.

Symbolic languages are unforgiving. Each sentence is a string of symbols with no spaces, and you have to get it exactly right, including capitalization.

It’s like a programming language, where one wrong letter can create a syntax error that crashes the whole program.

But the strictness has an advantage: as we’ll soon learn, it allows us to do computations with BOOL and prove that arguments are valid.

Don’t let small differences in the English fool you, like the word “and” appearing between the names. When we translate into BOOL, we just want something equivalent.

When translating into symbols, we want something equivalent which mimics the English when it can.

Saying “Raquel and Pia are guilty” is equivalent to saying the longer “Raquel is guilty and Pia is guilty.”

Our symbol “&” is only allowed to connect two sentences together. So technically we shouldn’t say P&Q&R.

BOOL has parentheses for grouping, so we could write (P&Q)&R to show that the first & connects P and Q together, and the second & connects the complex sentence (P&Q) with R.

Technically, “&” can only connect two sentences. But we will allow many &s to string together.

However, it is obvious that saying “Pia and Quinn are guilty, and Raquel is guilty” means the same thing as “Pia and Quinn and Raquel are guilty.” Both of those sentences mean that all three are guilty.

So we will allow many &s to string together. If English does not group them, then you shouldn’t in BOOL either.

BOOL mimics the English: if there is grouping in the original sentence, keep the grouping in BOOL.

But if English does group sentences clearly, we also need to group them in BOOL.

The symbol & is called conjunction, or ampersand, or just and for short.

Conjunction, ampersand, and: &.

Conjunction is a type of symbol called a connective, because it connects sentences to make new, longer sentences.

Here’s a helpful metaphor we will use a lot. Sentence letters like P, Q and R are called atomic sentences, or atoms, because they are the basic building blocks of BOOL.

If a sentence in English does not have another sentence as a part, like “Pia is guilty”, then it is atomic.

Connectives combine with atomic sentences to make complex sentences. Complex sentences are the molecules.

The connectives are the bonds that allow us to build bigger and bigger molecules.

The connectives are the bonds that allow us to build bigger and bigger complex sentences.

As you’ve already seen, connectives can connect any sentences, atomic or complex, which is how we build up ever bigger molecules like this: