Never mind the basics, here’s the syntax

Now that we have the very basics out of the way, we are ready to dive into the details of symbolic logic. In particular, the logic we’re going to look at is called sentential logic. You got a preview of the type of logic that was in Part One, but you were never equipped with the tools for how to evaluate such arguments. Remember when I said that logic concerns itself with form, and that form is governed by formal rules? This is where we’re headed. Ooooo yiss.

Cast of Characters

Sentential Logic

Also known as propositional calculus, if you want to sound smart. This is because the most atomic unit of this system is the complete sentence, such as “Becky is asleep,” “Snow is white,” etc. The words proposition and sentence will be used interchangeably here. We will stick to declarative sentences, i.e. sentences that are either true or false. So, we’re not going to deal with questions or imperatives (these are a bit more complicated, and require their own branch of logic to be evaluated). 

But there’s not a lot of fun to be had with just atomic propositions like the ones above; the real fun comes from compound/complex sentences, which are simply atomic propositions connected together by way of Boolean operators (e.g. ‘and,’ ‘not’). We can use the two propositions above to create a compound proposition: “Becky is asleep and snow is white.” So, a compound proposition consists of two or more stand-alone (atomic) propositions. Because the syntax of a natural language like English is a lot richer than our propositional logic (and in effect more redundant), there will be some surface differences between an English proposition and how it is symbolized in logic. You can think of natural language as a more compressed version of logical languages. Consider the English sentence “Becky and Daniel like trees”. That seems like a pretty concise sentence. But, because of the presence of the word ‘and’ (a boolean operator), it is actually a compound sentence. When decompressed into our logic, it consists of two sentences, i.e. “Becky likes trees,” AND “Daniel likes trees”. 

A Note on Convention

It can seem cumbersome to write out full English sentences, so we will use the following convention: we will symbolize each atomic proposition by a single letter. So “Becky likes trees” just becomes B, and “Daniel likes trees” becomes D. Capital letters will be used for specific instances of propositions and arguments (like the ones we’ve gone through so far). When we’re going more abstract, talking about argument form, we will use lower case letters (variables). The prevailing convention for symbolizing form is to use sequentially p, q, r, s, w, x, y, z. If that’s not enough, add whatever symbols you want. On my phone, there’s a hot dog emoji that I particularly enjoy: in it, the sausage is seen sleeping in its bun. You may use that. It’s fine. In fact, the last decade saw significant progress in the creation of sausage-based logics (see Wiener et al., 2012).

Boolean operators: Intro

Boolean operators are the glue that allows for atomic propositions to become compound propositions. They are called operators because they operate on a sentence; they’re called Boolean because they make the proposition either true or false (named after mathematician George Boole, a friend of the series). English has a wide array of operators, but in our logic(s), we’re only going to deal with 5 operators. These are:

Examples Translating English to logic

To get more acquainted with the operators, let’s try translating some English sentences into our logic. We will enlist some help from our friend, Becky. Remember: because these sentences are instances, we will simply denote each atomic proposition with a capital letter. Here’s a key for each of the atomic propositions we will use:

Okay, now let’s construct some English language sentences from these and then translate these to logic.

Note how the English word ‘but’ is treated as a conjunction in logic. I’m hoping that you’re getting a feel for how the very many natural language operators can be compressed into a small number of Boolean operators. (3) and (5) are especially interesting here:

In (3) the word ‘neither’ essentially means not p AND not q. But that can also be written as a disjunction of p or q wrapped inside a negation. This equivalence is a general rule called DeMorgan’s Law. We will revisit DeMorgan’s law once we get into constructing proofs.

(5) is meant to pump your intuitions about how you can write equivalent propositions in different ways and how you can define operators in terms of each other. The first expression (with the two conditionals in either direction) more accurately reflects how the English sentence is written. But it also goes to show exactly how a bidirectional statement works: ‘if and only if’ can be defined as a conjunction between two mirror images of the same conditional statement. 

But we still haven’t talked about how these operators work. To evaluate the truth and validity of arguments, we need to know how these 5 operators determine the truth/falsehood of whatever propositions they’re operating on. Prepare to look at some IKEA-grade tables of truth.

Truth tables: how the Boolean operators actually work

The Boolean operators are said to be truth-functional. What this means is that the truth value of the (compound) proposition they form can be worked out merely via the truth values of the atomic propositions they connect. There is a set of rules governing whether an operator is true or false based on which (and whether) the atomic components are true and/or false. For instance, a disjunction [p or q] requires only one of the components (p, q) to be true for the whole thing to be true. In a conjunction [p and q], both p and q must be true for the resultant compound proposition to be true.

Part Two Wrap-Up

What we have learned so far

In this section of the tutorial we have learned that sentential logic regards sentences as the most basic component. Sentences can be bound together to yield more complex sentences via Boolean operators. We’ve looked at some writing conventions: atomic sentences (instances) are denoted with capital letters, and variables (form) are denoted with lower case letters starting from p, q, r, etc. We learned that the 5 Boolean operators are the means by which we connect simple, atomic propositions into compound ones. Finally, we learned the rules of the Boolean operators, to be able to evaluate more compound propositions and (eventually) arguments.

Where we are going next