Formal Logic 101: Propositions, Negation, and Arrow Diagrams

Posted on July 30, 2012 by Anthony Ritz

The facts are the least important part of the LSAT.

It’s one of my favorite refrains, because it challenges students to rethink their approaches to the test and create real change in their results. And sure, it’s a slight exaggeration — the facts do matter, at least in some ways — but it’s more true than you might think. A successful approach to the LSAT starts with consideration of logical structure and runs through qualifiers, connecting words, and question types before ever arriving at the sort of fact-based considerations by which too many test takers live and die.

Ultimately, the LSAT is not a subject matter test but a logic test. It’s a test about arguments, how they’re formed, how they’re structured, and how they’re evaluated. In order to truly excel on the LSAT, then, one must first master arguments and the logical structure from which they are built. That underlying logical structure, the subject of today’s post, is known as Formal Logic.


These Propositions are True

The LSAT is made of arguments. Arguments, in turn, are made of true-or-false statements known as propositions. Almost anything can be a proposition, as long as ultimately it is either true or false but not both, and myriad such statements exist on the LSAT. The following examples are but the tiniest taste:

  • “No students with learning disabilities have yet enrolled in the school.”
  • “All members of Pteropsida are tracheophytes.”
  • “Used pop is on sale.”
  • “Without meaningful emotional connections to others we feel isolated.”

 

Simple propositions are generally denoted by single capital letters, like “A,” “B,” and “C.” When a specific and identifiable statement is described, though, we can use whatever letter or small set of letters will best describe the proposition. Thus, “it is raining” might be labeled “R,” and “there are special needs students enrolled at the school” could be denoted by “SN.” The goal is merely to shorthand the statement as quickly, easily, and rememberably as possible.

The benefits of labeling propositions in this way are that it abstracts away some of the factual details that may be intended to lead the test-taker astray and that it streamlines the proposition manipulations that are yet to come. After all, a pile of propositions only goes so far. In order to construct arguments, we need tools to manipulate our propositions, to alter them and to combine them. The tools that we need are called “operators,” and the first operator we’ll examine is known as “negation.”

 

This Part is Not Not About Negation!!

One of the simplest and most intuitive operators in formal logic is the “not” operator, also known as “negation.” We will denote negation by an exclamation point preceding the negated statement, so the negation of proposition A will be written “!A” — pronounced “not A.”

The not operator does exactly what you might expect — it flips the truth value of the affected proposition. If A is true, then !A is false; if A is false, then !A is true. These relationship can be conveniently shown in a chart called a truth table. The truth table for negation is:

\begin{tabular}{ l | r } A & !A \\ \hline T & F \\ F & T \\ \end{tabular}

What, then, can be done with a statement like !!A? If A is true, then !A is false, and !!A is !false, which is just true again. If A is false, the !A is true, and !!A flips back to false. So !!A, commonly known as a double negative, is just A (and of course that means that this section’s title is accurate; this part actually is about negation).

Ultimately, negation is a bit like a light switch. It’s fun for a while, but it doesn’t really “go” anywhere — it just toggles back and forth. We’ll need more if we want to build real arguments that link together whole chains of propositions.

 

Arrow \rightarrow Diagrams

…And what better way to get from one place to another than an arrow?

The formal logical concept known as “implication” can be stated in several ways. The following four expressions are wholly equivalent:

  •  A \rightarrow B
  • A implies B.
  • If A, then B.
  • If A is true, then B is true.

 

To reiterate: the statement A \rightarrow B means one thing and one thing only — if A is true, then B is true. It doesn’t tell you anything else.

At this point I generally invite students to help produce the truth table for the arrow diagram A \rightarrow B. Try your hand at it now.

…

…

…

The first line should be easy. If A is true, then B is… true.

\begin{tabular}{ l | r } A & B \\ \hline T & T \\ &\\ &\\ \end{tabular}

And if A is false, then B is…

\begin{tabular}{ l | r } A & B \\ \hline T & T \\ F & _\\ &\\ \end{tabular}

At this point I can usually get most students to respond that B should be false. Unfortunately, that’s wrong. The proposition A \rightarrow B tells you that if A is true then B is true — and nothing more. The fact that A implies B says nothing whatsoever about what happens if A is false. If A is false, then B could be false… or true. Accordingly, the full truth table for A \rightarrow B is:

\begin{tabular}{ l | r } A & B \\ \hline T & T \\ F & T\\ F & F\\ \end{tabular}

Why do I work so hard to trick students this way? Am I just that evil? Well, yes, but that’s not important right now. What is really important is that the LSAT can be evil in exactly the same way. If I can fool LSAT students into wrongly concluding that B is false, then no doubt the authors of the test can do so as well.

A less-abstract example is useful at this point. Suppose I tell you that if it’s raining then I’ll have my umbrella. We can express this statement logically as R \rightarrow U.

\begin{tabular}{ l | r } R & U \\ \hline T & T \\ F & T\\ F & F\\ \end{tabular}

If you ran across me some rainy day, you could be sure that my umbrella would be close at hand.

But suppose you met me on some other day — a day of sunshine and clear skies. Could you be sure that I would be umbrella-free? If you saw me with my umbrella anyway, would you get to call me a liar? “You lied to me! You promised that if it was raining then you’d have your umbrella, but here you have your umbrella and there’s not a cloud in the sky!”

I think not. I never promised that I would NOT have my umbrella if it was NOT raining, and that’s just not the same thing. Maybe the reason that I could be so sure that I’d have my umbrella anytime it rained is that I simply carry my umbrella at all times, regardless. Maybe I just really, really don’t want to get wet… or sunny.

 

But what other arrow diagram is equivalent to A \rightarrow B? How can arrow diagrams be combined? Will we ever finally see an LSAT question example? Does Anthony have his umbrella?

Find out next week, when Formal Logic returns!

 

To Be Continued…

 

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Comments

One Response to “Formal Logic 101: Propositions, Negation, and Arrow Diagrams”
  1. Amanda says:
    February 8, 2013 at 11:33 am

    Reading up on my formal logic before the big day!

    Reply

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