Logical Tutorial, Part 4.
Logical Symbols

(c) 2017 by Barton Paul Levenson

I owe much of the next several chapters to the brilliantly succinct exposition on the subject by Waner and Costenoble (2001). The various parts can be found in many texts on formal logic (for instance, Copi and Cohen 1998), but Waner and Costenoble's tutorial sums it up about as quickly as possible without leaving out anything important.

To understand formal logic, we have to master two techniques--the propositional calculus and the predicate calculus.

Don't panic. These fields have nothing to do with "calculus" in math, and they aren't as hard as they sound. In fact, they're pretty simple.

We'll handle the propositional calculus first.

Logical statements (as opposed to statements in general) are sentences which we can say are either true or false: "T" or "F." T and F are the only two "truth values" allowed in formal logic, so another way to define a logical statement is, "a sentence that can be given a truth value."

To say a sentence is a "logical statement" is not to praise it, or say it constitutes a logical argument. It is not even to say it is correct. It simply means the statement is either true or false--it has an unambiguous "truth value." These are examples of logical statements:

• My dog is fat.
• My cat is college-educated.
• If we spend too much on defense, the country will go bankrupt.
• All Armenians have read Saroyan.

These are examples of statements that don't qualify as logical statements:

• Clean your room. (Not a logical statement because it is not in the declarative mood, but the imperative. It is a command, not a statement about reality.)
• Your mother! (Not a logical statement because it isn't a complete sentence.)
• This sentence is a lie. (Not a logical statement because it can't clearly be called either true or false. If it's true, it's false, and if it's false, it's true. Its truth status is "indeterminate," and therefore it is not considered a logical statement.

We can replace particular statements with lower-case letters. Thus, for example:

p: My dog is fat.
q: My cat is college-educated.

We can then use p and q in formal logic using the notation of "propositional calculus."

Here is our first logical operator:

¬ ("not")

Placed in front of a logical statement, it negates the value of that statement (i.e., reverses its truth value). Thus:

¬p: My dog is not fat.
¬q: My cat is not college-educated.

The result of negating a statement can be reduced to a table which covers every possible case, a "truth table." Here's a truth table for p and its negation:

By definition, if p is true, ¬p is false, and if p is false, ¬p is true.

Here's another logical operator:

∧ ("and")

This operator is used between two logical statements. The result is called the conjunction of those statements. Thus p ∧ q means "my dog is fat AND my cat is college-educated." Here's a truth table for ∧ using p and q:

Both statements have to be true in order for their conjunction to be true.

For more than two statements, ∧ is "associative." This means you can shift a pair of parentheses around any two or more statements, however many you want, and it won't change the meaning. Say that statement r stands for "My car is sexy." Then:

Here's a third logical operator:

∨ ("inclusive or")

Used between statements, it means that if either one of the statements is true, or both are true, the whole phrase is true. This is a logical disjunction. Here's a truth table for ∨:

A disjunction like p ∨ q is read "p or q," and the word or is understood to mean the inclusive or. If p is true, or q is true, or both of them are true, then p ∨ q is true.

Just as with conjunction, disjunction is associative:

p ∨ q ∨ r = (p ∨ q) ∨ r = p ∨ (q ∨ r)

Given the three logical operators we now know (¬, ∧ and ∨), we can construct quite complicated logical statements. Let's say:

a: Colonel Mustard did the bloody deed.
b: Professor Plum knocked the old boy off.
c: Miss Scarlett done him in.

We can then say:

(a ∨ b) ∧ ¬c

Which we might translate as "Either Colonel Mustard did it or Professor Plum did it, but it was not Miss Scarlett!"

Another bit of terminology is that logical statements of the lower-case-letter type are "atomic statements" and more complicated ones built up using ¬, ∧, ∨ and/or ( ) are "compound statements."

In theory, a truth table can be constructed for any logical statement, no matter how complicated. Here's one for the "(a ∨ b) ∧ ¬c" we used above. As in algebra, work from the inside of the parentheses out and keep going until everything is covered and you've reproduced the whole sentence: