Symbolic Logic: An Introduction

The triple bar ()

The triple bar represents an “if and only if” statement. For example, let “I” represent “I am going to the concert” and let “Y” represent “You are going go the concert. To join the two:

                    I if and only if Y.
                    I  Y, which states “I am going to the concert if and only if you are going to the concert.”

The tilde

The tilde simply represents negation of a sentence. For example, to show the negation of “Bob likes carrots.” simply symbolize the sentence, (let’s use “C”) and add a tilde in front of it:

                    not-C
                    C, which states “Bob does not like carrots.”

Constructing sentences and arguments using connectives

In order to construct arguments in symbolic logic, you must construct symbolic logic sentences. Follow these rules to construct a sentence:

1. If “A” represents a statement in English, then “A” is a sentence [of logic].
2. If “A” is a sentence and “B” is a sentence, then “A  B” is a sentence.
3. If “A” is a sentence and “B” is a sentence, then “A v B” is a sentence.
4. If “A” is a sentence and “B” is a sentence, then “A  B” is a sentence.
5. If “A” is a sentence and “B” is a sentence, then “A  B” is a sentence.
6. If “A” is a sentence, then “A” is a sentence.
7. Any combination of rules 1-6 are sentences.
8. Nothing except those found in rules 1-8 are sentences.

The above rules work for any sentence letter of symbolic logic (as long as the letter represents a statement in English).
Note: when adding separate sentences of logic together, parentheses are often necessary. Use parentheses to separate separate sentences. For example if you want to conjoin “A v B” with “C v D”:

                    (A v B) and (C v D).
                    (A v B)  (C v D)

Examples of sentences of logic

A
A v B
(A C) F
E
[(W v R) (G J) (E T)]
etc.

The following are NOT sentences:

AS
WB
v D
P H
(F((D v R( 
etc.

IV. Arguments in Symbolic Logic

Now that we have the basics for constructing sentences of symbolic logic, we’re ready to symbolize arguments. Remember arguments must be composed of sentences and nothing else. (See rules 1-8 above for constructing sentences). There are certain rules in symbolic logic that allow you to manipulate sentences and arguments in order to fit your need. The following are rules of replacement.

Conjunction
A
B      
A  B

Addition
A      
A v B

note: when you have a single sentence letter, you may add to it whatever single sentence letter you like using the wedge. Yes, it seems strange, but it is logically valid!

Commutation
A B B  A
A v B  B v A

Association
A (B C) (A  B) C
A v (B v C) (A v B) v C

Implication
A B A v B

Double Negation
A A

De Morgan’s
(A B) A v B
(A v B) A B

Idempotence
A A  A
A A v A

Transposition
A  B B  A

Exportation
A  (B  C) (A  B)  C

Distribution
A  (B v C) (A  B) v (A  C)
A v (B  C) (A v B)  (A v C)

Equivalence
A  B (A  B)  (B  A) (A  B) v (A  B)

The first sentence of each rule (in front of the arrow) can be logically replaced by the second (after the arrow)  and vice versa. In other words, the two statements logically say exactly the same thing.

There are also several argument forms, or syllogisms of symbolic logic. A syllogism is just a sound argument in a particular form. The following is a list of some common syllogisms and argument forms:

Modus Ponens
A  B
A     
B

Modus Tollens
A  B
B     
A

Hypothetical Syllogism
A  B
B  C 
A  C

Disjunctive Syllogism
A v B
A      
B

A v B
B      
A