The Elements of Logic

 

Part I: Basics

 

1. What is logic?

 

Logic is the formal, dialectical science of necessary implication.  By a formal science, we mean that it is concerned, like mathematics, with the transformation of symbols in the discovery of truth.  Dialectical means that it concerns itself primarily with argument: a construction of formal logic is called an argument.  We will discover what is meant by "necessary implication" as we progress in our study.

 

A few logical statements in natural language are:  "If the shipment hasn't come by Friday, then it will be here on Monday."  "Either we are going to McDonalds, or we are going to Burger King."  "If you buy one, then you will get one free." 

 

Logic is vital in fields as diverse as law, philosophy, and computer programming; it is almost impossible to persue these fields without some knowledge of formal logic, and the deeper one's understanding of logic, the more successful will one be in any of these fields.

 

This textbook is concerned with propositional logic, the construction and evaluation of arguments.  There is another "logic", informal logic, which concerns itself with the discovery of argumentative fallacies.  While informal logic is valuable to beginners in argumentation, as Hobbes says, "for the perfect logician it is needless, for his science comprehends it all." This book does not cover informal logic.

 

Vocabulary:

 

Logic: The science of necessary implication.

Argument: A construction of formal logic.

 

2. Propositions

 

The simplest logical argument is the proposition.  A proposition is a positive statement about the world: "this car is red."  It is hard to lay down definite rules as to what statements are, and which are not, propositions, but briefly it may be said that a statement which can be neither confirmed nor denied is probably not a proposition.  A proposition can be true or false.

 

In the calculus, propositions are symbolized by the lowercase letters of the alphabet (p, q, r, etc.), called "atoms".  If you run out of letters, additional atoms can be created by using "primes", symbolized ('), e.g. x', y', c', etc. (pronounced "x-prime", "y-prime", "z-prime"), and if you run out of primes, you can add double primes, triple primes and so on.  In this manner, the number of propositions contained in an argument may be practically infinite.

 

A proposition reduces in the calculus to an atom.  For instance, in a particular argument, any of "John is at his desk," or "Monkeys enjoy bannanas" might reduce to p.

 

4. Denial

 

A proposition can be denied by using the symbol ~.  The symbol ~, in natural language translates most simply into "not" - a more accurate, if more baroque, reading might be "it is not the case that...".  Thus, let us have the proposition p, which in this case means "it is raining outside."  If this is denied by the use of ~, it would form the argument ~p, or in natural language "it is not the case that (~) it is raining outside (p)."

 

When two denials are applied to the same proposition within an argument, they cancel each other.  Thus, ~~p may be written as simply p.

 

A proposition is true if its denial is false, and false if its denial is true.

 

5. Conjunction

 

In addition to denial, two propositions may be conjoined in an argument by the symbol (.), which in natural language is to be read as "and".  If the proposition "the car is gassed up" is symbolized by p, and the proposition "we are ready for our trip" is symbolized by q, then the argument (p.q) would be read "The car is gassed up and we are ready for our trip."

 

Arguments can mix conjunction and denial, but a single proposition cannot be simultaneously asserted and denied: such an argument is said to be "invalid," and an invalid argument is false.

 

Valid: p.~q.

 

Using the example above, this would be read "the car is gassed up, and it is not the case that we are ready to go." ("Either the car is gassed up, or we are ready to go.")

 

Invalid: p.~p.

 

In natural language, this would be read "we are ready to go, and we are not ready to go" - an obvious absurdity.

 

5.1 "But" and "Although"

 

As an aside, conjunctive words like "but" and "although", are symbolized in logic as the conjunction . and should be read as "and."  Their natural function is to draw attention to the second statement in any pair, but this function is one of rhetoric.  In logical use, "and" is appropriate.

 

5.2 "Because" and "Therefore"

 

The word "because," and its sibling "therefore," which occur often in natural arguments, are not symbolized in propositional logic: where they cannot be reduced to a simple "and,"  to rewrite the argument such that it no longer requires "because" or "therefore" for its sense tends to make the argument plainer and more scientific.

 

6. Grouping

 

Parts of arguments may be grouped by the use of parenthesis.  A ~ placed outside of a group of parenthesis applies a denial to each proposition within the parenthesis.  i.e. ~(p.q) would read "it is not the case that p and q," and is equivalent to the statement "not p, and not q."  The argument ~(p.~q) would read "It is not the case that p and not q." 

 

By grouping propositions and their denials, and reducing them to natural language, we arrive at the use of conditionals such as "if p then q," which form the subject of the second part.

 

Part II: Conditionals

 

6. Conditionals

 

Conditionals are the reduction to natural language of complex arguments.  A conditional is a statement which tells us that a proposition is true under certain conditions e.g. "If you buy one, you will get one free."  There are four sets of conditionals, each of which corresponds to a pattern of conjunction and denial in the calculus.

 

I should note that in some versions of the propositional calculus, the conditionals "if...then" and "either...or" are given their own special shorthand symbols.  While this is legitimate, they can be expressed in the calculus using conjunction, denial and grouping, and these expressions should be thoroughly understood before the shorthand symbols are adopted.  The relations between them cannot be understood using the shorthand or natural language alone and these relations give us some of the most powerful tools in the inferrential science.

 

6.1. Neither / nor

 

The conditional "neither...nor" is applied to an argument in which both propositions are denied, either "~(p.q)" or simply "~p.~q".

 

6.2. Either / or

 

The conditional "either...or" is applied to an argument in which "neither p nor q" is denied.  It is formed as "~(~p.~q)", "it is not the case that neither p nor q is true": if it is not the case that neither p nor q is true, then it must be that "either p or q (or both) are true".

 

6.3. If / then

 

The conditional "if...then" is applied to arguments which deny that one proposition is true and another is not true.  ~(p.~q), "it is not the case that p and not q," which implies that "if p is true, then q is true."

 

7.  The Circle of Denials

 

It should be noted that each conditional pair can be derived from denying some other conditional.  The serial denial of each conditional argument in turn yields the pattern called "the Circle of Denials", and to understand the Circle of Denials is to have a complete understanding of the possible inferences in any argument. 

 

The conditional argument:

 

"neither p nor q" (~(p.q)) is derived from denying the argument "both p and q" (p.q);

 

"either p or q" (~(~p.~q)) is derived from denying "neither p nor q" (~p.~q);

 

"if p then q" (~(p.~q)) is derived from the denial of "either p or q." (~(~p.~q)).