Conjunction, Negation, and Disjunction

Abstract: The logical operations of conjunction, negation, and disjunction (alteration) are discussed with respect to their truth-table definitions.

Truth Functionality: In order to know the truth value of the proposition which results from applying an operator to propositions, all that need be known is the definition of the operator and the truth value of the propositions used.

Conjunction is a truth-functional connective similar to "and" in English and is represented in symbolic logic with the dot "

Ordinary language definition of the dot: a connective forming compound propositions which are true only in the case when both of the propositions joined by it are true.

One way of expressing this definition is by way of truth tables. Consider the following examples.

"John left and Carol arrived" can be symbolized as " J C " (i. e., (without the quotation marks), so long as we remember that the statement does not mean "Carol arrived after John left" which is a simple proposition).

There are four possible states of affairs which might have occurred with respect to John leaving and Carol arriving. These cases can be listed as follows in what is called a truth table.

p	q	p q

T	T	T
T	F	F
F	T	F
F	F	F

Other ordinary language conjoiners besides "and" include some uses of "but," "although," "however "yet," and "nevertheless."
The dot as a truth functional connective doesn't do everything that the "and" does in English. It might be thought of in terms of a "minimum common logical meaning" to conjoined statements.
1. I.e., the temporal or causal sequence "Bill tripped and fell" cannot be transposed as "Bill fell and tripped." The clauses cannot be interchanged.
2. Truth functional connectives are more limited than their corresponding English connectives: the whole meaning of the truth functional connective is given in its truth table.
3. So long as we do not expect more from truth-functional connectives, there should be few difficulties in translation.
Some characteristics of conjunction (in mathematical jargon) include:
1. associative—internal grouping is immaterial
  I. e.," [(p q) r] " is equivalent to " [p (q r)] ".
2. communicative—order is immaterial
  I. e., " p q " is an equivalent expression to " q p ".
3. idempotent—reduction of repetition
  I. e., " p p " is an equivalent expression to " p ".

Which brings us to an overdue additional convention: lower-case letters are variables, the small letters of the English alphabet usually beginning with letters after " p "(toward the end of the alphabet).
1. A variable is not a proposition, but is a "place holder" for any proposition.
2. Think of a variable as a "labeled box" which can be filled with any proposition, so long as we set up a correspondence between the "labeled box" and the variable.
3. E. g., just as "All S is P" is the form of statements like "All men are mortal" and "The whale is a mammal," " p q " is the form of statements like "John left and Carol arrived" and " J C " (which symbolizes the statement "John left and Carol arrived."
4. E. g., suppose Alice and Betty are in this room, but Charles is not. The form of the statement corresponding to each person being in the room is
  
  [(p q) r]
  
  and the statement "Alice is in this room and Betty is in this room, and Charles is in this room" can, itself, be symbolized as
  
  [(A B) C]
  
  The truth of the compound expression is analyzed by substituting in the truth values corresponding to the facts of the case, viz.,
  
  [(T T) F]
  
  so by the meaning of the " " the compound statement resolves to being false by the following step-by-step analysis in accordance with the truth table for conjunction:
  [(T T) F]
  [( T ) F]
  [ T F]
  F

Disjunction (or as it is sometimes called, alternation) is a connective which forms compound propositions which are false only if both statements (disjuncts) are false.

The connective "or" in English is quite different from disjunction. "Or" in English has two quite distinctly different senses.
1. The exclusive sense of "or" is "Either A or B (but not both)" as in "You may go to the left or to the right." In Latin, the word is "aut."
2. The inclusive sense of "or" is "Either A or B {or both)." as om "John is at the library or John is studying." In Latin, the word is vel."
It is the second sense that we use the "vel" or "wedge" symbol:
" "

The truth table definition of the wedge is

p	q	p q

T	T	T
T	F	T
F	T	T
F	F	F

Consider the statement, "John is at the Library or he is Studying." If, in this example, John is not at the library and John is not studying, then the truth value of the complex statement is false:

F F
F

Another truth functional operator is negation: the phrase "It is false that …" or "not" inserted in the appropriate place in a statement.

The phrase is usually represented by a minus sign " - " or a tilde "~"
For example, "It is not the case that Bill is a curious child" can be represented by "~B".

The truth table for negation is as follows:

p	~ p

T	F
F	T

The general principles that govern parentheses for grouping are as follows.

A " ~ " standing in front of a letter negates only that proposition, while a " ~ " in front of an expression in parentheses negates the whole compound statement within those parentheses.

Note the difference between: ~ A B and ~( A B ).

Each occureence of a connective has associated with it a set of parentheses which indicate what it is connecting.

Hence, ( A

B )

C is quite different from A

( B

C )

E. g., let A and B be false, and let C be true. The resolution of the truth value of these expressions would be as follows.

( A

B )

C
( F

F )

T
F

T
T

( B

C )
F

( F

T )
F

T
F