Conjunction
Within propositional logic, the term conjunction is used in the following ways: (1) as a particular type of propositional statement that is the result of combining component propositions, and (2) as the logical operator "and" used to form a conjunctive statement.
Some background concerning propositional logic is necessary in order to develop a precise definition of conjunction. As stated above, conjunction is defined in terms of propositional statements. Within so-called classical "Aristotelian" logic, a proposition is defined as a linguistic formation used to communicate information within certain constraints. Those constraints are the "principle of the excluded middle," which states that a proposition must either be true or false, and the "principle of the excluded contradiction," which states that a proposition cannot be both true and false. It should be noted that the words proposition and statement are usually used interchangeably within propositional logic.
A further principle regarding propositions is that they may be combined according to certain rules to yield compound propositions. This is where conjunction comes in: it is used as a logical operator to combine two or more propositions to form a compound proposition. In propositional logic the word "and" is used synonymously with conjunction. In everyday conversation the word "and" is used in much the same way as it is defined for use in classical logic. Suppose a boy tells his mother before going off to school that "I made my bed and I fed the goldfish." If the boy is telling the truth in this statement then both component statements must be true; that is, he did make his bed (true) and he did feed the goldfish (true). However, if the boy lies about even one of the two tasks then his original conjunctive statement is false. Borrowing from these common sense ideas, it is then the task of propositional logic to precisely define the meaning of conjunction as a logical operator.
Though the ancient Greeks first developed a formal, propositional logic (carried out with ordinary conversational statements), it was only with German mathematician and philosopher Gottfried Leibniz (1646-1716) that the need for a totally "symbolic" logic was seriously considered. Starting with Leibniz's work, propositions and notation in logic would eventually be represented with symbols instead of just words. Hence, one may consider symbolic propositional logic where the symbol "" (and sometimes the dot "⋅") is used to represent "and" or "conjunction." The conjunction of the two propositions A and B to form a third proposition may then be written completely symbolically as A B or A ⋅ B, where the statements A and B are called conjuncts. The truth or falsity (called the truth-value) of the conjunctive statement A B is dependent solely upon the corresponding truth-values of statements A and B. This situation is best illustrated via the use of a truth table, as shown below:
This table demonstrates the four possible combinations of truth/falsity for statement A (first column), statement B (second column), and the resulting truth-value for the conjunction A B (third column), where T = true and F = false. As demonstrated by the truth table, a conjunction is true only if its two component propositions are true. Although the truth table defines conjunction as a binary operation upon truth-values, it may be applied to "n" number of statements for integer values of n 2. For example, A B C is a valid statement. To determine its truth-value evaluate the truth-value of one component conjunction, say A B, and in turn use its truth-value in a conjunction with statement C. This process could be represented symbolically as (A B) C, where the conjunctive statement in brackets (A B) has its truth-value calculated first, and then the truth-value of the entire compound statement decided.
Of course conjunction is not the only logical operator. Conjunction is used along with other logical operators, like "or," "implication," "equivalent," and "not" to form compound statements. For example, "~" (not) operates on a statement to yield the opposite truth-value. That is to say, if statement A is true then ~A (read "not A") is false; likewise if A is false then ~A is a true statement. Hence, the logical operators "" and "~" can be used to form compound statements. The statement A (~A) is an entirely permissible statement within symbolic propositional logic. Furthermore, it happens to be false for each possible truth-value of its component statements. That is, by the principles of the excluded middle and excluded contradiction, statement A must be either true or false (but not both true and false). If statement A is true then ~A is false and by the truth table for conjunction A (~A) is false. On the other hand, if A is false then (according to the truth table) A (~A) is false. One can imagine that by using these simple logical operators, large and complex compound statements can be formed.
The use of conjunction in this article has been repeatedly associated with propositional logic, the origins of which can be traced back to the ancient Greeks and especially to Greek philosopher Aristotle (384 b.c.-322 b.c.), which uses statements, truth-values, and logical operators (like conjunction) to perform deductive reasoning. However, conjunction is used as a symbolic logical operator in other types of logic that may generally be called "non-Aristotelian" because of differences from classical propositional logic. For example, "m-valued" logics have been constructed that violate the principle of the excluded middle (i.e., two possible truth values for a statement are replaced with "m number" of truth-values, where m is an integer 3). Obviously those sorts of logical systems would dramatically alter the truth table for conjunction (if a conjunction operator was retained at all).
It should also be noted that the term conjunction is used in Boolean algebra, named after English mathematician and logician George Boole (1815-1864), a sort of "algebraic logic" involving notations and methods that represents relationships among classes or sets. The connective symbol ⊕ is called the conjunction and represents the additive operation.
This is the complete article, containing 990 words
(approx. 3 pages at 300 words per page).
