Books > Mathematics >
Bivalent Logic ©2012
A clear and succinct introduction to the two-valued logic pioneered by Aristotle. Associated topics covered include: the combinatorial basis of truth tables, Boolean Vector Arithmetic (an efficient implementation of truth tables utilizing matrices), rules of inference, the deductive argument form, as well as a brief logical analysis of Descartes' most famous argument.
...excerpt from Bivalent Logic:
Table of Contents
¶1 Introduction to Bivalence
¶2 Logical Operators and Sentence Types
¶3 Truth Tables and Combinatorics
¶4 Introduction to Boolean Vectors
¶5 Restricting Result Sets to Zeros and Ones
¶6 Boolean Vector Arithmetic: Reversal
¶7 Boolean Vector Arithmetic: + and ×
¶8 BVA: + and × (continued)
¶9 Boolean Vector Arithmetic: - and ÷
¶10 BVA: - and ÷ (continued)
¶11 Rule of Bivalence (Biv)
¶12 Rule of Double Negation (DN)
¶13 Commutation (Com) and DeMorgan (DeM)
¶14 The Deductive Argument Form
¶15 Conj, Simp, and DS
¶16 MP, MT, Trans, Impl, and Bicon
¶17 Examples using MP, MT, Trans, Impl, and Bicon
¶18 Rules of Inference as Tautologies
¶19 The Standardization of Vectors
¶20 Association (Assoc) and Distribution (Dist)
¶21 HS, Exp, and CD
¶22 Descartes' Premises
¶23 Deducing that 'I am'
¶1 Either it is or it is not. It cannot both be and not be. As long as we are agreed on these two points you may want to continue, otherwise it is not recommended. What was stated above is two of Aristotle’s three laws of thought, which are the basis of bivalent logic. That is, logic in which every statement is assigned exactly one of two possible truth values. Restating the laws in terms of bivalence: it is either true or it is false, it cannot be both. 'It' may refer to several things: the existence of something, the occurrence of an event, a sentence uttered or thought, for example. However, through the lens of logic all of these are seen in the form of statements. A statement is a sentence which asserts something, like “we hit the deck,” in contrast to non-statements such as questions [should we hit the deck?], opinions [we should not hit the deck], and commands [hit the deck!]. What distinguishes statements is the fact that they can be assigned a truth value. Either we actually did hit the deck and the statement “we hit the deck” is true (T), or we did not hit the deck and the statement “we hit the deck” is false (F). To do bivalent logic, we must bear these two distinctions – statements/non-statements, true/false – always in mind.
¶2 The five basic logical operators are 'not' (¬), 'and' (∧), 'or' (∨), 'if...then...' (→), and 'if and only if' (↔). Of these five, only 'not' is a unary operator, since it operates on a single statement. To give an example, if we let A equal 'we hit the deck', then ¬A is the negation of A, 'we did not hit the deck'. Verily, each operator corresponds to a sentence type. As we just saw, the ¬ operator corresponds to negation. The other four operators are binary, since they operate on at least two statements. Since ¬A asserts the opposite of A, it is considered a different statement than A: viz. the statement ¬A ≠ A is true. We can thus utilize A and ¬A to build the other sentence types. The ∧ operator corresponds to conjunction, A∧¬A 'we hit the deck AND we did not hit the deck'. In this case A and ¬A are called conjuncts. The ∨ operator correponds to disjunction, A∨¬A, 'we hit the deck OR we did not hit the deck', in which case A and ¬A are disjuncts. The → operator corresponds to conditional, A→¬A, 'IF we hit the deck THEN we did not hit the deck', in which case A is the antecedent and ¬A is the consequent. The conditional sentence type is also known as implication, and can alternately be read 'A implies ¬A'. Finally, the ↔ operator corresponds to biconditional, A↔¬A, 'we hit the deck IF AND ONLY IF we did not hit the deck'. Before we can understand biconditional we must understand three further sentence types formed from the modification of the conditional: converse, inverse, and contrapositive. Using our previous conditional (A→¬A), the converse is ¬A→A, the inverse is ¬A→¬¬A, and the contrapositive is ¬¬A→¬A. The converse is formed by reversing the roles of the components, consequent becomes antecedent and antecedent becomes consequent. The inverse is formed by negating both components, and the contrapositive (a combination of the other two) is the inverse of the converse. That being said, we can now define the biconditional. Put simply, it is shorthand notation for the conjunction of a conditional and its converse, which in the case of A↔¬A translates into (A→¬A)∧(¬A→A).
¶3 Now we will evaluate the sentences above for truth or falsity. For this exercise let us say that in reality we truly did hit the deck. Knowing that, it is immediately evident that A is true and ¬A is false. In bivalent terms we say that A has truth value T and ¬A has truth value F. Here we come upon a proper definition of negation: a statement that is false when its component is true and true when its component is false. We represent this definition graphically utilizing a table containing truth values, called a truth table.
To understand why a truth table contains the number of rows and columns that it does we must turn to the fundamental counting principle (also known as the product rule of combinatorics): if a task consists of k parts, the first part of which is performable in n1 ways, the second in n2 ways, and on up to the kth part, which is performable in nk ways, then the total number of ways to complete the task is exactly n1×n2×n3×...×nk. Since every component statement must be either true or false, there are two possible truth values for each component under consideration (n = 2); it follows that if we have 1 component (i.e. A), then any number of statements involving it alone will have 21 possible concurrent truth values. We can now explain the above truth table. It has two columns because we are evaluating two statements, A and its negation; it has two rows because the two statements contain only 1 component. Generalizing this result for truth tables involving any number of components we obtain the rule: a truth table involving k components will require precisely 2k rows. To see this rule in action let us introduce a new component, B. Remember, B can stand for any statement, so we need not explicitly state it. Here is the truth table for the conjunction of A and B:
|A||B||A ∧ B|
We can see that since we have two components we now have 2² rows. Using this we can properly define conjunction: a statement which is true when both of its conjuncts are true (row one) and is false otherwise (rows two through four).
Go to Top