PHIL-57 Lecture 02 Fundamentals of Logic

Binary Logic

A logical system based on two “truth values,” true or false, that obey the Three Laws of Thought.

The Three Laws of Thought

Identity, Non-Contradiction, Excluded Middle

Law of Identity

A is A (& A is not B); A thing is identical only to itself

Low of Non-Contradiction

A and not-A cannot both be true; A statement and its contradictory cannot both be true.

Law of the Excluded Middle

A or not-A must be true; either a statement or its contradictory must be true.

Many-Valued Logics

Logical systems based on more than two truth values, intended to allow (for example) reasoning about propositions that lack meaning (exist in an undefined state), or definiteness (are ambiguous or contingent), or precision (proportions that use vague terms like “tall,” which may apply to a greater or lesser degree). Systems that deal with vague propositions are often called “fuzzy logic.”

Contradictories

Every proposition has a contradictory, formed by prefixing it with the logical operator “not,” which reverses its truth value. Thus, the contradictory of proposition “A” is “not-A.” A pair of contradictories is exclusive and exhaustive, i.e., they cannot both be true and they cannot both be false (one must be true, the other false). If “A” is true, “not-A” must be false; if “A” is false, “not-A” must be true.

Contraries

Two propositions that are exclusive, i.e., they cannot both be true, but are not exhaustive, i.e., they can both be false. Contraries are generally terms that fall at opposite ends of a common scale, e.g., “hot” and “cold” on the temperature scale. If an object is at the “hot” end of the temperature scale, it cannot be at the “cold” end and vice versa. But an object need not be at either extreme, but may instead occupy an intermediate position on the scale: warm, room temperature, or cool.

Common Sense Realism

a metaphysical position that posits the existence of an objective external world apart from the subjective private world of out thoughts and feelings. We exist in this world, able to experience it though our senses of sight, hearing, smell, taste, and touch, and can affect it through our actions, but the facts that describe reality are independent of and unaffected by our subjective states.

Correspondence Theory of Truth

Propositions about reality are true if and only if they correspond to a fact about reality. Facts remain facts whether or not they are believed or known.

Sentences vs. Propositions

Sentences are units in a language. Propositions are the true or false content of sentences and are language-independent, i.e., the fact that “grass is green” is the same regardless of the language or the choice of words used to express this fact.

Empiricism

An epistemological position about human knowledge that holds all truths are either analytic or synthetic.

Analytic Truths

are certain, but only reflect conventions of human linguistic systems. The truths of mathematics are analytic. For example, a mathematical truth about the number of parallel lines that can be drawn through a point not on a given line (0, 1, or an infinite number ) depends on the conventions adopted by a geometric system

Synthetic Truths

are propositions about reality. These are never certain but always subject to correction or replacement if new factual information is discovered. Synthetic truths about reality are uncovered and systematized through science, and grounded at bottom in experience.