LOGIC A BASIC OVERVIEW WHAT IS LOGIC Logic
LOGIC A BASIC OVERVIEW
WHAT IS LOGIC? • Logic is mankind’s attempt to justify its thinking, and thus it is an unfinished, ongoing process that currently has many different branches, which in turn have many different definitions. • In Western thinking, logic starts with the Greeks who created a systemic study of the form of verbal arguments. In fact the word “logic” is derived from the Greek word for “what is spoken. ” • In the simplest, modern terms logic is structured thought or reason.
FORMAL LOGICAL SYSTEMS • Consistency – no theorem of the system of logic contradicts another • Validity – the system’s rules of proof never allow a false inference from true premises • Completeness – in order to be a theorem, a formula must be provably true • Soundness – only true formulas are used as theorems in a system
DEDUCTIVE REASONING • It is the process of reasoning from one or more statements to reach a logically certain conclusion. • It is called deduction because it starts with relatively broad premises and reaches a relatively narrow conclusion – thus it has deducted (think “reduced”) the relative size of the relevant premises to create a more specific conclusion. • Deductive arguments are often syllogistic in nature in that they use a logical formula called a syllogism to reach a conclusion.
BASIC SYLLOGISMS 1 PART • A syllogism is a combination of two broad statements that lead to a conclusion • Major Premise: a general statement • Minor Premise: a related, more specific statement • Therefore the Conclusion: a statement derived from the previous statements • Each statement follows a basic structure • A: All A are B (Aa. B) Universal Affirmative • E: No A are B (Ae. B) Universal Negative • I: Some A are B (Ai. B) Particular Affirmative • O: Some A are not B (Ao. B) Particular Negative
BASIC SYLLOGISMS II PART • The statements and conclusion of the syllogism are made up of three parts • S: the subject of the conclusion • P: the predicate of the conclusion • M: the middle term does not appear in the conclusion, but it does appear in both premises • The major premise links M and P mortal Example: All M are P • The minor premise links S and M men Example: All S are M All Greeks are • The conclusion links S and P All men are Example: All S are P All Greeks are mortal
BASIC SYLLOGISMS III PART EXAMPLES • Celarent (EAE-1) • No reptiles have fur (Me. P) • All snakes are reptiles (Sa. M) • ∴ No snakes have fur (Se. P) • Darii (AII-1) • All rabbits have fur (Ma. P) • Some pets are rabbits (Si. M) • ∴ Some pets have fur (Si. P)
BASIC SYLLOGISMS IV PART EXAMPLES • Ferioque (EIO-1) • No homework is fun. (Me. P) • Some reading is homework (Si. M) • ∴ Some reading is not fun (So. P) • Baroco ( AOO-2) • All informative things are useful (Pa. M) • Some websites are not useful (So. M) • ∴ Some websites are not informative (So. P)
BASIC SYLLOGISMS V PART EXAMPLES • Bocardo (OAO-3) • Some cats have no tails (Mo. P) • All cats are mammals (Ma. S) • ∴ Some mammals have no tails (So. P) • Darapti (AAI-3) • All squares are rectangles (Ma. P) • All squares are rhombuses (Ma. S) • ∴ Some rhombuses are rectangles (Si. P)
INDUCTIVE REASONING • It is the process of reasoning from one or more statements of evidence to reach a logically probable conclusion. • It is called induction because it starts with relatively specific premises and reaches a relatively broad conclusion – thus it has induces (think “expands”) the relative size of the relevant premises to create a larger conclusion. • Inductive arguments are often less formal in nature and rely on the degree of support provided by inductive probability. • It is important to note that inductive conclusions are NOT CERTAIN.
INDUCTIVE PROBABILITY PART 1 • While deduction relies on the axiomatic circumstance of its premises to produce valid conclusion, induction relies on the quality of its observations. • Simple Induction: the inductive conclusion proceeds from an observation about a sample of a population. • Example: All observed crows are black, therefore all crows are black. • Statistical Syllogism: the inductive conclusion is a statement of a statistical likelihood about a member of an observed population. • Example: 66 members of a class of 100 are girls, therefore there is a 2 in 3 chance a member of the class randomly selected will be a girl.
INDUCTIVE PROBABILITY PART II • Generalization: the inductive conclusion about a population is derived from a sample of that population. • Example: There are twenty students in a class and the teacher calls four random names – three boys and a girl. It can be inferred that there are fifteen boys and five girls in the class. • Argument from Analogy: the inductive conclusion about one population is derived from observing a similar population. • Example: Chickens that have feathers, beaks, and toes have been observed to lay eggs; therefore, penguins that have feathers, beaks, and toes will also lay eggs.
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