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Chapter 3: Logic
Logic: science of correct reasoning
Aristotle (384­322 BC): "Father of Logic"; searched for universal truths that were irrefutable. Aristotelian or syllogistic logic is a deductive process of arriving at a conclusion.
Deductive Reasoning: reasons from the general to the specific; reasons from a given set of accepted premises to a conclusion which must be true if the premises are true.
ex. All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
Inductive Reasoning: reasons from specific to general by looking at cases; the conclusion is probably true.
ex.
1, 10, 100, 1000, ______, ______
Intuitive Reasoning: based on sense, feeling, hunch, insight; the conclusion is probably true.
ex.
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Flaws of Aristotelian Logic: no systematic and orderly scheme to enable rapid deductions; limited use of symbols hindered further development.
Leonard Euler (1701­1783): used diagrams to determine validity of syllogisms. George Boole (1815­1864): "Father of Symbolic Logic", developed logic as an abstract mathematical system consisting of undefined terms, operations (conjunction, disjunction, negation), and rules for using the operations.
Boole's Basic Idea: If symbols could be used to represent simple propositions, then reasoning could be reduced to manipulations of symbols. This "algebra of logic" or Boolean algebra is used in the design of computers!
Other Famous Logicians:
Augustus De Morgan (1806 ­1872)
John Venn (1834 ­1923)
Rev. Charles Lutwidge Dodgson (1832 ­ 1898)
Symbolic logic is covered in Sections 3.1 ­ 3.6.
Syllogistic logic is covered in Section 3.7.
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Section 3.1: Statements, Negations, and Quantified Statements (Objectives 1­ 5, 7)
Statement: a sentence that is either true or false.
ex. The moon is made of blue cheese.
False
Use lower case letters, such as p, q, r, and s to represent them.
Negating Simple Statements
Negate means change the truth value. The negation of p is denoted ~p.
ex. p: I enjoy teaching.
~p: I do not enjoy teaching.
True
False
q: The President is not in Tallahassee. True
~q: The President is in Tallahassee.
False
To negate a simple statement, insert or remove "not". Trivially, insert "It is not true that ……" in front of the statement.
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Negating Quantified Statements Universal Quantifiers
all
each
every
no
none
Existential Quantifiers
some
at least one
there exists
ex. Some people are lonely.
Some people are not lonely.
True
Still true!!
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Negate the following:
All cats are black. _____________________________
Some people are lazy. _____________________________
Some people do not eat pizza. __________________________ No Americans are patriotic. ____________________________
None of my students are tardy. __________________________
Some days are exhausting. _____________________________
All my dreams are nightmares. __________________________
Today is a holiday.
_____________________________
See text ex. 1 ­ 4.
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