Introduction to Formal Logic

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# Introduction to Formal Logic - PowerPoint PPT Presentation

Introduction to Formal Logic. CSC 333. Why isn’t English a programming language?. Ambiguity! Words with double meanings. Meanings of clauses depend on Punctuation Spoken emphasis Context Order of words Lack of precision And so forth. Formal Logic. Eliminates ambiguity.

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### Introduction to Formal Logic

CSC 333

Why isn’t English a programming language?
• Ambiguity!
• Words with double meanings.
• Meanings of clauses depend on
• Punctuation
• Spoken emphasis
• Context
• Order of words
• Lack of precision
• And so forth
Formal Logic
• Eliminates ambiguity.
• Provides a means of conveying information clearly.
• Uses capital letters to represent statements. For example, A could represent “CSC 333 is fun” and B could represent “CSC 333 is easy.”
• Uses logical connectives to connect statements.
• Λ means “and”.
Is A Λ B true?
• If CSC 333 is indeed fun and CSC 333 is easy, then we could agree that A Λ B is true.
• If CSC 333 isn’t fun, but it’s easy, is A Λ B true?
• If CSC 333 is hard, but it’s fun, is A Λ B true?
• If CSC 333 is boring and you can’t pass it, is A Λ B true?
Is A Λ B true?
• A Λ B is true only when A is true and B is true.
• The truth value of A Λ B really depends on whether the truth value of A and the truth value of B, which can vary depending on the student, the teacher, the book, and other variables.
• We can show the situations that determine the truth or falsity of A Λ B in a truth table.
Is A Λ B true?
• A truth table should contain all possible combinations of truth values for its statement letters.
• For example,

AB

t t

t f

f t

f t

Is A Λ B true?
• We can see that all possible combinations of truth values are represented.

AB A Λ B

t tt

t f f

f t f

f t f

• The table shows what we already know! That A Λ B is true only when both A is true and B is true.
Terms
• Proposition
• A statement (basic logical component) that is either true or false.
• Logical connective
• Binary - Logical operator connecting propositions.
• Conjunction (and); disjunction (or); implication (if-then); equivalence (is the same as).
• Unary – Logical operator acting on one logical component.
• negation (not).
• Compound statement
• A statement composed of multiple propositions connected by logical connectives.
More terms
• Antecedent
• Consequent
• wff – well-formed formula
• Tautology
• Hypothesis
• Algorithm
Truth tables
• Each row in a table shows a unique combination of truth values.
• A truth table must contain all possible combinations of truth values.
• A table with n statements must have how many rows?
• 2n
Truth tables
• Consider the truth table on page 7.
• Note that there are 2 statements and, thus, 22 rows.
• Also, note the role that precedence plays in interpretation of the wff.
• In English, how can we summarize this situation? When is the wff true?
To be continued . . .
• Notable terms:
• Modus ponens
• Modus tollens
• Valid argument
• Equivalence rules
• De Morgan’s laws
• Hypothetical syllogism
• Quantifiers