introduction to formal logic
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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.

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why isn t english a programming language
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
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 b true
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 b true1
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 b true2
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 b true3
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
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
More terms
  • Antecedent
  • Consequent
  • wff – well-formed formula
  • Tautology
  • Contradiction
  • Hypothesis
  • Algorithm
truth tables
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 tables1
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
To be continued . . .
  • Notable terms:
    • Modus ponens
    • Modus tollens
    • Valid argument
    • Equivalence rules
    • De Morgan’s laws
    • Hypothetical syllogism
    • Quantifiers
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