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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