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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
- Contradiction
- 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

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