AN INTRODUCTION TO LOGIC

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AN INTRODUCTION TO LOGIC

• by
• Jerry K. Adams
• Fall 2004

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SENTENCES

• TYPES
• Interrogative
• Asks a question.
• Imperative
• Gives a command.
• Exclamatory
• expresses surprise. 
• Declarative
• gives information.

• EXAMPLES
• What is logic? Why should we study it?
• Sit down.  Shut up.  Listen carefully.
• Wow!  Real Nice Car! Boy, shes pretty!
• The car is fast.  Today is Tuesday. He drives to
  work.

Only declarative sentences can become STATEMENTS.
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Statements

• Statement a declarative sentence that can be
  determined to be true or false.
• The earth revolves around the sun. (True)
• Two is equal to five. (False)
• This statement is false.
• Not a valid statement.
• It cannot be classified as true or false.
• Negation of a statement a statement that has
  the opposite truth value of the given statement.
• It is false if the given statement is true.
• It is true if the given statement is false.
• They cannot both be true nor can both be false.

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Symbolic Logic

• Logic examines the truth of statements and
  validity of arguments.
• Symbolic logic uses letters in place of simple
  statements.
• This allows us to avoid conclusions based on
  biases or connotations inherent is some words.
• Usually the following letters are used
• p, q, r, s, t, etc.

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Quantifiers

• Universal refers to every element in the set.
• Indicated by the use of words such as all,
  every, none, not any
• Existential refers to one or more of the
  elements in the set. (There is at least one thing
  that satisfies the conditions.)
• Indicated by the use of words such as some,
  there exists at least one

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Negation Of QuantifiersNegation of an
existential statement will be a universal
statement and vice-versa.

• Statement
• Some A are B
• Some A are not B
• All A are B
• No A is B

• Negation
• No A is B
• All A are B
• Some A are not B
• Some A are B

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Negation Of QuantifiersExamples

• All students like hamburgers
• Negation At least one student does not like
  hamburgers.
• Or Some students do not like hamburgers
• Some people like mathematics
• Negation No people like mathematics
• Or All people dislike mathematics

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TRUTH TABLES

• Truth Tables are used to show ALL possible
  true-false patterns for a statement.
• First make a title row with a column for each
  different simple statement represented by a
  single letter.
• The additional number of columns depends on the
  structure of the sentence.
• The next step is to determine the number of
  additional rows needed in the table.
• The number of rows depends on the number of
  simple statements in the total statement.
• The number of rows 2 n where n is the number of
  simple statements.
• Examples are shown in the next slide.

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Examples Of Row Calculations
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Truth Tables Listing all the possible
combinations of truth values

• First divide the number of truth value rows by 2.
• Place a T in that many rows of the first column.
• Place an F the remaining rows in that column.
• Divide the previous number by 2 again.
• In the second column, place a T in that many
  rows.
• Next, place an F in the same number of rows below
  the ones with T in them.
• Repeat the process in step II, alternating the T
  and F until the simple statement columns are
  filled.
• The last simple statement column will repeat T F
  T F all the way down the column.

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Examples of Table Set Up

• Two Statements Three Statements Four
  Statements

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Truth Table of the negation of one simple
statement P
One statement needs 2 rows A tilde ( ) is used
to indicate the negation of a statement. P can be
either True or False, so T is placed in the first
row and F in the second row.
When P is True, its negation, P, is False, so an
F is placed in the first row of the second
column. When P is False, its negation P, is
True, so a T is placed in the second row of the
second column.
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Compound Statements

• In logic (and in English) simple statements are
  combined to form compound statements.
• They are connected using conjunctions,
  disjunctions, implications, etc.
• The sky is blue and the sun is shining.
• I will go to the game or I will give you my
  ticket.
• In logic AND is called a conjunction, and OR is
  called a disjunction
• BUT is also a conjunction.

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Basic Truth Tables AND ( the Conjunction )

• A compound AND statement is true only when both
  statements are true.

• Example
• Let P be I ate spinach.
• Let Q be I won the race.
• I ate spinach and I won the race.
• Would be true only if I did both.
• Symbolically it would be written

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Basic Truth Tables OR (the Disjunction)

• A compound OR statement is true when at least
  one part is true and false only when both
  statements are false.

• Example
• Using P and Q from before.
• I ate spinach OR I won the race.
• Would be true as long as I did at least one of
  them.
• Symbolically it would be written

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Proper Use of OR

• When used formally, especially in Mathematics and
  Logic, OR actually means one or the other or
  both.
• This is referred to as the inclusive OR.
• In everyday usage, many people use OR to mean
  one or the other but not both.
• This is called the exclusive OR.
• Make sure you use OR properly!

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Basic Truth Tables If , then (the
Conditional )

• A Conditional ( or Implication) statement is
  true except when the conditions are met but the
  promise is not kept.

• Example
• Using P and Q from before.
• If I ate spinach THEN I won the race.
• Would be true unless I ate spinach but lost the
  race.
• Symbolically it is written

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Basic Truth Tables If and only if the
(Bi-conditional )

• A Bi-conditional (OR Equivalence) statement is
  true when both parts are true or both are false.
• It is true when the statements have the same
  truth value.
• Also referred to as the Double Implication.

• Example
• Using P and Q from before.
• I ate spinach, if and only if, I won the race.
• Would be true only if I did both.
• Symbolically it would be written

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TRUTH TABLE COMPOUND STATEMENT

• Create a truth table for
• Start with the table for P and Q.
• Then negate the next column.
• (Swap T s and F s.)

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RELATED STATEMENTS

• Any conditional has three other conditionals
  related to it.
• Conditional If P, then Q P ? Q
• If I win, then I will celebrate.
• Converse if Q, the P Q ? P
• If I celebrate, then I won.
• Inverse if not P, then not Q P ? Q
• If I dont win, then I dont celebrate.
• Contrapositive If not Q, then not P Q ? P
• If I dont celebrate, then I did not win.

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TRUTH TABLE Contrapositive

• Create the table by adding appropriate columns.

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EQUIVALENT STATEMENTS

• Statements are equivalent if they have the same
  truth tables (same last column).
• A conditional and its contrapositive are
  equivalent. (Compare the last column in each.)