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The Founda+ons: Logic and

Proofs

Chapter 1 , Part I: Proposi+onal Logic

Edited by Shih-Tsung Liang

Chapter Summary

• Proposi+onal Logic ( 命題邏輯)
  • The Language of Proposi+ons
  • Applica+ons
  • Logical Equivalences
• Predicate Logic (述詞邏輯)
  • Predicates(述詞) and Quan+fiers(量詞)
  • Logical Equivalences (等價)
  • Nested Quan+fiers
• Proofs
  • Rules of Inference
  • Proof Methods
  • Proof Strategy

Proposi+ons

• A proposi&on (命題) is a declara+ve sentence (陳述句) that is

either true or false.

• Examples of proposi+ons:

a) The Moon is made of green cheese.

b) Trenton is the capital of New Jersey.

c) Toronto is the capital of Canada.

d) 1 + 0 = 1

e) 0 + 0 = 2

• Examples that are not proposi+ons.

a) Sit down!

b) What +me is it?

c) x + 1 = 2

d) x + y = z

Construc+ng Proposi+ons

• Proposi+onal Variables: p , q, r , s , …
• The proposi+on that is always true is denoted by T and the proposi+on that is always false is denoted by F.
• Compound Proposi+ons; constructed from logical connec+ves and other proposi+ons - Nega+on (否定) ¬ - Conjunc+on (合取) ∧ - Disjunc+on (分取) ∨ - Implica+on (蘊涵) → - Bicondi+onal (雙條件；若且唯若) ↔

Conjunc+on (合取)

• The conjunc&on of proposi+ons p and q is denoted by p ∧ q and has this truth table:
• Example : If p denotes “I am at home.” and q denotes “It is raining.” then p ∧ q denotes “I am at home and it is raining.”

p q p ∧ q

T T T T F F F T F F F F

Disjunc+on(分取)

• The disjunc&on of proposi+ons p and q is denoted by p ∨ q and has this truth table:
• Example : If p denotes “I am at home.” and q denotes “It is raining.” then p ∨ q denotes “I am at home or it is raining.”

p q p ∨ q

T T T T F T F T T F F F

Implica+on(蘊涵)

• If p and q are proposi+ons, then p → q is a condi&onal statement or implica&on which is read as “if p, then q ” and has this truth table:
• Example : If p denotes “I am at home.” and q denotes “It is raining.” then p → q denotes “If I am at home then it is raining.”
• In p → q , p is the hypothesis ( antecedent or premise ) and q is the

conclusion (or consequence ).

p q p → q

T T T T F F F T T F F T

Understanding Implica+on

• In p → q there does not need to be any connec+on between

the antecedent or the consequent. The “meaning” of p → q

depends only on the truth values of p and q.

• These implica+ons are perfectly fine, but would not be used in ordinary English. - “If the moon is made of green cheese, then I have more money than Bill Gates. ” - “If the moon is made of green cheese then I’m on welfare.” - “If 1 + 1 = 3, then your grandma wears combat boots.”

Different Ways of Expressing p → q

if p, then q p implies q

if p, q p only if q

q unless ¬p q when p

q if p q when p

q whenever p p is sufficient for q

q follows from p q is necessary for p

a necessary condi:on for p is q

a sufficient condi:on for q is p

Converse (逆命題), Contraposi+ve(對位命題), and Inverse(反命題)

• From p → q we can form new condi+onal statements.
  • q → p is the converse of p → q
  • ¬ q → ¬ p is the contraposi:ve of p → q
  • ¬ p → ¬ q is the inverse of p → q Example : Find the converse, inverse, and contraposi+ve of “It raining is a sufficient condi+on for my not going to town.” Solu:on: converse : If I do not go to town, then it is raining. inverse : If it is not raining, then I will go to town. contraposi:ve : If I go to town, then it is not raining.

Expressing the Bicondi+onal

• Some alterna+ve ways “ p if and only if q ” is

expressed in English:

• p is necessary and sufficient for q
• if p then q , and conversely
• p iff q

Truth Tables For Compound Proposi+ons

• Construc+on of a truth table:
• Rows
  • Need a row for every possible combina+on of values for the atomic proposi+ons.
• Columns
  • Need a column for the compound proposi+on (usually at far right)
  • Need a column for the truth value of each expression that occurs in the compound proposi+on as it is built up. - This includes the atomic proposi+ons

Equivalent (等價)Proposi+ons

• Two proposi+ons are e quivalent if they always

have the same truth value.

• Example : Show using a truth table that the

implica+on is equivalent to its contraposi+ve.

Solu:on:

p q ¬ p ¬ q p → q ¬ q → ¬ p

T T F F T T T F F T F F F T T F T T F F T T T T

Using a Truth Table to Show Non- Equivalence

Example : Show using truth tables that neither

the converse nor inverse of an implica+on are

not equivalent to the implica+on.

Solu:on:

p q ¬ p ¬ q p → q ¬ p →¬ q q → p

T T F F T T T T F F T F T T F T T F T F F F F T T F T T