Logic Study Guide: Chapters 2.2 - 3.4

I'll create a comprehensive study guide for your logic exam covering chapters 2.2-3.4. This will include all important concepts, symbols, and tables that you'll need to memorize.

Table of Contents

1. Conditional Statements (Ch 2.2)

2. Valid and Invalid Arguments (Ch 2.3)

3. Predicates and Quantified Statements I (Ch 3.1)

4. Predicates and Quantified Statements II (Ch 3.2)

5. Statements with Multiple Quantifiers (Ch 3.3)

6. Arguments with Quantified Statements (Ch 3.4)

1. Conditional Statements (Ch 2.2)

1.1 If-Then Statements (p → q)

Truth Table for p → q:

Key points:

• A conditional statement is false ONLY when the hypothesis (p) is true and the conclusion (q) is false

• When the hypothesis is false, the conditional is automatically true (called "vacuously true" or "true by default")

1.2 Order of Operations

1. Negation (~)

2. Conjunction (∧) and Disjunction (∨)

3. Conditional (→) and Biconditional (↔)

1.3 Equivalent Forms of Conditional

• p → q ≡ ~p ∨ q (conditional equals negation of hypothesis OR conclusion)

• p → q ≡ ~(p ∧ ~q) (conditional equals negation of "hypothesis AND NOT conclusion")

1.4 Negation of Conditional

• ~(p → q) ≡ p ∧ ~q (negation of conditional equals "hypothesis AND NOT conclusion")

1.5 Contrapositive, Converse, and Inverse

• Contrapositive: ~q → ~p (logically equivalent to original)

• Converse: q → p (NOT logically equivalent to original)

• Inverse: ~p → ~q (NOT logically equivalent to original)

• The contrapositive is logically equivalent to the original conditional

• The converse and inverse are logically equivalent to each other

1.6 "Only If" Statements

• "p only if q" means p → q

• "p only if q" is equivalent to "If p then q" or "If not q then not p"

1.7 Biconditional (p ↔ q)

Truth Table for p ↔ q:

• p ↔ q ≡ (p → q) ∧ (q → p)

• "p if and only if q" means p ↔ q

1.8 Necessary and Sufficient Conditions

• "q is a sufficient condition for p" means q → p

• "q is a necessary condition for p" means p → q

2. Valid and Invalid Arguments (Ch 2.3)

2.1 Valid vs. Invalid Arguments

• Valid argument: It's impossible for all premises to be true and the conclusion false

• Invalid argument: There is at least one case where all premises are true and the conclusion is false

2.2 Rules of Inference (Valid Argument Forms)

Modus Ponens (Affirming):

• Premises: p → q, p

• Conclusion: q

Modus Tollens (Denying):

• Premises: p → q, ~q

• Conclusion: ~p

Generalization:

• Premise: p

• Conclusion: p ∨ q

Specialization:

• Premise: p ∧ q

• Conclusion: p (or q)

Elimination:

• Premises: p ∨ q, ~p

• Conclusion: q

Transitivity:

• Premises: p → q, q → r

• Conclusion: p → r

Proof by Division into Cases:

• Premises: p ∨ q, p → r, q → r

• Conclusion: r

Contradiction Rule:

• Premise: ~p → c (where c is a contradiction)

• Conclusion: p

2.3 Common Fallacies (Invalid Argument Forms)

Converse Error:

• Premises: p → q, q

• Invalid conclusion: p

Inverse Error:

• Premises: p → q, ~p

• Invalid conclusion: ~q

3. Predicates and Quantified Statements I (Ch 3.1)

3.1 Predicates

• A predicate is a statement containing variables

• P(x) represents "x has property P"

• The truth value depends on the value assigned to the variable

• The truth set is the set of all values that make the predicate true

3.2 Quantifiers

Universal Quantifier (∀):

• ∀x, P(x) means "for all x, P(x) is true"

• Read as "for every," "for each," "for any," or "for all"

• True only if P(x) is true for every x in the domain

Existential Quantifier (∃):

• ∃x such that P(x) means "there exists at least one x such that P(x) is true"

• Read as "there exists," "there is," or "for some"

• True if P(x) is true for at least one x in the domain

3.3 Universal Conditional Statements

• ∀x, if P(x) then Q(x)

• Equivalent forms:

  • ∀x ∈ D, if P(x) then Q(x) ≡ ∀x ∈ {x | P(x)}, Q(x)

  • "All P are Q"

3.4 Implicit Quantification

• "If a number is an integer, then it is rational" implicitly means "For all numbers, if a number is an integer, then it is rational"

4. Predicates and Quantified Statements II (Ch 3.2)

4.1 Negation of Quantified Statements

Negation of Universal Statement:

• ~(∀x, P(x)) ≡ ∃x such that ~P(x)

• "Not all are" ≡ "Some are not"

Negation of Existential Statement:

• ~(∃x such that P(x)) ≡ ∀x, ~P(x)

• "None are" ≡ "All are not"

4.2 Negation of Universal Conditional Statements

• ~(∀x, if P(x) then Q(x)) ≡ ∃x such that P(x) ∧ ~Q(x)

• "Not all P are Q" ≡ "Some P are not Q"

4.3 Relationship Between Quantifiers and Logical Operators

• ∀ is related to ∧ (conjunction)

• ∃ is related to ∨ (disjunction)

4.4 Contrapositive, Converse, and Inverse of Universal Statements

Original: ∀x, if P(x) then Q(x)

• Contrapositive: ∀x, if ~Q(x) then ~P(x)

• Converse: ∀x, if Q(x) then P(x)

• Inverse: ∀x, if ~P(x) then ~Q(x)

4.5 Vacuous Truth

• A universal statement ∀x ∈ D, if P(x) then Q(x) is vacuously true if P(x) is false for every x in D

5. Statements with Multiple Quantifiers (Ch 3.3)

5.1 Order of Quantifiers

• In statements with multiple quantifiers, imagine actions performed in the order the quantifiers appear

• ∀x, ∃y such that P(x,y) means "for every x, there exists at least one y such that P(x,y)"

• ∃y, ∀x such that P(x,y) means "there exists at least one y such that for every x, P(x,y)"

• Order matters when different types of quantifiers are used (∀ and ∃)

• Order doesn't matter when same type of quantifiers are used (∀ and ∀, or ∃ and ∃)

5.2 Negating Statements with Multiple Quantifiers

• ~(∀x, ∃y such that P(x,y)) ≡ ∃x, ∀y such that ~P(x,y)

• ~(∃x, ∀y such that P(x,y)) ≡ ∀x, ∃y such that ~P(x,y)

5.3 Formal Logical Notation

• ∀x ∈ D, P(x) can be written as ∀x (x ∈ D → P(x))

• ∃x ∈ D such that P(x) can be written as ∃x (x ∈ D ∧ P(x))

6. Arguments with Quantified Statements (Ch 3.4)

6.1 Universal Instantiation

• If ∀x, P(x) is true and a is a particular element in the domain, then P(a) is true

6.2 Universal Modus Ponens

• Premises: ∀x, if P(x) then Q(x); P(a) for a particular a

• Conclusion: Q(a)

6.3 Universal Modus Tollens

• Premises: ∀x, if P(x) then Q(x); ~Q(a) for a particular a

• Conclusion: ~P(a)

6.4 Testing Validity with Diagrams

• For statements like "All A are B":

  • Draw a Venn diagram with the A circle inside the B circle

• For statements like "No A are B":

  • Draw a Venn diagram with A and B circles not overlapping

6.5 Common Errors in Arguments with Quantified Statements

• Converse Error:

  • Premises: ∀x, if P(x) then Q(x); Q(a)

  • Invalid conclusion: P(a)

• Inverse Error:

  • Premises: ∀x, if P(x) then Q(x); ~P(a)

  • Invalid conclusion: ~Q(a)

Important Symbols to Memorize

Key Truth Tables to Memorize

Conditional (p → q):

Biconditional (p ↔ q):

Order of Operations:

1. Negation (~)

2. Conjunction (∧) and Disjunction (∨)

3. Conditional (→) and Biconditional (↔)

Good luck with your exam! Make sure to practice applying these concepts by working through sample problems from your textbook.