Logic Study Guide: Chapters 2.2 - 3.4

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37 Terms

1
Conditional Statement
An assertion of the form 'If p then q', expressed as p → q.
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2
Truth Table
A table showing all possible truths of a statement and its components.
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3
Vacuously True
A statement that is considered true because its hypothesis is false.
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4
Contrapositive
The statement ~q → ~p, which is logically equivalent to the original conditional.
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5
Modus Ponens
A valid argument form: If p → q and p are true, then q is true.
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6
Predicates
Statements containing variables, represented as P(x), where the truth depends on the variable.
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7
Universal Quantifier (∀)
Indicates that a statement is true for all elements in a domain.
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8
Existential Quantifier (∃)
Indicates that there exists at least one element in a domain for which a statement is true.
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9
Negation of Universal Statement
~(∀x, P(x)) ≡ ∃x such that ~P(x), meaning 'not all P are true'.
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10
Biconditional Statement
A statement of the form p ↔ q, meaning 'p if and only if q'.
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11
Valid Argument
An argument where it is impossible for all premises to be true and the conclusion false.
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12
Order of Quantifiers
The sequence of quantifiers matters when different types are used; order doesn't matter with the same type.
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13
Universal Instantiation
If ∀x, P(x) is true, then for any particular element a, P(a) is also true.
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14
Inverse Error
An invalid conclusion resulting from asserting that if p → q and ~p, then ~q is true.
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15
Biconditional Truth Table
A table showing the truth values of the biconditional statement p ↔ q.
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16
Element of (∈)
Symbol indicating that an element belongs to a set.
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17
Subset (⊆)
Symbol indicating that one set is contained within another.
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18

Converse

The statement q → p, which is NOT logically equivalent to the original conditional p → q.

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19

Inverse

The statement ~p → ~q, which is NOT logically equivalent to the original conditional p → q but is equivalent to the converse.

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20

Modus Tollens

A valid argument form: If p → q and ~q are true, then ~p must be true.

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21

Universal Modus Ponens

A valid argument form combining universal instantiation with modus ponens: ∀x, if P(x) then Q(x); P(a) for a particular a; therefore Q(a).

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22

Universal Modus Tollens

A valid argument form combining universal instantiation with modus tollens: ∀x, if P(x) then Q(x); ~Q(a) for a particular a; therefore ~P(a).

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23

Converse Error

An invalid conclusion resulting from asserting that if p → q and q, then p is true.

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24

Transitivity

A valid argument form: If p → q and q → r, then p → r.

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25

Necessary Condition

"q is a necessary condition for p" means p → q (p cannot be true without q also being true).

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26

Sufficient Condition

"q is a sufficient condition for p" means q → p (if q is true, then p must be true).

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27

Only If

"p only if q" means p → q.

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28

Truth Set

The set of all values that make a predicate true.

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29

Negation of Existential Statement

~(∃x such that P(x)) ≡ ∀x, ~P(x), meaning "there does not exist any x such that P(x)" is equivalent to "for all x, P(x) is false."

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30

Negation of Universal Conditional

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

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31

Conditional Equivalence

p → q ≡ ~p ∨ q (a conditional statement is equivalent to the disjunction of the negation of its hypothesis and its conclusion)

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32

Multiple Quantifier Negation

The negation of ∀x, ∃y such that P(x,y) is ∃x, ∀y such that ~P(x,y)

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33

Proof by Division into Cases

A valid argument form: p ∨ q, p → r, q → r; therefore r.

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34

Contradiction Rule

If ~p leads to a contradiction, then p must be true.

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35

Explicit vs. Implicit Quantification

Explicit quantification uses quantifier symbols; implicit quantification is understood from context (e.g., "If a number is even, then it is divisible by 2" implicitly means "For all numbers, if a number is even, then it is divisible by 2").

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36

Vacuous Truth of Universal Statements

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

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37

Order of Operations in Logic

  1. Negation (~), 2) Conjunction (∧) and Disjunction (∨), 3) Conditional (→) and Biconditional (↔)

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