
Rules of Inference (definition)
Rules that permit valid inferences from statements assumed as premises. There are 23 total: 9 elementary argument forms, 10 replacement rules, and 4 instantiation/generalization rules.

Difference between elementary argument forms and replacement rules
 The first 9 elementary rules can be applied only to whole lines of a proof. For example, A can be inferred from A • B only if A • B constitutes a whole line.
 Any of the last 10 replacement rules can be applied either to whole lines or to parts of lines.

Difference between a valid and invalid argument
 An argument is valid if and only if it has no substitution instances with true premises and a false conclusion.
 An argument is invalid if it has at least one substitution instance with true premises and a false conclusion.

When are two statements materially equivalent?
Two statements are materially equivalent (≡) if they have the same truth value. This means that either they are both true or both false…and that they materially imply each other.

Replacement (definition)
 The exchange of a component of one statement only by a statement that is known (by one of the 10 replacement rules) to be logically equivalent to that component. Unlike substitution instances, it isn’t necessary to replace every occurrence of the component being replaced.
 Ex: A ⊃ B can be replaced by ~B ⊃ ~A

Substitution instance (definition)
 A substitution of statements for statement variables, which can be done as long as it is substituted for every occurrence of that statement variable.
 Ex: A ⊃ B is a substitution instance of p ⊃ q

Natural deduction (definition)
A method of proving the validity of a deductive argument by using the rules of inference. It’s a complete system, in that it is compact and readily mastered, and with it, one can construct a formal proof of validity for any valid truth functional argument.

Formal proof of validity (definition)
A sequence of statements, each of which is either a premise of that argument or follows from preceding statements of the sequence by an elementary valid argument, such that the last statement in the sequence is the conclusion of the argument whose validity is being proved.

Rule of replacement (definition)
A rule that permits us to infer from any statement the result of replacing any component of that statement by any other statement that is logically equivalent to the component replaced.

4 Quantification Rules
 ~(x)Mx ⇔ (∃x)~Mx
 The statement “Not everything is mortal” is logically equivalent to the statement “There is at least one thing that is not mortal.”
 (x)Mx ⇔ ~(∃x)~Mx
 The statement “Everything is mortal” is logically equivalent to the statement “There is not at least one thing that is not mortal.”
 ~(x)~Mx ⇔ (∃x)Mx
 The statement “Nothing is not mortal” is logically equivalent to the statement “Something is mortal.”
 (x)~Mx ⇔ ~(∃x)Mx
 The statement “Everything is not mortal” is logically equivalent to the statement “There is not at least one thing that is mortal.”

Express the following quantification rule in predicate logic: The statement “Not everything is mortal” is logically equivalent to the statement “There is at least one thing that is not mortal.”
~(x)Mx ⇔ (∃x)~Mx

Express the following quantification rule in predicate logic: The statement “Everything is mortal” is logically equivalent to the statement “There is not at least one thing that is not mortal.”
(x)Mx ⇔ ~(∃x)~Mx

Express the following quantification rule in predicate logic: The statement “Nothing is not mortal” is logically equivalent to the statement “Something is mortal.”
~(x)~Mx ⇔ (∃x)Mx

Express the following quantification rule in predicate logic: The statement “Everything is not mortal is logically equivalent to the statement “There is not at least one thing that is mortal.”
(x)~Mx ⇔ ~(∃x)Mx

Express the following quantification rule in English: ~(x)Mx ⇔ (∃x)~Mx
The statement “Not everything is mortal” is logically equivalent to the statement “There is at least one thing that is not mortal.”

Express the following quantification rule in English: (x)Mx ⇔ ~(∃x)~Mx
The statement “Everything is mortal” is logically equivalent to the statement “There is not at least one thing that is not mortal.”

Express the following quantification rule in English: ~(x)~Mx ⇔ (∃x)Mx
The statement “Nothing is not mortal” is logically equivalent to the statement “Something is mortal.”

Express the following quantification rule in English: (x)~Mx ⇔ ~(∃x)Mx
The statement “Everything is not mortal is logically equivalent to the statement “There is not at least one thing that is mortal.”

Association (Replacement Rule)
 [p v (q v r)] ⇔ [(p v q) v r]
 [p • (q • r)] ⇔ [(p • q) • r]
 In a conjunction or disjunction of three true statements, it does not matter if you group the first two or the last two.

Double Negation (Replacement Rule)
 p ⇔ ~~p
 Any statement is logically equivalent to the negation of the negation of that statement.

Commutation (Replacement Rule)
 (p v q) ⇔ (q v p)
 (p • q) ⇔ (q • p)
 The order of the elements of a conjunction or disjunction does not matter.

Material Implication (Replacement Rule)
 (p ⊃ q) ⇔ (~p v q)
 Either the antecedent (p) is false or the consequent (q) is true.

Transposition (Replacement Rule)
 (p ⊃ q) ⇔ (~q ⊃ ~p)
 Any conditional statement is logically equivalent to the conditional statement asserting that the negation of its consequent implies the negation of its antecedent.

Tautology (Replacement Rule)
 p ⇔ (p v p)
 p ⇔ (p • p)
 Any statement is logically equivalent to the conjunction or disjunction of itself.

Exportation (Replacement Rule)
 [(p • q) ⊃ r] ⇔ [p ⊃ (q ⊃ r)]
 If two propositions conjoined imply a third, that is logically equivalent to asserting that if the first one is true, then the truth of the second must imply the truth of the third.

Material Equivalence (Replacement Rule)
 (p ⇔ q) [(p ⊃ q) • (q ⊃ p)]
 (p ⇔ q) [(p • q) v (~q • ~p)]
 The assertion that two statements are materially equivalent is logically equivalent to asserting that they are both true or both false…and that they materially imply each other.

De Morgan’s Theorems (Replacement Rule)
 ~(p • q) ⇔ (~p v ~q)
 ~(p v q) ⇔ (~p • ~q)
 The negation of a conjunction is logically equivalent to the disjunction of the negations of the conjuncts. The negation of a disjunction is logically equivalent to the conjunction of the negations of the disjuncts.

Distribution (Replacement Rule)
 [p • (q v r)] ⇔ [(p • q) v (p • r)]
 [p v (q • r)] ⇔ [(p v q) • (p v r)]
 The conjunction of one statement with the disjunction of two other statements is logically equivalent to either the conjunction of the first with the second or the conjunction of the first with the third. The disjunction of one statement with the conjunction of two others is logically equivalent to the conjunction of the disjunction of the first with the second and the disjunction of the first with the third.


Constructive Dilemma
 (p ⊃ q) • (r ⊃ s)
 p v r
 ∴ q v s








The fallacy of affirming the consequent

The fallacy of denying the antecedent

Tautology (definition of tautological statement form)
 A statement form with only true substitution instances.
 Ex: p v ~p
 Ex: (p ⊃ q) ⊃ [p ⊃ (p ⊃ q)]
 Ex: A truth table in which the conclusion column has only T values.

Contradiction (definition of contradictory statement form)
 A statement form with only false substitution instances.
 Ex: p • ~p
 Ex: A truth table in which the conclusion column has only F values.

Contingent (definition of contingent statement form)
 A statement form with some true and some false substitution instances.
 Ex: p • q
 Ex: A truth table in which the conclusion column has both T and F values.

Tautology (definition of tautological statement)
 A statement whose specific form is a tautology.
 Ex: (A ⊃ B) ⊃ [A ⊃ (A ⊃ B)]

3 Meanings of Tautology
 A statement form with only true substitution instances: (p ⊃ q) ⊃ [p ⊃ (p ⊃ q)]
 A statement whose specific form is a tautology: (A ⊃ B) ⊃ [A ⊃ (A ⊃ B)]
 The particular logical equivalence: p ⇔ (p v p) p ⇔ (p • p)

The predicate logic form of “The common cold is never fatal.”
 Cx: x is the common cold.
 Fx: x is fatal.
 (x) (Cx ⊃ ~Fx)

The predicate logic form of “There is not a single witch or wizard who went bad who wasn’t in Slitherin.”
 Bx: x is a witch or wizard who went bad.
 Sx: x is a witch or wizard who was in Slitherin.
 (x) (Bx ⊃ Sx)

The predicate logic form of “Some humans are not mortal.”
 Hx: x is a human.
 Mx: x is mortal.
 (∃x) (Hx • ~Mx)

The predicate logic form of “Some humans are mortal.”
 Hx: x is a human.
 Mx: x is mortal.
 (∃x) (Hx • Mx)

The predicate logic form of “No humans are mortal.”
 Hx: x is a human.
 Mx: x is mortal.
 (x) (Hx ⊃ ~Mx)

The predicate logic form of “All humans are mortal.”
 Hx: x is a human.
 Mx: x is mortal.
 (x) (Hx ⊃ ~Mx)

Three laws of thought
 The principle of identity: p ⊃ p is always true (a tautology)
 The principle of contradiction: p • ~p is always false
 The principle of the excluded middle: p v ~p is always true (a tautology)

The principle of identity
 p ⊃ p
 always true (a tautology)

The principle of contradiction

The principle of the excluded middle
 p v ~p
 always true (a tautology)

Proposition (aka Statement)
A sentence with a truth value (true or false).

Sound argument
A deductive argument that is valid and has true premises.

Inference
A process by which one proposition is arrived at and affirmed on the basis of some other proposition or propositions.

Valid argument
In this type of argument, if the premises are true, then the conclusion must be true.

Enthymeme
An argument based on an unstated proposition.

Euphemisms
Gentle words that are used to refer to harsh realities.

Two things that disagreements can be about.
 Beliefs (about facts)
 Attitudes

Validity
A formal characteristic of arguments that refers to the relation between propositions (premises and conclusion).

Logic
The study of the methods and principles used to distinguish correct from incorrect reasoning.

Explanation
A group of statements from which some event (or thing) to be explained can logically be inferred and whose acceptance removes or diminishes the problematic character of that event (or thing).

Argument
Any group of propositions (premises and conclusion) of which one (conclusion) follows from the others.

Structure of definitions: Extension
Also called denotation, and it includes the objects to which a term may be correctly applied.

Structure of definitions: Intension
Also called connotation, and it includes the set of attributes shared by all and only those objects to which a general term refers.

Five rules for good definitions
 Should state essential attributes
 Must not be circular
 Must not be too broad or narrow
 Should not rely on ambiguous, obscure, or figurative language
 When possible, should not be negative

Three kinds of intensional definitions
 Synonymous
 Operational
 Definitions by genus and difference

A synonymous definition
A definition involving two words with the same meaning.

An operational definition
A definition in which the meaning is tied to a clearly describable set of actions or operations.

A definition by genus and difference
A definition involving a division into subclasses (genus) and the attribute (difference) that differentiates members of the classes.

Three different senses of intension
 Subjective
 Objective
 Conventional

Subjective intension of a word
A type of intension in which the set of all attributes the speaker believes to be possessed by objects denoted by that word.

Objective intension of a word
A type of intension in which the total set of characteristics shared by all objects in the word’s extension.

Conventional intension of a word
A type of intension of stable meaning by implicit agreement to use the same criterion for deciding whether an object is part of the term’s extension.

Four forms of language
 Declarative: A statement. (“Logic is easy.”)
 Exclamatory: An exclamation. (“You’ve got to be kidding!”)
 Imperative: A command. (“Trust me.”)
 Interrogative: A question. (“Why should I?”)

Five major functions (uses) of language
 Informative
 Expressive
 Directive
 Ceremonial
 Performative

What is the purpose of the informative function of language?
 To report facts.
 Example: “It is raining.”

What is the purpose of the expressive function of language?
 To express attitudes or beliefs.
 Example: “That’s great!”

What is the purpose of the directive function of language?
 To guide, command, or direct the behavior of others.
 Example: “Turn left at the next light.”

What is the purpose of the ceremonial function of language?
 To fulfill social obligations.
 Example: “How do you do?”

What is the purpose of the performative function of language?
 To perform the action that it describes.
 Example: “I now pronounce you man and wife.”
 Example: “I accept your offer.”
 Example: “I apologize.”
 Example: “I congratulate you on your engagement.”
 Example: “I promise you that I will be there.”

Five kinds of definitions
 Stipulative
 Lexical
 Precising
 Theoretical
 Persuasive

Stipulative definition
A definition that is deliberately assigned.

Lexical definition
A definition that reports an already established meaning.

Precising definition
A definition that eliminates ambiguity or vagueness.

Theoretical definition
A definition that encapsulates a larger understanding.

Persuasive definition
A definition used to resolve a dispute by influencing attitudes or stirring emotions.

Rhetorical question
Suggests or assumes an answer that is made to serve as the premise of an argument.

Vague term
A term for which there are borderline cases to which the term might or might not apply.

Deductive argument
A type of argument in which the conclusion is necessarily true if the premises are true.

Ambiguous term
A type of term for which there is more than one distinct meaning.

Two terms for definitions and their uses

Definiendum
The part of a definition that contains the symbol being defined.

Definiens
The part of a definition that contains the symbol (or group of symbols) used to explain the meaning of the symbol being defined.

Inductive argument
A type of argument in which the conclusion is probably true if the premises are true.

Three types of disputes
 Genuine – about beliefs or attitudes
 Merely verbal – arising from the unrecognized use of ambiguous terms
 Apparently verbal but really genuine – in which a real difference remains even after apparent ambiguity has been eliminated

Conclusion
A proposition to which other propositions in the argument are claimed to give support.

Premise
A proposition that provides support for a conclusion.

Denotative definition structure
A type of definition structure that employs techniques that identify the extension of the term being defined.

Two types of denotative definition structures

Ostensive definition structure
A definition structure that points at or demonstrates what it defines.

Quasiostensive definition structure
A definition structure that includes a descriptive phrase about what it defines.

Six conclusion indicators (card 1 of three)
 for this reason
 which points to the conclusion that
 which allows us to infer that
 we may infer that
 therefore
 hence

Seven conclusion indicators (card 2 of three)
 thus
 for these reasons
 it follows that
 I conclude that
 which shows that
 so
 which entails that

Eight conclusion indicators (card 3 of three)
 so
 accordingly
 in consequence
 consequently
 which means that
 proves that
 as a result
 which implies that

Seven premise indicators (card 1 of two)
 since
 because
 in view of the fact that
 for
 as
 follows from
 as shown by

Seven premise indicators (card 2 of two)
 inasmuch as
 as indicated by
 the reason is that
 for the reason that
 may be inferred from
 may be derived from
 may be deduced from

Implies / entails
A relationship between statements (premises and conclusions).

Truth and falsity
Formal characteristics of individual propositions.

Translate the following into sentential logic: “Smith is either the owner or the manager.”
O v M

Translate the following into sentential logic: “You will do poorly on the exam unless you study.”
P v S

Translate the following into sentential logic: “If you don’t study, you will do poorly on the exam.”
~S ⊃ P

Translate the following into sentential logic: “Either Fillmore or Harding was the greatest U.S. president.”
F v H

Translate the following into sentential logic: “Neither Fillmore nor Harding was the greatest U.S. president.”
 ~(F v H)
 or it can be expressed as (~F • ~H)

Translate the following into sentential logic: “Jamal and Derek will not both be elected.”
 ~(J • D)
 or it can be expressed as (~J • ~D)

Translate the following into sentential logic: “I will study hard and pass the exam or I will fail the exam.”
(S • P) v F

Translate the following into sentential logic: “I will study hard and I will either pass the exam or fail it.”
S • (P v F)

