Intro to Logic

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.

• 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.

• 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.

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.

• 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

• 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

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.

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.

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.

• ~(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.”

~(x)Mx ⇔ (∃x)~Mx

(x)Mx ⇔ ~(∃x)~Mx

~(x)~Mx ⇔ (∃x)Mx

(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.”

The statement “Everything is mortal” is logically equivalent to the statement “There is not at least one thing that is not mortal.”

The statement “Nothing is not mortal” is logically equivalent to the statement “Something is mortal.”

The statement “Everything is not mortal is logically equivalent to the statement “There is not at least one thing that is mortal.”

• [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.

• p ⇔ ~~p
• Any statement is logically equivalent to the negation of the negation of that statement.

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

• (p ⊃ q) ⇔ (~p v q)
• Either the antecedent (p) is false or the consequent (q) is true.

• (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.

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

• [(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.

• (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.

• ~(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.

• [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.

• p
• ∴ p v q

• (p ⊃ q) • (r ⊃ s)
• p v r
• ∴ q v s

• p
• q
• ∴ p • q

• p v q
• ~p
• ∴ q

• p ⊃ q
• p
• ∴ q

• p • q
• ∴ p

• p ⊃ q
• q ⊃ r
• ∴ p ⊃ r

• p ⊃ q
• ~q
• ∴~p

• p ⊃ q
• p ⊃ (p • q)

• p ⊃ q
• q
• ∴ p

• p ⊃ q
• ~p
• ∴ ~q

• 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.

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

• 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.

• A statement whose specific form is a tautology.
• Ex: (A ⊃ B) ⊃ [A ⊃ (A ⊃ B)]

• 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)

• Cx: x is the common cold.
• Fx: x is fatal.
• (x) (Cx ⊃ ~Fx)

• Bx: x is a witch or wizard who went bad.
• Sx: x is a witch or wizard who was in Slitherin.
• (x) (Bx ⊃ Sx)

• Hx: x is a human.
• Mx: x is mortal.
• (∃x) (Hx • ~Mx)

• Hx: x is a human.
• Mx: x is mortal.
• (∃x) (Hx • Mx)

• Hx: x is a human.
• Mx: x is mortal.
• (x) (Hx ⊃ ~Mx)

• Hx: x is a human.
• Mx: x is mortal.
• (x) (Hx ⊃ ~Mx)

• 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)

• p ⊃ p
• always true (a tautology)

• p • ~p
• always false

• p v ~p
• always true (a tautology)

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

A deductive argument that is valid and has true premises.

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

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

An argument based on an unstated proposition.

Gentle words that are used to refer to harsh realities.

• Beliefs (about facts)
• Attitudes

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

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

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).

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

Also called denotation, and it includes the objects to which a term may be correctly applied.

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

• 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

• Synonymous
• Operational
• Definitions by genus and difference

A definition involving two words with the same meaning.

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

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

• Subjective
• Objective
• Conventional

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

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

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.

• 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?”)

• Informative
• Expressive
• Directive
• Ceremonial
• Performative

• To report facts.
• Example: “It is raining.”

• To express attitudes or beliefs.
• Example: “That’s great!”

• To guide, command, or direct the behavior of others.
• Example: “Turn left at the next light.”

• To fulfill social obligations.
• Example: “How do you do?”

• 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.”

• Stipulative
• Lexical
• Precising
• Theoretical
• Persuasive

A definition that is deliberately assigned.

A definition that reports an already established meaning.

A definition that eliminates ambiguity or vagueness.

A definition that encapsulates a larger understanding.

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

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

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

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

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

• Definiendum
• Definiens

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

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

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

• 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

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

A proposition that provides support for a conclusion.

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

• Ostensive
• Quasi-ostensive

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

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

• for this reason
• which points to the conclusion that
• which allows us to infer that
• we may infer that
• therefore
• hence

• thus
• for these reasons
• it follows that
• I conclude that
• which shows that
• so
• which entails that

• so
• accordingly
• in consequence
• consequently
• which means that
• proves that
• as a result
• which implies that

• since
• because
• in view of the fact that
• for
• as
• follows from
• as shown by

• 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

A relationship between statements (premises and conclusions).

Formal characteristics of individual propositions.

O v M

P v S

~S ⊃ P

F v H

• ~(F v H)
• or it can be expressed as (~F • ~H)

• ~(J • D)
• or it can be expressed as (~J • ~D)

(S • P) v F

S • (P v F)