Logic and Critical Reasoning

Standard Conditional

Expressed in if/then form: If P, then Q, where antecedent P is sufficient and consequent Q is necessary.

Conditional

P → Q. States that if the antecedent is true, the consequent must be true; structure matters over content.

dependence –

a series of arguments in which the conclusion of a prior argument

becomes a premise of a subsequent argument

independence

a single argument in which the premises provide independent

support for a single conclusion; if one of the premises were omitted, the other(s) would

continue to support the conclusion in the same way

Nonstandard conditionals

P on the condition Q.

P implies that Q.

Q, provided that P

Sufficient Condition

A guarantee. If it occurs, the necessary condition must occur (P in P → Q).

Necessary Condition

A requirement or prerequisite that must be present for the sufficient condition to occur (Q in P → Q).

Antecedent

The "if" clause of a conditional specifying the sufficient condition.

Consequent

The "then" clause of a conditional specifying the necessary condition.

Arrow Notation

P → Q means "If P, then Q," with the sufficient condition on the left and the necessary on the right.

"If"

Signals a sufficient condition; introduces the antecedent.

"Only If"

Signals a necessary condition; "P only if Q" translates to P → Q.

"If P, Then Q" and "P Only If Q"

Logically equivalent statements that both translate as P → Q.

Ad hominem fallacy

trying to undermine the truth of a position by attacking the person who is advancing it.

Antecedent

the 'if' clause of a conditional; the clause that specifies a sufficient condition of the conditional's consequent.

Appeal to authority

an informal fallacy that involves relying on authority figures to substantiate a position outside of their area of

Appeal to ignorance:

an informal fallacy, also known as ignoratio elenchi, that can take one of two forms. In the first, one believes a claim to be true because it hasn't been proven false.

In the second, one believes that a claim is false because it hasn't been proven true.

Appeal to irrelevant emotions:

an effort to convince you of a claim by playing on your emotions, rather than by offering facts and evidence that bear on the truth of the claim.

Argument

: a chain of thought in which reasons are offered in support of a particular conclusion.

Biconditional

a claim that supplies a condition that is both necessary and sufficient for some-thing; an "if and only if' sentence.

Conditional

an 'if-then' sentence.

Consequent

the 'then' clause of a conditional; it specifies a necessary condition of the conditional's antecedent.

Fallacious

the feature of exhibiting or having committed a fallacy.

Fallacy

kind of poor reasoning. A formal fallacy is an argument form all of whose instances are invalid. Informal fallacies are other kinds of mistakes in reasoning.

Fallacy of affirming the consequent

any argument of the form: if P, then Q; Q is true; there-fore, P is true.

Fallacy of denying the antecedent:

any argu nent of the form: if P, then Q; P is false; there fore, Q is false.

Hasty generalization:

illicitly drawing a general lesson from only a small handful of cases.

Straw man fallacy:

a form of reasoning that epicts a position in a way that makes it easy t efute, thereby diverting attention from the rea

Sufficient condition: a guarantee.

"Unless"

Means "if not." Q unless P translates as "If not P, then Q."

Example of Unless

"The carnival cannot proceed unless the clown gets better" translates as: If the clown does not get better, the carnival cannot proceed.

Another Unless Example

"Jill does not drive unless her sister gets tired" translates as: If her sister does not get tired, Jill does not drive.

Biconditional

A claim that supplies both necessary and sufficient conditions, combining two simple conditionals.

"If and Only If"

Signals a biconditional; means P → Q and Q → P.

Biconditional Symbol

P ↔ Q represents a biconditional, also expressed as (P → Q) & (Q → P).

Contrapositive

Found by reversing and negating the original conditional: P → Q becomes ¬Q → ¬P.

Contrapositive Example

If Jill drives, her sister is tired (P → Q) becomes: If Jill's sister is not tired, Jill does not drive (¬Q → ¬P).

Conditional Content vs. Structure

Logical form depends on structure rather than actual content.

Argument

A series or chain of statements where premises support a conclusion.

Premise

A reason or statement offered in support of an argument's conclusion.

Conclusion

The statement that premises are intended to support.

Argument Structure

Represented with premises followed by a conclusion, such as P1, P2, therefore C.

Identifying a Conditional in an Argument

An if/then conditional usually functions as a premise when supporting another statement.

Implicit Premise

A premise left unstated because it is considered self-evident or understood.

Implicit Premise Example

"Mammals are warm-blooded, so whales must be warm-blooded" relies on the unstated assumption that whales are mammals.

Complex Argument

A longer argument containing intermediate conclusions.

Intermediate Conclusion

The conclusion of an earlier argument that becomes a premise supporting a later conclusion.

Complex Argument Example

In a chain about frogs and vertebrates, "Fernando is an amphibian" serves as an intermediate conclusion.

Validity

An argument property where, assuming premises are true, the conclusion must be true.

Validity Test

Imagine all premises are true and ask whether the conclusion could still be false.

Soundness

An argument status achieved when it is valid and all its premises are actually true.

Valid but Unsound

A valid argument that remains unsound because one or more premises are false.

Modus Ponens

Form: If P, then Q; P; Therefore Q. Always valid (affirming the antecedent).

Modus Ponens Example

If Bono gave glasses to the Pope, the Pope tried them on. Bono did. Therefore, the Pope tried them on.

Modus Tollens

Form: If P, then Q; Not-Q; Therefore not-P. Always valid (denying the consequent).

Modus Tollens Example

If objective ethics existed, people would agree. People do not agree. Therefore, no objective ethics exist.

Hypothetical Syllogism

Form: If P, then Q; If Q, then R; Therefore, if P, then R. Always valid.

Conditional Chain

Multiple linked conditional premises, such as P → Q, Q → R, and R → S, leading to P → S.

Non-Commutativity of Conditionals

P → Q does not equal Q → P; reversing changes meaning.

Commutative Operations

Operations where reversing order doesn't change the result, like 3+2=2+3 or P&Q=Q&P.

Non-Commutative Operations

Operations where reversing order changes the result, like 3-2 ≠ 2-3 or P→Q ≠ Q→P.

Compound Conditional

A conditional with a two-part antecedent and/or consequent, such as P → (Q & R).

Splitting a Compound Conditional

P → (Q & R) splits into P → Q and P → R because the consequent contains a conjunction.

Splitting an Alternative Antecedent

(P ∨ Q) → R splits into P → R and Q → R.

Combining a Conjunction in the Antecedent

(P & Q) → R keeps P and Q together as both are required antecedents.

Combining a Disjunction in the Consequent

P → (Q ∨ R) means if P occurs, at least one of Q or R occurs.

Disjunction

An "or" statement represented by P ∨ Q.

Disjuncts

The individual statements in a disjunction (P and Q in P ∨ Q).

Inclusive Or

Means P, or Q, or both (represented as P ∨ Q).

Exclusive Or

Means P or Q, but not both.

False Dichotomy

An argument presenting limited options while neglecting other plausible alternatives.

Disjunctive Syllogism

Form: P ∨ Q; ¬P; Therefore Q (also called argument by elimination).

Disjunctive Syllogism Example

Either P or Q is true. P is false. Therefore, Q must be true.

Dilemma

Form: P ∨ Q; P → R; Q → S; therefore R ∨ S.

Horns of a Dilemma

The two alternatives in the first premise of a dilemma.

Escaping Between the Horns

Rejecting the first premise of a dilemma.

Taking the Dilemma by Its Horns

Rejecting one of the conditional premises in a dilemma.

Negation

Stating that something is not the case, represented by ¬ or ~.

Double Negation

Cancels out: ¬¬P = P.

Conjunction

An "and" statement represented by P & Q where both parts must be true.

Conjuncts

The individual statements joined by "and" in a conjunction.

Negating a Conjunction

¬(A & B) equals ¬A ∨ ¬B (turns into "or" and negates each part).

Negating a Disjunction

¬(A ∨ B) equals ¬A & ¬B (turns into "and" and negates each part).

Dependence Pattern

A series of arguments where a prior conclusion becomes a subsequent premise.

Independence Pattern

A single argument where multiple premises independently support one conclusion.

Vertical Pattern

A series where an intermediate conclusion supports the main conclusion.

Horizontal Pattern

A single argument where two or more premises independently support one conclusion.

Conjoint Premises

Premises that must work together to support a conclusion.

Multiple Conclusions

When one premise or argument supports two or more conclusions.

Enthymeme

An argument with an implicit premise or conclusion.

Deductive Arguments

Evaluated using validity and soundness (modus ponens, modus tollens, etc.).

Non-Deductive Arguments

Evaluated using strength and cogency (analogy, generalization, IBE).

Strong Non-Deductive Argument

An argument where true premises would make the conclusion highly likely.

Cogent Argument

A strong non-deductive argument with all true premises.

Inference to the Best Explanation

Inductive reasoning where a hypothesis is supported because it explains observations best.

Form of Inference to the Best Explanation

Observations, strong hypothesis explanation, and lack of better alternatives lead to a probable conclusion.

Evaluating Inference to the Best Explanation

Assess whether an explanation is explanatory, simple, powerful, falsifiable, plausible, deep, modest, and conservative.

Explanatory

Accounts for all relevant observations.