Deductive Logic, Truth Functions, Causality, and Informal Fallacies

Comprehensive vocabulary flashcards covering causality and determination (Anscombe), methods of deduction and formal proofs of validity (Copi, Cohen, and Rodych), sentential connectives and tautologies (Suppes), and informal fallacies (Acuña).

Causality (Anscombe's Account)

The view that causality consists essentially in the derivativeness of an effect from its causes (arising out of or coming of them), rather than being defined by necessity, universality, or exceptionless generalizations.

Necessitating Cause

A cause $C$ of an effect $E$ such that, on the occasion, if $C$ occurs it is certain to cause $E$ unless something prevents it; it is not possible that $C$ occurs and does not cause $E$, given nothing interferes.

Non-Necessitating Cause

A cause that can fail of its effect without the intervention of anything to frustrate it, such as radioactive material causing a Geiger counter to trigger a bomb in Feynman's example.

Determined Result (Anscombe)

A result that is related to an antecedent range of possibilities such that something antecedent narrows that range down, disallowing all but one possibility prior to the event occurring.

Physical Indeterminism (Anscombe)

The philosophical thesis that not all physical effects are necessitated by their causes.

Formal Proof of Validity

A sequence of statements each of which is either a premise of a given argument, or follows from the preceding statements by one of the rules of inference, where the last statement in the sequence is the conclusion of the argument.

Natural Deduction

A method of proving the validity of a deductive argument by using rules of inference successively to deduce the conclusion from the premises step by step.

Elementary Valid Argument Form

Any one of a set of specified simple valid argument forms that serve as rules of inference and may therefore be used in constructing a formal proof of validity.

Rules of Inference

Valid argument forms or logical equivalences stipulated to ensure valid inferences from one or more statements to an inferred statement in a proof.

Statement Column

The column on the left side of a formal proof in which all premises, assumed statements, and inferred statements are written and numbered.

Justification Ledger

The column on the right side of a formal proof in which the abbreviation for the rule of inference used and the line number(s) of the statement(s) from which each line was inferred are written.

Modus Ponens (M.P.)

An elementary valid argument form of the structure: $p \supset q$, $p$, therefore $q$.

Modus Tollens (M.T.)

An elementary valid argument form of the structure: $p \supset q$, $\sim q$, therefore $\sim p$.

Hypothetical Syllogism (H.S.)

An elementary valid argument form of the structure: $p \supset q$, $q \supset r$, therefore $p \supset r$.

Disjunctive Syllogism (D.S.)

An elementary valid argument form of the structure: $p \lor q$, $\sim p$, therefore $q$.

Constructive Dilemma (C.D.)

An elementary valid argument form of the structure: $(p \supset q) \cdot (r \supset s)$, $p \lor r$, therefore $q \lor s$.

Absorption (Abs.)

An elementary valid argument form of the structure: $p \supset q$, therefore $p \supset (p \cdot q)$.

Simplification (Simp.)

An elementary valid argument form stating that from a conjunction $p \cdot q$, one may validly infer its left conjunct, $p$.

Conjunction (Conj.)

An elementary valid argument form stating that if two propositions $p$ and $q$ are known to be true separately, they can be jointly asserted as the conjunction $p \cdot q$.

Addition (Add.)

An elementary valid argument form stating that from a true proposition $p$, one may validly infer the disjunction $p \lor q$ for any proposition $q$.

Rule of Replacement

A principle of inference permitting the replacement of all or part of a truth-functional statement by any other statement that is logically equivalent to it without changing the statement's truth value.

Kurt Gödel

Austrian logician (1906–1978) who published the Incompleteness Theorems in 1931, proving that consistent axiomatic systems powerful enough to describe elementary number theory contain true but unprovable propositions.

Gödel's First Incompleteness Theorem

The theorem proving that for any consistent axiomatic system powerful enough to describe the arithmetic of natural numbers, there exist undecidable propositions $B$ such that neither $B$ nor $\sim B$ is derivable in the system.

Sentential Connectives

Words or symbols with precise rules of usage—specifically 'not', 'and', 'or', 'if… then…', and 'if and only if'—used to combine or modify sentences into compound structures.

Truth-Functional Connective

A connective where the truth value of the compound sentence is strictly a function of the truth values of its component sentences.

Atomic Sentence

A sentence that contains no sentential connectives.

Tautology

A compound sentence that is true independently of the truth values of its component atomic sentences, yielding true ($T$) for every row of its truth table.

Tautological Implication

The relation whereby a sentence $P$ tautologically implies a sentence $Q$ if and only if the conditional $P \rightarrow Q$ is a tautology.

Tautological Equivalence

The relation between two sentences that tautologically imply each other, meaning the biconditional $P \leftrightarrow Q$ is a tautology and their truth table columns match row for row.

Non Sequitur

A blanket term meaning 'it does not follow,' referring to any fallacious argument where the premise set fails to support the claim of the conclusion.

Argumentum ad Hominem

A fallacy committed when an argument is rejected by attacking the person who composed or offered it—such as pointing out a character flaw or evil intentions—rather than evaluating the argument on its own merits.

Argumentum ad Baculum

An appeal to force; a fallacy committed when an arguer uses threats, intimidation, or strong-arm tactics to reject an argument or compel agreement.

Argumentum ad Misericordiam

An appeal to pity; a fallacy where one attempts to support or refute an argument by appealing to sympathy, compassion, or emotional misery.

Argumentum ad Ignorantiam

An argument from ignorance; a fallacy committed when one infers that a conclusion is true simply because it has not been disproved, or false because it has not been proved.

Argumentum ad Populum

An appeal to popularity or traditional institutions; a fallacy where the soundness of an argument is grounded on widespread popular belief or tradition.

Argumentum ad Vericundiam

An appeal to authority; a fallacy committed when an argument relies on an authority that is illegitimate, irrelevant, or not a genuine expert on the subject matter.

Strawman Fallacy

A fallacy committed when an arguer deliberately misrepresents or distorts an opponent's argument to make it weaker, passes off the weak substitute as genuine, and attacks that substitute.

Red Herring Fallacy

A fallacy committed when the discussion is diverted away from the original issue to a different topic, causing the original argument to be neglected.

False Analogy

A fallacy where an inaccurate or misleading comparison between two things is drawn, and a conclusion is inferred as though the analogy were exact.

Slippery Slope Fallacy

A fallacy consisting in the claim that allowing a first step will inevitably trigger a chain of uncontrollable events leading to a disastrous final outcome, without evidence that such consequences will occur.

Fallacy of Complex Question

A fallacy committed by posing a question that illicitly presupposes or smuggles in the truth of an unproven hidden conclusion.

Petitio Principii

Begging the question; a circular reasoning fallacy in which the conclusion to be proved is smuggled into the premises or repeated as evidence.

Fallacy of False Dilemma

A fallacy where an argument forces an opponent to accept one of only two alternatives when, in reality, other viable possibilities exist.