Glossary

Every concept in the curriculum.

129 defined terms, grouped by the unit that introduces them. Each links to the units and lessons where it's introduced in context.

Foundations of Logical Reasoning · 8

ArgumentA set of statements in which one or more premises are offered in support of a conclusion.ConclusionThe statement that an argument claims to establish on the basis of its premises.FormalizationThe process of translating natural-language arguments into a structured symbolic or semi-symbolic form.Indicator WordsWords or phrases such as 'therefore,' 'because,' and 'since' that signal the presence of an argument and mark premises or conclusions.Inductive StrengthA property of inductive arguments in which the premises make the conclusion probable but not certain.InferenceThe reasoning step that connects premises to a conclusion.PremiseA statement offered as evidence or a reason in support of an argument's conclusion.ValidityThe property of an argument whose conclusion cannot be false while all its premises are true.

Definitions and Conceptual Precision · 8

CircularityA defect in which the definiens depends on the very concept it is supposed to explain — directly or through near-synonyms.DefiniendumThe term being defined — the word or phrase whose meaning the definition is trying to clarify.DefiniensThe part of the definition that does the defining — the explanation or account that supplies meaning to the definiendum.Function of a DefinitionThe job a definition is intended to perform in a specific context, such as reporting usage, sharpening a concept, or fixing a technical meaning.Genus and DifferentiaA classical pattern that defines a term by naming a broader class (genus) and the distinguishing feature (differentia) that sets the target apart from other members of that class.Kinds of DefinitionsDifferent categories of definitions such as lexical, stipulative, precising, theoretical, and persuasive, each suited to different tasks.Necessary and Sufficient ConditionsA necessary condition is one that must hold for the term to apply; a sufficient condition is one whose holding guarantees the term applies. A good definition typically states conditions that are both.ScopeThe range of cases a definition includes or excludes — neither so broad that it admits non-instances, nor so narrow that it excludes real ones.

Natural Deduction: Validity and Formal Proof · 5

CounterexampleA situation in which the premises of an argument are all true while the conclusion is false.EntailmentA relation in which the premises, taken together, guarantee the conclusion.ProofA rule-governed derivation showing that a conclusion follows from a set of premises.SoundnessA deductive argument is sound when it is valid and all of its premises are true.SubproofA nested section of a proof used to track assumptions and scope in conditional or indirect derivations.

Categorical Logic: Terms, Classes, and Syllogistic Form · 11

A, E, I, O PropositionsThe four standard forms of categorical proposition: A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative).Categorical PropositionA proposition asserting inclusion or exclusion between two classes, namely the subject class and the predicate class.Categorical SyllogismA deductive argument consisting of exactly two categorical premises and a categorical conclusion involving exactly three terms.DistributionA term is distributed in a categorical proposition if the proposition refers to every member of the class named by that term.Existential ImportThe question of whether a proposition, especially a universal one, carries the claim that its subject class has at least one member.Major, Minor, and Middle TermsThe major term is the predicate of the conclusion; the minor term is the subject of the conclusion; the middle term appears in both premises but not the conclusion.Predicate TermThe term in a categorical proposition that names the class asserted to contain, exclude, or partially overlap with the subject class.Quantity and QualityQuantity is whether a proposition is universal (about every member) or particular (about at least one member); quality is whether it is affirmative or negative.Square of OppositionA diagram that represents the logical relationships among A, E, I, and O propositions sharing the same subject and predicate terms.Subject TermThe term in a categorical proposition that names the class about which something is being asserted.Venn DiagramA diagram using overlapping circles to represent classes; shading indicates an empty region and an 'x' indicates at least one member.

Propositional Logic: Form, Connectives, and Valid Inference · 10

Atomic StatementA declarative sentence that is not further analyzed at the propositional level and is represented by a single sentence letter.Compound StatementA statement formed from one or more simpler statements by the use of logical connectives.ContradictionA statement that is false under every possible truth-value assignment to its atomic parts.Inference RuleA schematic pattern that licenses the derivation of a conclusion from one or more premises, such as modus ponens or disjunctive syllogism.Logical ConnectiveAn operator such as negation, conjunction, disjunction, conditional, or biconditional that forms a compound proposition from simpler ones.Logical EquivalenceA relation between two statements that are true under exactly the same truth-value assignments.Main ConnectiveThe connective with the widest scope in a compound statement, which determines the statement's overall logical form.TautologyA statement that is true under every possible truth-value assignment to its atomic parts.Truth FunctionalityThe property that the truth value of a compound statement is completely determined by the truth values of its component parts.Truth TableA systematic listing of every possible assignment of truth values to the atomic parts of a compound statement together with the resulting value of the whole.

Predicate Logic: Quantifiers, Predicates, and Formal Structure · 10

Domain of DiscourseThe set of objects that the variables of a predicate-logic formula are taken to range over in a given interpretation.Existential QuantifierThe quantifier ∃x, read 'there exists x such that,' which claims that the formula it binds holds for at least one object in the domain.GeneralizationA proof move that goes from a claim about an individual to a quantified claim; universal generalization is allowed only when the individual was arbitrary, and existential generalization is always allowed when a specific instance has been found.Individual ConstantA symbol that names a specific object in the domain of discourse, typically written with lowercase letters like a, b, c.InstantiationA proof move that goes from a quantified claim to a claim about a specific individual; universal instantiation picks any individual, while existential instantiation introduces a fresh name for a witness.PredicateAn expression that attributes a property to an object or a relation among several objects; a one-place predicate applies to a single object, while a relational predicate applies to two or more.QuantifierAn operator that binds a variable and states how much of the domain it ranges over; the universal quantifier ∀ means 'for all,' and the existential quantifier ∃ means 'for some' or 'there exists.'Scope and BindingThe scope of a quantifier is the portion of a formula it governs; a variable occurrence is bound if it falls within the scope of a quantifier using the same variable, and free otherwise.Universal QuantifierThe quantifier ∀x, read 'for all x,' which claims that the formula it binds holds for every object in the domain.VariableA symbol like x, y, or z that does not name a specific individual but can range over the domain when bound by a quantifier.

Modal Logic: Necessity, Possibility, and Counterfactuals · 8

Accessibility RelationA relation between possible worlds that says which worlds count as 'available' from a given world when evaluating modal operators.Counterfactual ConditionalA conditional of the form 'if it had been the case that P, it would have been the case that Q', evaluated by looking at the nearest possible worlds where P is true.De Dicto vs De ReDe dicto modal claims are about the modality of a whole proposition; de re modal claims are about a modal property attributed to a particular thing.Modal DualityThe equivalence between box and diamond via negation: the box of P is equivalent to not the diamond of not P, and vice versa.NecessityA proposition is necessary when it is true in every possible world; written as the box operator in front of the proposition.PossibilityA proposition is possible when it is true in at least one possible world; written as the diamond operator in front of the proposition.Possible WorldA complete way things could consistently be, usually represented as a point in a model at which every proposition has a definite truth value.Rigid DesignatorA term that picks out the same object in every possible world in which that object exists, as opposed to terms whose reference can shift across worlds.

Fallacies and Reasoning Errors · 7

Diagnosis and RepairThe practice of pairing any fallacy identification with a description of what change would make the reasoning acceptable.Fundamental FallacyA broad family of inferential failure that can generate several more specific named fallacies.Option-Space FailureA family of errors in which relevant alternatives, constraints, or strategy revisions are ignored.Relevance FailureA family of errors in which the offered consideration does not bear on the conclusion in the required way.Scope or Structure FailureA family of errors caused by misreading logical form, quantifier scope, class structure, or formal relation.Species of a FallacyA more specific recurring error pattern that instantiates a broader fundamental inferential defect.Support MiscalibrationA family of errors in which the degree or type of support claimed is stronger, weaker, or otherwise different from what the evidence actually warrants.

Inductive Logic: Evidence, Generalization, and Causal Support · 7

Analogical ReasoningAn inference that supports a conclusion about one case because it is relevantly similar to another case.Causal InferenceReasoning that moves from evidence to a claim about what caused a given outcome.Confounding VariableA third factor that influences both the supposed cause and the supposed effect, producing a correlation that does not reflect direct causation.DefeasibilityThe feature of an argument whose support can be weakened or defeated by new evidence.Inductive StrengthThe degree to which premises make a conclusion probable or well-supported without guaranteeing it.RepresentativenessThe extent to which a sample reflects the broader population it is used to support claims about.Sample SizeThe number of observed cases in the evidence base from which a generalization is drawn.

Bayesian Probability: Updating Belief with Evidence · 7

Base RateThe background prevalence or prior frequency relevant to the hypothesis being assessed.Bayes FactorThe ratio of likelihoods under two rival hypotheses, P(E | H1) / P(E | H2), which captures how strongly evidence favors one over the other.CalibrationThe property of having stated confidence match long-run accuracy — 70%-confident predictions should come true about 70% of the time.False Positive RateThe probability that a test or indicator yields a positive result when the hypothesis is false, written P(E | not-H).LikelihoodThe probability of the observed evidence on the assumption that a given hypothesis is true, written P(E | H).Posterior ProbabilityThe revised degree of confidence in a hypothesis after incorporating new evidence, written P(H | E).Prior ProbabilityThe degree of confidence assigned to a hypothesis before the new evidence is taken into account.

Abductive Logic: Arguments to the Best Explanation · 8

Argument to the Best ExplanationA form of reasoning in which we infer that one hypothesis is currently the best explanation of the evidence when compared with rivals.Best of a Bad LotA concern that the 'best explanation' might still be poor if the real explanation was never among the considered candidates.CoherenceThe degree to which a hypothesis fits with well-supported background knowledge and with other accepted claims.Explanatory FitHow closely a hypothesis matches the specific features of the observations, as opposed to merely being consistent with them.Explanatory ScopeThe range of evidence or observations a hypothesis successfully explains.HypothesisA candidate explanation proposed to account for the observations.ObservationA fact or data point that calls for explanation.SimplicityA virtue of a hypothesis that explains the observations without introducing unnecessary assumptions or entities.

Logic of Problem Solving: Goals, Constraints, and Strategy · 7

ConstraintA limitation, requirement, or condition that shapes which solutions are acceptable — time, budget, rules, resources, or policies.Decision RevisionUpdating a plan when new information shows that the original path is incomplete, inefficient, or blocked.Goal StateThe outcome or condition the reasoner is trying to reach, stated clearly enough to tell whether a proposed solution would actually produce it.Problem StateThe current situation that must be understood before a reasonable plan can be formed, including what is known, what is unknown, and what has already been tried.ResourceAnything the reasoner can draw on to move toward the goal, including time, money, tools, information, or help from others.Revision TriggerA specific observation or event that would signal the plan needs to change.Strategy SelectionThe process of comparing available approaches and choosing the one that best fits the problem, given the goals and constraints.

Decision Theory: Choosing Under Uncertainty · 9

Diminishing Marginal UtilityThe property that successive units of a good produce smaller and smaller increases in utility; the tenth dollar gained matters less to you than the first.DominanceOption A dominates option B when A yields an outcome at least as good as B in every possible state of the world, and strictly better in at least one state.Expected ValueThe probability-weighted average of an action's possible outcomes, computed as EV = sum over outcomes of probability times payoff.Marginal ReasoningEvaluating decisions by asking about the incremental costs and benefits of small changes rather than about total averages.Opportunity CostThe value of the best alternative you give up when you choose one option over another.Preference OrderingA ranking of alternatives that a decision maker holds, ideally satisfying completeness (every pair is comparable) and transitivity (if A is preferred to B and B to C, then A is preferred to C).Risk versus UncertaintyRisk refers to situations where the probabilities of outcomes are known or estimable; uncertainty refers to situations where those probabilities themselves are unknown or contested.Sunk CostA cost that has already been incurred and cannot be recovered regardless of future action.UtilityA numerical measure of how much an outcome is worth to a particular agent, calibrated so that higher numbers always correspond to more preferred outcomes.

Set Theory and Relations: The Structure Underneath Logic · 9

CardinalityA measure of the size of a set; two sets have the same cardinality when there exists a bijection between them, and a set is countable when it has the same cardinality as the natural numbers.Cartesian ProductThe Cartesian product A × B of two sets is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.Element (Member)An object belonging to a set; if x is an element of the set A, we write x ∈ A, and if it is not, we write x ∉ A.Equivalence RelationA relation R on A that is reflexive, symmetric, and transitive; each equivalence relation partitions A into disjoint equivalence classes.FunctionA function f: A → B is a relation from A to B such that for every a ∈ A there is exactly one b ∈ B with (a, b) ∈ f; we write f(a) = b for this unique value.Power SetThe power set of a set A, written P(A) or 2^A, is the set whose elements are all the subsets of A, including the empty set and A itself.RelationA binary relation from A to B is a subset R ⊆ A × B; elements (a, b) ∈ R are usually written a R b.SetA well-defined collection of distinct objects, considered as a single mathematical object; the objects are called elements or members of the set.SubsetA set A is a subset of a set B, written A ⊆ B, if every element of A is also an element of B; A is a proper subset if in addition A ≠ B.

Hermeneutics: Arguing for an Interpretation · 15

Authorial IntentThe meaning or purpose that the author intended to communicate, as recoverable from the text, the author's other writings, and the historical situation.Burden of Proof in InterpretationThe principle that the person proposing a non-literal reading bears the burden of showing why the literal reading fails and why the proposed alternative is better supported by the text.Contextual ConstraintThe principle that valid interpretations must be consistent with the text's linguistic, historical, genre, and situational context.Contextual MeaningThe meaning of a word, phrase, or passage as determined by its surrounding linguistic and situational context rather than by dictionary definition alone.Genre ExpectationThe interpretive norms associated with a text's genre — poetry invites figurative reading; legal statutes demand literal reading; philosophical treatises require careful attention to defined terms.Hermeneutic CircleThe principle that understanding a whole text requires understanding its parts, and understanding each part requires understanding the whole.HermeneuticsThe theory and practice of interpretation, especially of texts, arguments, and communicative acts.Horizon of UnderstandingThe totality of assumptions, experiences, and knowledge that a reader brings to a text, shaping what they are able to see in it.Interpretive ArgumentA reasoned case for one reading of a text over another, supported by evidence from the text, its context, its genre, and authorial intent.Interpretive HonestyThe commitment to letting a text say what it says, even when the implications are uncomfortable, rather than distorting its meaning to fit the interpreter's preferences.Literal PresumptionThe default assumption that a text means what its words ordinarily mean unless there is specific evidence to the contrary.Metaphorical EscapeThe interpretive move of declaring a text metaphorical or non-literal specifically because the interpreter finds the literal meaning uncomfortable, implausible, or inconvenient.PresuppositionA background assumption that a text takes for granted without explicitly stating or defending it.Principle of CharityThe interpretive norm of choosing the strongest or most reasonable reading of a passage when multiple interpretations are possible.Verbal DisputeA disagreement that arises from different uses of a word rather than from a genuine difference in belief or judgment.