PhD — Logic & Philosophy Benchmark
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100-question multiple-choice exam — 5 sections — scored /100. Calibrated at doctoral-level difficulty in analytic philosophy, formal logic, philosophy of language, epistemology, and metaphysics. The most intellectually demanding benchmark in the NotesXML suite.
Introduction
The PhD Logic & Philosophy Benchmark tests whether a model has internalized the deep structural reasoning of advanced academic discourse. Questions reference specific theorems, seminal papers, and philosophical positions that only appear in graduate coursework or primary source literature. At this level, a model cannot answer correctly by pattern-matching common phrases — it must correctly apply Kripke semantics, navigate Gödel's incompleteness theorems, distinguish Frege's Sinn from Bedeutung, and understand why Gettier's 1963 paper fundamentally changed epistemology.
Relationship to Standard Benchmarks
MMLU philosophy questions typically cover introductory undergraduate material. The PhD Benchmark opens at the graduate level and goes beyond, covering material that would appear on a comprehensive examination required for doctoral candidacy. Reference: Hendrycks et al. (2020), arxiv.org/abs/2009.03300.
Assessment Overview
The PhD Benchmark is a 100-question, multiple-choice examination across five sections, temperature fixed at 0.1. A traditional letter grade (A+ through F) is also reported alongside the competency level.
| Section | Topic Area | Questions |
|---|---|---|
| I | Logic, Language & Core Epistemology | Q1–Q20 |
| II | Language, Science & Metaphysics | Q21–Q40 |
| III | Metaphysics, Mind & Ethics | Q41–Q60 |
| IV | Advanced Metaphysics, Mind & Language | Q61–Q80 |
| V | Formal Logic, Ethics & Contemporary Themes | Q81–Q100 |
Section I — Logic, Language & Core Epistemology (Q1–Q20)
| Q | Question (abbreviated) | Answer |
|---|---|---|
| 1 | In Kripke semantics for modal logic, the characteristic axiom of S4 (□P → □□P) corresponds to which property of the accessibility relation? (A) Reflexivity (B) Symmetry (C) Transitivity (D) Seriality | C |
| 2 | Primary consequence of the Downward Löwenheim-Skolem Theorem? (A) Every first-order theory with an infinite model has a countable model, raising the Skolem Paradox (B) First-order logic is structurally incomplete (C) PA cannot prove its own consistency (D) Second-order logic is compact | A |
| 3 | In Naming and Necessity, Kripke argues proper names are: (A) Abbreviated definite descriptions (B) Rigid designators (C) Cluster concepts (D) Fictional entities | B |
| 4 | Quine's “Two Dogmas of Empiricism” attacks: (A) The analytic-synthetic distinction and reductionism (B) A priori knowledge and deduction (C) Verification and falsificationism (D) Scientific realism and instrumentalism | A |
| 5 | Putnam's “Twin Earth” (XYZ vs. H2O) supports: (A) Logical behaviorism (B) Type-identity theory (C) Semantic externalism (D) Eliminative materialism | C |
| 6 | Theorem showing truth in arithmetic cannot be defined within that arithmetic: (A) Gödel's Completeness Theorem (B) Church's Theorem (C) Tarski's Undefinability Theorem (D) Craig Interpolation Theorem | C |
| 7 | Grice's “conversational implicature” relies on speakers following: (A) Compositionality (B) The Cooperative Principle (C) The Verification Principle (D) The Principle of Charity | B |
| 8 | Davidson's “Anomalous Monism”: mental events are physical, but: (A) No strict deterministic laws connect the mental and physical (B) Mental events only understood via first-person phenomenology (C) Physical events are causally inert (D) They exist in separate ontological domains | A |
| 9 | Gettier's 1963 paper presented cases where a subject has JTB but lacks knowledge due to: (A) Epistemic luck (B) Lack of justification (C) Absolute certainty but false belief (D) Reliable faculties but false belief | A |
| 10 | Compactness Theorem: (A) A set of first-order sentences has a model iff every finite subset has a model (B) Every valid formula can be derived in finite steps (C) No system can be both complete and consistent (D) Every recursively enumerable set is decidable | A |
| 11 | David Lewis's “Modal Realism”: (A) Possible worlds are abstract constructs (B) Possible worlds are maximally consistent propositions (C) Possible worlds exist as concretely as the actual world, spatiotemporally isolated (D) Possible worlds exist only as mental constructs | C |
| 12 | Frege's Sinn vs. Bedeutung: (A) Sinn is the psychological image; Bedeutung is the definition (B) Sinn is the mode of presentation; Bedeutung is the actual object (C) Sinn applies to predicates; Bedeutung to proper names (D) Sinn is the truth-value; Bedeutung is the proposition | B |
| 13 | Moore's “Open Question Argument” is directed against: (A) Ethical non-cognitivism (B) Ethical naturalism (C) Moral error theory (D) Divine command theory | B |
| 14 | Husserl's epoche requires: (A) Deconstructing language origins (B) Bracketing the existence of the natural world (C) Engaging in existential action (D) Reducing mental states to neurobiology | B |
| 15 | Russell's Theory of Descriptions solved: (A) Non-referring expressions (“The present King of France is bald”) (B) The problem of induction (C) Paradoxes of material implication (D) The demarcation problem | A |
| 16 | The Barcan Formula (◊∃xFx → ∃x◊Fx) implies: (A) Everything that exists necessarily exists (B) If it's possible something is F, there actually exists something possibly F (C) Identity is necessary (D) Predicates cannot be modalized | B |
| 17 | Kant's view on space and time: (A) Empirical concepts from sensory experience (B) Absolute, mind-independent realities (C) Relational properties between objects (D) Pure forms of sensible intuition | D |
| 18 | Goodman's “grue” predicate demonstrates: (A) Color perception is culturally relative (B) Syntactic structure alone cannot determine which hypotheses are confirmed (C) Falsification is superior to verification (D) Deductive logic is flawed for future events | B |
| 19 | Frank Jackson's “Knowledge Argument” (Mary the color scientist) is intended to refute: (A) Physicalism (B) Dualism (C) Panpsychism (D) Functionalism | A |
| 20 | Implication of the Church-Turing Thesis: (A) First-order logic is decidable (B) Any algorithmic calculation can be performed by a Turing machine (C) Quantum computers solve NP-hard problems in polynomial time (D) Human consciousness is fundamentally computational | B |
Section II — Language, Science & Metaphysics (Q21–Q40)
| Q | Question (abbreviated) | Answer |
|---|---|---|
| 21 | Wittgenstein's “Private Language Argument” asserts: (A) Mental states are entirely private (B) A language comprehensible to only one person is logically impossible (C) Meaning is determined by private intentions (D) Scientific language must be divorced from ordinary language | B |
| 22 | Carnap's “internal” vs. “external” questions: (A) Internal=psychological; external=sociological (B) Internal are within a framework and scientifically answerable; external concern framework choice and are pragmatic (C) Internal=a priori; external=a posteriori (D) Internal=syntax; external=semantics | B |
| 23 | Searle's “Chinese Room” is directed against: (A) Possibility of AGI (B) The Turing Test and Strong AI (C) Connectionist models (D) Computational theory of vision | B |
| 24 | Mackie's “Argument from Queerness” supports: (A) Moral realism (B) Error theory (C) Constructivism (D) Expressivism | B |
| 25 | Heidegger's fundamental characteristic of Dasein: (A) Rationality (B) Being-in-the-world (C) Abstract mathematical thought (D) Subject-object dualism | B |
| 26 | In ZFC, logically equivalent to the Axiom of Choice: (A) Continuum Hypothesis (B) Zermelo's Well-Ordering Theorem (C) Axiom of Foundation (D) Generalized Continuum Hypothesis | B |
| 27 | Kuhn's “incommensurability” implies: (A) Rival paradigms cannot be objectively compared (B) Progress is cumulative but older theories lack precision (C) Scientists from different disciplines cannot collaborate (D) Empirical data strictly determines theory choice | A |
| 28 | Goldman's Reliabilism: belief is justified iff: (A) Subject can articulate logical steps (B) It coheres with existing beliefs (C) Produced by a cognitive process reliably yielding true beliefs (D) Derived from a priori foundational axioms | C |
| 29 | S5 characteristic axiom added to S4 (equivalence relation): (A) ◊P → □◊P (B) □P → □□P (C) □P → P (D) P → ◊P | A |
| 30 | Endurantism vs. Perdurantism debate concerns: (A) Existence of abstract objects (B) Persistence through time and temporal parts (C) Relationship between mind and body (D) Whether causation is physical or counterfactual | B |
| 31 | Benacerraf's 1973 dilemma is between: (A) Formalism and Intuitionism (B) Platonist semantics for mathematical truth and a causal theory of knowledge (C) Incompleteness of arithmetic and consistency of set theory (D) Constructive mathematics and classical logic | B |
| 32 | Sartre's “Bad Faith” (mauvaise foi): (A) Projecting faults onto society (B) Denying one's absolute freedom by adopting the role of a determined entity (C) Relying on religious dogma for scientific questions (D) Failure of communication between subjects | B |
| 33 | Frege introduced Sinn/Bedeutung primarily to solve the puzzle of: (A) Identity statements of the form a=a vs. a=b (B) Logical form of universal quantification (C) The Liar paradox (D) Ontological status of mathematical entities | A |
| 34 | Gödel's Second Incompleteness Theorem: (A) Every truth is provable in second-order logic (B) First-order logic is undecidable (C) No consistent, sufficiently strong system can prove its own consistency (D) Axiom of Choice is independent of ZF | C |
| 35 | Moore's “Naturalistic Fallacy” is: (A) Defining “good” in terms of natural properties (B) Assuming nature is teleologically ordered (C) Deriving an “is” from an “ought” (D) Believing human psychology is exempt from physical laws | A |
| 36 | Quine's “Epistemology Naturalized” argues epistemology should be replaced by: (A) Cartesian foundationalism (B) The empirical study of how humans process sensory input to form beliefs (C) Sociological analysis of power (D) Formal logic and probability theory | B |
| 37 | “Multiple Realizability” is primarily used against: (A) Eliminative Materialism (B) Type-Identity Theory (C) Functionalism (D) Property Dualism | B |
| 38 | Intuitionistic Logic (Brouwer/Heyting) rejects: (A) Law of Non-Contradiction (B) Modus Ponens (C) Law of the Excluded Middle (D) De Morgan's Laws | C |
| 39 | McTaggart argued time is unreal because: (A) The A-series is inherently contradictory and the B-series cannot account for genuine change (B) Time is a physiological illusion (C) Relativity proves simultaneity is subjective (D) Entropy is a statistical artifact | A |
| 40 | Duhem-Quine thesis asserts: (A) Falsification is logically impossible (B) Hypotheses cannot be tested in isolation, only as part of a holistic web of background beliefs (C) Scientific progress operates through revolutionary paradigm shifts (D) Unobservable theoretical entities do not exist | B |
Section III — Metaphysics, Mind & Ethics (Q41–Q60)
| Q | Question (abbreviated) | Answer |
|---|---|---|
| 41 | Russell's resolution to “The present King of France is not bald” involves: (A) Scope of the definite description (B) Intension vs. extension (C) Performative vs. constative use (D) Analytic vs. synthetic nature | A |
| 42 | Foucault's epistème: (A) Qualitative sociological research method (B) Unconscious historical a priori grounding knowledge in a given epoch (C) Innate linguistic structures (D) Ethical imperative to deconstruct hierarchical relations | B |
| 43 | Cantor's Theorem: (A) There are exactly two sizes of infinity (B) The power set of any set has strictly greater cardinality (C) No set can be a member of itself (D) Real numbers can be placed in 1:1 correspondence with integers | B |
| 44 | Davidson's “Swampman” argues for: (A) Historical externalism regarding meaning and mental content (B) Logical possibility of philosophical zombies (C) Necessity of a soul for consciousness (D) Epiphenomenalism | A |
| 45 | Epistemic Contextualism resolves skeptical paradoxes by arguing: (A) Skeptical hypotheses are logically incoherent (B) Truth conditions of knowledge attributions shift with conversational context (C) We have an innate faculty of common sense (D) Knowledge only requires true belief | B |
| 46 | “Frege-Geach Problem” is directed at: (A) Moral Realism (B) Divine Command Theory (C) Expressivism / Non-cognitivism (D) Kantian Constructivism | C |
| 47 | Meinong's ontology is famous for asserting: (A) Only physical objects exist (B) There are objects that don't “exist” spatio-temporally but still have being (C) Reality is composed entirely of mental events (D) Existence is a first-order predicate | B |
| 48 | Tarski's Convention T requires all instances of: (A) X is true iff p (B) X is true if it corresponds to a physical fact (C) X is true if provable (D) X is true if coherent with maximal beliefs | A |
| 49 | Frankfurt's counterfactual-intervener thought experiments refute: (A) The psychological theory of determinism (B) The Principle of Alternative Possibilities (C) First-order vs. second-order distinctions (D) Compatibilism | B |
| 50 | The “Species Problem” primarily concerns: (A) Ethical implications of genetic engineering (B) Difficulty of a universal, objective definition of species (C) Mathematical modeling of population genetics (D) Conflict between evolution and teleology | B |
| 51 | Sosa's “apt” belief (Virtue Epistemology): (A) True because it coheres with foundational beliefs (B) True, justified, immune from Gettier cases (C) Accurate because of its adroitness (D) Socially verified by epistemic peers | C |
| 52 | Paul Cohen's “forcing” originally proved: (A) Completeness of first-order logic (B) Independence of the Continuum Hypothesis and Axiom of Choice from ZFC (C) Every consistent system contains undecidable propositions (D) Consistency of Peano Arithmetic | B |
| 53 | Churchland's Eliminative Materialism: (A) Mental states are identical to neurobiological processes (B) Folk psychology is radically false; beliefs/desires will be eliminated by neuroscience (C) Mind is software on brain hardware (D) Mental properties supervene on physical without reduction | B |
| 54 | Deflationary Theory of Truth (Horwich): (A) Truth is correspondence to physical facts (B) “Is true” adds no substantive property; “P is true” is equivalent to asserting P (C) Truth is relative to conceptual framework (D) Only verified empirical statements have truth-values | B |
| 55 | Van Inwagen's Mereological Nihilism: (A) Abstract objects do not exist (B) Nothing exists; the universe is an illusion (C) Composite objects don't exist except for living organisms; only “simples arranged X-wise” (D) Universe composed of non-physical monads | C |
| 56 | Modal logic system T adds which axiom to system K? (A) □P → □□P (B) □P → P (C) P → □◊P (D) ◊P → □◊P | B |
| 57 | Merleau-Ponty's Phenomenology of Perception argues: (A) Body is a machine directed by rational soul (B) Perception is fundamentally an active, embodied engagement with the world (C) The phenomenal world is constructed by linguistic categories (D) Consciousness reduces to physiological stimuli | B |
| 58 | Horgan & Timmons's “Moral Twin Earth” critiques: (A) Cornell Realism (moral naturalism) (B) Moral Error Theory (C) Emotivism (D) Divine Command Theory | A |
| 59 | Anscombe's Intention: intentional actions are those subject to: (A) Being caused by a prior “volition” (B) The agent articulating neurological mechanisms (C) A specific sense of the question “Why?” to which an answer gives a reason, not a physical cause (D) Conformity to categorical imperatives | C |
| 60 | Graham Priest's Dialetheism requires paraconsistent logic because: (A) All contradictions are trivially true (B) Truth is a two-sided social construct (C) There are true contradictions; some statements are both true and false (D) Neither P nor not-P can be proven in certain systems | C |
Section IV — Advanced Metaphysics, Mind & Language (Q61–Q80)
| Q | Question (abbreviated) | Answer |
|---|---|---|
| 61 | “Truthmaker Objection” most frequently deployed against: (A) Eternalism (B) Presentism (C) Growing Block Theory (D) Four-Dimensionalism | B |
| 62 | “Combination Problem” is the most severe challenge to: (A) Epiphenomenalism (B) Panpsychism (C) Logical Behaviorism (D) Functionalism | B |
| 63 | Kripkenstein's rule-following paradox (“quus”) demonstrates: (A) Mathematical addition is empirical (B) No objective fact about past usage determines one meant “plus” rather than “quus” (C) Children learn mathematics via conditioning (D) Private languages are possible with perfect memory | B |
| 64 | Williamson's “Knowledge First” epistemology: (A) Knowledge is highly subjective (B) Belief and justification are basic building blocks (C) Knowledge is unanalyzable and fundamental; belief understood in terms of knowledge (D) Epistemology precedes ontology | C |
| 65 | Craig's Interpolation Theorem: if A entails B, there exists C such that: (A) Every valid formula derives without Cut (B) A entails C, C entails B, using only vocabulary common to A and B (C) Any proof can be normalized intuitionistically (D) Truth is conserved across syntactic transformations | B |
| 66 | Worrall's Structural Realism reconciles: (A) Quantum mechanics and general relativity (B) The “No Miracles” argument and the Pessimistic Meta-Induction (C) Logical positivism and sociological constructivism (D) The macro/micro observational dichotomy | B |
| 67 | Bernard Williams's “Jim and the Indians” argues Utilitarianism fundamentally violates a person's: (A) Negative liberty (B) Categorical imperatives (C) Integrity (D) Natural rights | C |
| 68 | David Lewis's counterfactual theory of causation: “C causes E” analyzed as: (A) A necessary connection in nature (B) Regularity and constant conjunction (C) If C had not occurred, E would not have occurred (D) Transfer of a conserved physical quantity | C |
| 69 | Plantinga's “Free Will Defense” relies on: (A) Privation theory of evil (B) Transworld depravity (C) Soul-making theodicy (D) Deterministic compatibilism | B |
| 70 | Axiom of Constructibility (V=L) implies: (A) No uncomputable functions (B) Axiom of Choice is false (C) The Generalized Continuum Hypothesis is true (D) Every set is equivalent to a physical construct | C |
| 71 | Husserl's Lebenswelt (Life-world) refers to: (A) The biological domain studied by natural sciences (B) The pre-theoretical world of everyday human experience grounding scientific abstraction (C) The metaphysical realm of pure Ideas (D) Unconscious drives motivating human behavior | B |
| 72 | Austin's illocutionary vs. perlocutionary act: (A) Illocutionary=spoken; perlocutionary=written (B) Illocutionary=semantic meaning; perlocutionary=syntax (C) Illocutionary is what is done in saying something; perlocutionary is the psychological/behavioral consequences (D) Illocutionary=always true; perlocutionary can be false | C |
| 73 | “Ability Hypothesis” (Lewis/Nemirow) response to Mary's Room: when Mary sees red she: (A) Discovers a new non-physical fact (B) Realizes an illusion (C) Acquires new abilities (to recognize, imagine, remember red), not new propositional knowledge (D) Reprograms her syntactic engine | C |
| 74 | C.I. Lewis's strict implication is logically equivalent to: (A) not-(P and not-Q) (B) □(P → Q) (C) ◊P → ◊Q (D) P ↔ Q | B |
| 75 | Rawls's Reflective Equilibrium: (A) Distributing economic goods equally (B) Reaching psychological peace about moral decisions (C) Justifying moral principles by mutually adjusting considered judgments and theoretical principles (D) Balancing powers in government | C |
| 76 | Kaplan's logic of indexicals introduces: (A) Primary vs. secondary intension (B) Rigid vs. non-rigid designators (C) Character (linguistic rule) vs. content (proposition expressed) (D) Locutionary vs. illocutionary acts | C |
| 77 | Łoś's Theorem (Fundamental Theorem of Ultraproducts): (A) Every first-order theory with infinite model has models of all infinite cardinalities (B) A formula is true in an ultraproduct iff the set of indices where it is true belongs to the ultrafilter (C) Two models are elementarily equivalent iff isomorphic (D) Any finitely satisfiable set is globally satisfiable | B |
| 78 | Clark & Chalmers's “Extended Mind” thesis argues for: (A) Panpsychism (B) Active externalism — cognitive processes can genuinely extend into the environment (C) Global workspace theory (D) Elimination of propositional attitudes | B |
| 79 | “Grounding” (Fine, Schaffer) best defined as: (A) Physical mechanisms by which one event causes another (B) A primitive, non-causal relation of metaphysical determination (the “in-virtue-of” relation) (C) Logical deduction of a theorem from an axiom (D) Epistemological foundation justifying basic beliefs | B |
| 80 | Van Fraassen's Constructive Empiricism: the aim of science is: (A) To discover the literal truth about observable and unobservable entities (B) To formulate theories that are empirically adequate, without requiring belief in unobservables (C) To continuously falsify hypotheses (D) To construct mathematical models with no physical interpretation | B |
Section V — Formal Logic, Ethics & Contemporary Themes (Q81–Q100)
| Q | Question (abbreviated) | Answer |
|---|---|---|
| 81 | Relevance logics reject: (A) Modus Tollens (B) The Law of Identity (C) The Principle of Explosion / Ex Falso Quodlibet (D) Double Negation Elimination | C |
| 82 | Miranda Fricker's “testimonial injustice”: (A) A speaker lies under oath (B) A hearer lacks conceptual vocabulary (C) A speaker receives deflated credibility due to systemic identity prejudice (D) A marginalized group is excluded from research | C |
| 83 | Habermas's “Communicative Action” oriented toward: (A) Strategic manipulation to achieve political goals (B) Reaching uncoerced consensus among rational subjects (C) Deconstructing unconscious biases in bourgeois language (D) Maximizing utility through perfect information | B |
| 84 | In Two-Dimensional Semantics, “primary intension” determines referent relative to: (A) The actual world considered as counterfactual (B) Scenarios considered as actual (epistemic possibilities) (C) Causal history of the concept's use (D) The speaker's physiological brain state | B |
| 85 | Boyd and Cornell Realists: moral properties are: (A) Platonically abstract objects (B) Fictions projected by human emotions (C) Irreducible natural properties featuring in causal-explanatory theories of human behavior (D) Reducible to physics and chemistry | C |
| 86 | “Free Logic” modifies classical logic by: (A) Rejecting Excluded Middle (B) Allowing non-denoting terms; dropping non-empty domain assumption (C) Using truth degrees between 0 and 1 (D) Removing quantification rules | B |
| 87 | Brentano's “Intentionality”: defining characteristic of all mental phenomena: (A) Ability to exercise free will (B) Directedness toward / “aboutness” of an object (C) Presence of phenomenal “feel” (qualia) (D) Computational processing of formal syntax | B |
| 88 | Bayesian Conditionalization: (A) Prior probabilities distributed equally (B) Mathematical rule for updating degrees of belief upon receiving new, certain evidence (C) Hypothesis acceptable only if it conditionalizes a universally quantified law (D) Probability of conjunction cannot exceed probability of conjuncts | B |
| 89 | Ryle's “Ghost in the Machine” critique: Descartes committed a: (A) Naturalistic fallacy (B) Modal fallacy (C) Category mistake (D) Paradox of self-reference | C |
| 90 | Ontological Pluralism (Turner, McDaniel): (A) Universe contains infinite physical substances (B) There are different, fundamentally distinct ways or modes of being (existence is not univocal) (C) Multiple competing theories can be equally true (D) Properties exist independently of the objects that instantiate them | B |
| 91 | Gentzen's Hauptsatz (Cut-Elimination): (A) Every first-order sentence set is decidable (B) Any formula derivable in the sequent calculus can be derived without the Cut rule — proofs are analytic (C) Mathematical induction cannot be formalized in first-order logic (D) No formal system can prove its own consistency | B |
| 92 | Popper's criterion of demarcation: scientific iff: (A) Verified by repeated observation (B) Derived from a priori principles (C) Falsifiable (D) Endorsed by the prevailing paradigm | C |
| 93 | Derrida's Deconstruction targets: (A) Radical skepticism about the external world (B) Logocentrism and the metaphysics of presence (C) Strict separation of church and state (D) Over-reliance on formal logic for humanistic problems | B |
| 94 | Frankfurt's “wanton” is: (A) Acts purely from malice (B) Determined by physical laws (C) Has first-order desires but lacks second-order volitions (D) Strictly follows Kantian imperatives | C |
| 95 | Second-order logic differs from first-order because it: (A) Cannot quantify over relations (B) Lacks a complete and sound effective proof system (C) Is strictly decidable (D) Satisfies Compactness and Löwenheim-Skolem | B |
| 96 | Kalam Cosmological Argument crucially relies on: (A) Time is an illusion (B) An actual infinite series of past events is logically impossible (C) Universe is in eternal static equilibrium (D) God is an abstract object | B |
| 97 | Rosenthal's HOT theory: mental state is conscious iff: (A) It causes outward behavioral reaction (B) It possesses intrinsic phenomenal quality (C) It is the object of a higher-order thought to the effect that one is in that state (D) It is globally broadcast to multiple cognitive subsystems | C |
| 98 | Epistemic Internalism (accessibilist): the factors that justify a belief must be: (A) Cognitively accessible to the believer upon reflection (B) Innate and generated entirely within the mind (C) Produced by a reliable mechanism, regardless of believer's awareness (D) Subjective and relative to emotional states | A |
| 99 | Putnam's “Meaning ain't in the head” directly opposes: (A) Fregean/Cartesian internalist views where internal mental states determine reference (B) Quine's ontological relativity (C) Kripke's causal-historical theory (D) Wittgenstein's meaning-as-use doctrine | A |
| 100 | Lindström's Theorem characterizes first-order logic as most expressive satisfying both: (A) Modus Ponens and Modus Tollens (B) Axiom of Choice and Continuum Hypothesis (C) The Compactness Theorem and the Downward Löwenheim-Skolem Theorem (D) Church's Thesis and Turing equivalence | C |
Scoring Rubric
| Score | Percentage | Competency Level |
|---|---|---|
| 0–39 | 0–39% | Below PhD Level |
| 40–59 | 40–59% | Advanced Undergraduate / MA Level |
| 60–69 | 60–69% | MA Graduate Level |
| 70–79 | 70–79% | PhD Candidate Level |
| 80–89 | 80–89% | PhD Proficient |
| 90–100 | 90–100% | PhD Mastery / Expert |
Summary
The PhD Benchmark is the hardest test in the NotesXML suite and is designed to stress-test models rather than validate them. Most current small on-device models score below 60 on this assessment. Larger Premium-tier models begin to approach the MA Graduate level. A model reaching PhD Candidate level (70+) is genuinely sophisticated and can be trusted to assist with graduate-level academic work, literature reviews, and analytical writing in philosophy and formal logic.
References
- Kripke, S. (1980). Naming and Necessity. Harvard University Press.
- Gödel, K. (1931). On Formally Undecidable Propositions.
- Gettier, E. (1963). Is Justified True Belief Knowledge? Analysis, 23(6), 121–123.
- Tarski, A. (1944). The Semantic Conception of Truth. Philosophy and Phenomenological Research, 4(3).
- Hendrycks, D. et al. (2020). arxiv.org/abs/2009.03300
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