Paper VI — Philosophy: References
90 SOURCES · 30 QUESTIONS · ALL FREELY AVAILABLE
Each source carries a technical description and a plain one; authors’ own pages and encyclopaedia entries are linked where they exist. Every address on this page was verified live at publication. The paper itself: Paper VI.
1. Is logical entailment a transitive relation? Defend your answer.
The relation itself analysed — model-theoretic and proof-theoretic accounts — with the structural rules, cut included, made explicit.
In plain terms: The standing scholarly account of what 'follows from' means, where chaining inferences is one of the rules under examination.
A second full treatment: necessity, formality, and the substitutional and semantic definitions compared.
In plain terms: A free companion survey of the same question from an independent editorial tradition.
The formal machinery free and complete: sequent calculi where transitivity appears as cut, and its eliminability is a theorem.
In plain terms: An open logic textbook in which the chaining rule can be inspected — and seen proved dispensable yet harmless.
Background: Logical consequence · Transitive relation
2. Do Gödel's incompleteness theorems bear on the thesis that the mind is a machine?
Gödel's incompleteness theorems
The theorems stated with their exact hypotheses — consistency, recursive axiomatisability — on which every anti-mechanist reading turns.
In plain terms: The careful free statement of what Gödel proved, before anyone claims it proves minds outrun machines.
The computational theory of mind
The mechanist thesis itself, precisely formulated — what 'the mind is a machine' must mean for the theorems to touch it.
In plain terms: The free entry on the claim under attack, so the Gödelian argument has a definite target.
The Lucas–Penrose line and its standard rebuttals — the consistency-knowledge gap — with Gödel's own cautious disjunction.
In plain terms: The free scholarly account of the famous argument, its holes, and what Gödel himself was willing to conclude.
Background: Gödel's incompleteness theorems · Mechanism (philosophy)
3. Can there be vague objects, or only vague descriptions?
The semantic, epistemic, and ontic options mapped — including Evans's argument against vague identity and its replies.
In plain terms: The free survey of whether fuzziness lives in the world or only in our words — the question's exact terrain.
The engine of the problem: tolerance principles and the heap, forcing a choice among semantic theories.
In plain terms: The free entry on the grain-by-grain argument that makes vagueness everyone's problem.
Where vague objects would live: boundary-drawing, arbitrariness, and the problem of the many for tables and cats.
In plain terms: The free examination of everyday things — whose edges are precisely where the dispute becomes concrete.
Background: Vagueness · Sorites paradox
4. Is it possible to define truth non-circularly for a language containing its own truth predicate?
The classical result: truth definable only in a richer metalanguage, with undefinability for the object language itself.
In plain terms: The free account of Tarski's bargain — truth can be defined, but only from one language up.
The obstruction itself: self-applied truth generating contradiction, and the space of revisionary escapes.
In plain terms: The free entry on the sentence that started it all — 'this sentence is false' — and every known way out.
The constructive programme: Kripke fixed points and axiomatic systems housing a language's own truth predicate at known cost.
In plain terms: The free survey of the workaround — truth kept in-house by paying carefully itemised logical prices.
Background: Tarski's undefinability theorem · Liar paradox
5. What does the Löwenheim–Skolem theorem show about the determinacy of mathematical reference?
The determinacy question head-on: countable models of set theory and what they do and do not show about reference.
In plain terms: The free treatment of the unsettling theorem — theories of the uncountable satisfied by countable worlds.
The framework in which the theorem lives: satisfaction, elementary equivalence, and the limits of first-order expressive power.
In plain terms: The free entry on the mathematics of interpretation — where the gap between theory and intended model is defined.
The theorem in its exact mathematical statement, upward and downward, with the cardinality machinery visible.
In plain terms: The free mathematical reference stating precisely the result the philosophy then interrogates.
Background: Löwenheim–Skolem theorem · Skolem's paradox
6. Are there absolutely undecidable mathematical propositions?
Gödel's own distinction: relative undecidability from the theorems against the disputed absolute kind, and his optimism about new axioms.
In plain terms: The free account of the man who proved undecidability yet doubted any question was undecidable absolutely.
The leading candidate examined: CH's independence, the axiom programmes, and the case for and against its absoluteness.
In plain terms: The free deep survey of the one problem most often nominated as forever unanswerable — and the campaigns to answer it.
Papers and slides in remembrance of Solomon Feferman
The sceptical corpus free in one place — including the argument that CH is not a definite mathematical problem at all.
In plain terms: A great logician's open archive, where the question 'is it even well-posed?' is pressed hardest.
Background: Independence (mathematical logic) · Continuum hypothesis
7. Do the paradoxes of quantum mechanics pose a threat to classical logic?
Quantum logic and probability theory
The challenge in its strongest form: the non-distributive lattice of projections and Putnam's proposal to revise logic itself.
In plain terms: The free entry on the one serious claim that physics might overturn logic — and what that claim actually amounts to.
The paradoxes at source — superposition, measurement — and the interpretive options that leave classical logic untouched.
In plain terms: The free overview showing the strangeness can be lodged in physics or semantics rather than in logic.
The sharpest quantum paradox stated with full care — and derived without any tampering with classical inference.
In plain terms: The free authoritative account of quantum theory's deepest shock, reasoned throughout in ordinary logic.
Background: Quantum logic · Complementarity (physics)
8. Is set-theoretic Platonism compatible with our capacity to know mathematical facts?
Platonism in the philosophy of mathematics
The position and its epistemic wound: Benacerraf's access problem for causally inert abstracta, with the live responses.
In plain terms: The free entry on believing in a mathematical heaven — and explaining how earthbound creatures could know it.
The compatibility question located among the schools — what each rival buys in epistemology by what it forgoes in ontology.
In plain terms: The free map of the whole debate, on which the Platonist's predicament is one marked position.
Fictionalism in the philosophy of mathematics
The foil: an epistemology with no access problem because there is nothing to access — sharpening what Platonism must explain.
In plain terms: The free entry on the rival that treats mathematics as useful fiction — the cleanest measure of Platonism's cost.
Background: Philosophy of mathematics · Set theory
9. What, if anything, does reverse mathematics teach us about foundations?
The programme entire: calibrating theorems against the Big Five subsystems, and what the calibration means philosophically.
In plain terms: The free entry on mathematics run backwards — asking of each theorem exactly which axioms it secretly demands.
The technical apparatus indexed: subsystems of second-order arithmetic and the equivalences that define the hierarchy.
In plain terms: A free working reference for the machinery — the ladder of systems on which theorems are placed.
The foundational stakes: reverse mathematics as the instrument measuring how much of Hilbert's aim survives Gödel.
In plain terms: The free account of the grand project whose partial salvage is precisely what the calibrations reveal.
Background: Reverse mathematics · Second-order arithmetic
10. Can mathematical facts explain physical phenomena?
Indispensability arguments in the philosophy of mathematics
The explanatory turn documented: cicadas and honeycombs as cases where mathematics seems to do the explaining, and what follows.
In plain terms: The free entry where the question's examples live — prime-numbered life cycles explained by arithmetic itself.
The applicability problem in context: what any account of mathematical objects owes to mathematics' physical grip.
In plain terms: The free overview of the standing puzzle behind the question — why abstract truths bear on concrete happenings at all.
The criteria of explanation itself — causal, unificationist, and beyond — against which 'mathematical explanation' must qualify.
In plain terms: The free standard for what explaining is, so the mathematical cases can be judged rather than merely admired.
Background: The Unreasonable Effectiveness of Mathematics in the Natural Sciences · Explanation
11. Is chemistry reducible to physics? State the criterion of reduction you use.
The reduction debate in its home discipline: molecular structure, emergence, and what quantum mechanics leaves underived.
In plain terms: The free scholarly account of chemistry's contested independence from the physics beneath it.
The criteria themselves — Nagelian bridge laws to functional reduction — which the question demands be stated.
In plain terms: The free menu of what 'reducible' can mean, from which an answer must first choose.
The periodic table and the physics that drives it
The reductionist case at maximum strength: chemical periodicity computed from relativistic quantum mechanics.
In plain terms: The free scientific exhibit for the prosecution — how much of chemistry physics can actually deliver.
Background: Philosophy of chemistry · Reductionism
12. Does teleological explanation have an ineliminable role in biology?
Teleological notions in biology
The question entire: function talk, selected effects, and whether purposive idiom is eliminable or load-bearing.
In plain terms: The free scholarly treatment of biology's 'in order to' — shorthand, or something science cannot do without.
From enzymatic adaptation to allosteric transitions (Nobel lecture)
The scientist's resolution in practice: teleonomy — apparent purpose grounded in molecular mechanism and selection.
In plain terms: The free lecture from the biologist who coined the working compromise — purpose-like, without purpose.
The reducing base: selection histories as the naturalistic cash value of functional ascription.
In plain terms: The free entry on the mechanism that lets 'for the sake of' be translated into 'because it was favoured'.
13. What is a law of nature, and could the laws have been otherwise?
The candidate analyses — Humean best systems, necessitation, anti-reductionism — each fixing differently whether laws could differ.
In plain terms: The free survey of what a law of nature is, where each answer settles the 'could they have been otherwise' by itself.
The contingency question's grammar: nomological against metaphysical necessity, and where laws sit between them.
In plain terms: The free entry sorting the senses of 'could' — without which the second half of the question cannot be posed.
The stress test: hedged laws of the special sciences, straining every clean analysis of lawhood.
In plain terms: The free study of laws that hold 'other things equal' — the awkward cases any definition must survive.
Background: Scientific law · Counterfactual conditional
14. Is there a coherent notion of metaphysical modality distinct from logical and physical modality?
The question's own taxonomy: whether metaphysical necessity is a genuine grade between logical and nomic, and how to earn it.
In plain terms: The free entry asking exactly this — is there a middle kind of 'must', neither logic's nor physics'?
The semantics that gives the notion content — and the realist, ersatzist, and fictionalist accounts of what worlds are.
In plain terms: The free survey of the device — ways things might have been — on which the coherence claim leans.
The test of distinctness: whether conceivability or essence gives access to a necessity beyond logic and law.
In plain terms: The free entry on how anyone could know the middle 'must' — where the notion earns its keep or fails to.
Background: Possible world · Saul Kripke
15. Compare the Russellian and Strawsonian treatments of definite descriptions.
The comparison itself: Russell's quantificational analysis against Strawson's presupposition account, with the referential–attributive aftermath.
In plain terms: The free scholarly staging of the exact debate — does 'the present King of France is bald' say something false, or fail to say?
The primary text: descriptions analysed away into quantifiers, with scope distinctions doing the work.
In plain terms: The free original — the most celebrated paper in analytic philosophy, making its case in fourteen pages.
The doctrine in its epistemic setting: knowledge by acquaintance and by description, motivating the analysis.
In plain terms: Russell's free classic showing why the theory mattered — how we speak of things we have never met.
Background: Definite description · Theory of descriptions
16. Can there be time without change, or change without time?
Both directions of the question treated: Shoemaker's changeless-time worlds and relationist denials of time without change.
In plain terms: The free entry where the thought experiments live — frozen regions, and whether the clock still ticks.
The primary text binding time to change: the A-series argued essential and contradictory at once.
In plain terms: The free original claiming time requires change so strictly that time itself cannot be real.
The other relatum analysed: what change is — property variation across times — and its classical puzzles.
In plain terms: The free companion on change itself, without which neither half of the question is fixed.
Background: Philosophy of space and time · A series and B series
17. Is the principle of bivalence defensible in the face of the future?
The debate entire: bivalence against openness, from Łukasiewicz's third value to Ockhamist and supervaluationist rescues.
In plain terms: The free survey of tomorrow's sea battle — whether statements about it are already true or false today.
The primary text: chapter nine's sea battle, where the challenge to bivalence was first raised.
In plain terms: The free original — the ancient page from which the entire problem descends.
The formal instruments: branching-time semantics in which the bivalence options become precise theories.
In plain terms: The free entry on the logic of 'will be' — the machinery that turns the puzzle into exact alternatives.
Background: Problem of future contingents · Principle of bivalence
18. What does quantum entanglement imply for our concept of causation?
Quantum entanglement and information
The phenomenon and its causal awkwardness: correlations without signalling, and what dependence survives.
In plain terms: The free entry on the correlations that puzzle causation — connected outcomes with nothing sent between.
The experimental and theoretical record: local causality's precise failure, quantified and reviewed.
In plain terms: The free authoritative survey of what the experiments actually rule out — and what they leave standing.
The concept under revision: interventionist and signalling analyses of causation confronting the quantum correlations.
In plain terms: The free examination of what 'cause' should mean once the correlations are admitted into evidence.
Background: Quantum entanglement · Bell's theorem
19. Is there a livable interpretation of probability that is neither frequentist nor subjectivist?
Interpretations of probability
The third ways themselves: propensities, best-system chances, and logical probability, audited against the classic pair.
In plain terms: The free survey where the question's candidates — chance as neither counting nor credence — are laid out and tested.
Probability Theory: The Logic of Science (open manuscript)
The objective-Bayesian construction in full: probability as extended logic, constrained by consistency rather than opinion.
In plain terms: The famous free book arguing probability is reasoning itself — neither frequencies nor mere personal betting odds.
The subjectivist baseline made precise — what any third interpretation must differ from, and the objective-prior middle ground.
In plain terms: The free entry on the credence view, so the sought alternative has a definite rival to beat.
Background: Probability interpretations · Propensity probability
20. Can modal logic tell us anything about reality, or only about our language?
The systems and their semantics: what the box and diamond formalise, and how axioms answer to frame conditions.
In plain terms: The free entry on the logic of necessity — the machinery whose worldly import the question weighs.
The metaphysical crux: whether the semantics' worlds are discoveries about reality or bookkeeping for our idioms.
In plain terms: The free survey of what, if anything, the formalism's 'worlds' actually are.
Modal logic: a contemporary view
A second full treatment, with correspondence theory exhibiting how axioms mirror structural claims.
In plain terms: A free companion account of the same machinery from an independent editorial tradition.
Background: Modal logic · Kripke semantics
21. Is the a priori / a posteriori distinction coherent and explanatory? Draw it or reject it.
A priori justification and knowledge
The distinction drawn with modern care — experience-independence, defeasibility — and the challenges to its coherence.
In plain terms: The free entry doing exactly what the question demands: drawing the line, and defending that there is one.
The analytic/synthetic distinction
The companion cut and its Quinean crisis — the rejection option's strongest source, with the replies.
In plain terms: The free account of the neighbouring distinction whose collapse would drag this one with it.
The primary architecture: the a priori made explanatory — conditions of possible experience — at the doctrine's source.
In plain terms: The free original in which the distinction was first made to carry a whole philosophy.
Background: A priori and a posteriori · Analytic–synthetic distinction
22. What is the best response to the underdetermination of theory by evidence?
Underdetermination of scientific theory
The thesis in its varieties — holist and contrastive, transient and permanent — with the standard responses ranked.
In plain terms: The free survey of the worry that evidence can never single out one theory, and the ways science answers it.
The response's toolbox: how evidence bears differentially on rivals — Bayesian and qualitative measures of support.
In plain terms: The free entry on how data actually discriminate — the machinery any good answer will invoke.
The origin and its temper: holism with 'good sense' as arbiter — an answer already present at the source.
In plain terms: The free account of the physicist who raised the problem — and never thought it made theory choice arbitrary.
Background: Underdetermination · Duhem–Quine thesis
23. Are laws of nature relations among universals, or regularities, or neither?
The question's exact menu: Armstrong's necessitation among universals, Humean regularity, and the anti-reductionist third option.
In plain terms: The free entry laying out precisely the three answers the question names — and the costs of each.
The best-system regularity account at full strength, inside the Humean programme that motivates it.
In plain terms: The free study of the regularity view's greatest architect — laws as the best summary of what happens.
The universals account given its full hearing — the Dretske–Tooley–Armstrong view and its inference problem.
In plain terms: A free companion survey where the 'relations among universals' answer receives its best defence.
Background: Universal (metaphysics) · David Hume
24. Does the success of science give us reason to believe its unobservable posits?
The no-miracles argument and its pessimistic-induction counter — the question's exact dialectic, fully sourced.
In plain terms: The free survey of the master argument: success would be miraculous unless the invisible posits are real.
The moderating position: commitment to structure surviving theory change, dodging the induction while keeping success explained.
In plain terms: The free entry on the compromise — believe the equations' relations, hold the entities lightly.
The strongest denial: empirical adequacy as science's aim, with success explained without unobservable belief.
In plain terms: The free account of van Fraassen's alternative — accept the theory, believe only what could be seen.
Background: Scientific realism · Instrumentalism
25. Is identity across time a further fact, or fixed by qualitative continuity?
The question posed in its own terms: persistence puzzles, and whether identity outruns continuity's fixing.
In plain terms: The free entry on the ship rebuilt plank by plank — and whether anything over and above the changes decides its fate.
The reductionist machinery: perdurance and stage theory, on which continuity is all the fact there is.
In plain terms: The free account of the view that things are stretched through time like events — dissolving the 'further fact'.
The underlying tension — one thing, incompatible properties, different times — that persistence theories exist to resolve.
In plain terms: The free companion on why survival through change was ever puzzling in the first place.
Background: Identity (philosophy) · Perdurantism
26. What is a natural kind, and does the periodic table exhibit them?
The concept audited — essentialism, clusters, promiscuous realism — with the chemical elements as the paradigm test.
In plain terms: The free survey of nature's alleged joints, where the elements are everyone's best example.
The case study from inside: elements individuated by nuclear charge, isotopes, and what the table's order reflects.
In plain terms: The free entry on whether chemistry's great chart carves nature or organises our bookkeeping.
The periodic table and the physics that drives it
The physical underwriting: periodicity from quantum structure — the strongest case that the kinds are found, not made.
In plain terms: The free scientific exhibit that the table's pattern is written into physics itself.
Background: Natural kind · Periodic table
27. Can there be a consistent version of verificationism, and would it be plausible?
The doctrine at source and under strain: verifiability's formulations and the self-application problem.
In plain terms: The free history of the movement that staked meaning on testability — and struggled to state the stake testably.
The refinement record: from strict verifiability to confirmability and translatability — the consistency question's actual history.
In plain terms: The free account of the retreats and repairs — each weaker criterion bought at a price in plausibility.
The most sophisticated attempt: Carnap's frameworks and tolerance — verificationism made consistent by going internal.
In plain terms: The free study of the movement's subtlest mind, whose late version is the question's best candidate.
Background: Verificationism · Logical positivism
28. What does it mean to say a physical theory is background-independent, and is that a virtue?
The concept's philosophical core: diffeomorphism invariance and what it means for spacetime to be no fixed stage.
In plain terms: The free entry on Einstein's own puzzle — the argument that made the stage itself part of the play.
The case for background independence
The virtue defended: relational principles as methodology for quantum gravity, with the definitional work done explicitly.
In plain terms: The free manifesto arguing that abolishing fixed backgrounds is not just a feature but the way forward.
The arena of the dispute: background independence as a divide among research programmes, with the counter-considerations.
In plain terms: The free survey of the field where the virtue is contested — strings against loops, stage against story.
Background: Background independence · Hole argument
29. Is arithmetic analytic, synthetic a priori, or empirical?
Kant's philosophy of mathematics
The synthetic a priori option at source: arithmetic grounded in the form of intuition, with modern assessments.
In plain terms: The free entry on Kant's middle answer — necessary truths that still say something substantial.
The analytic option's career: Frege's programme, Russell's paradox, and Hume's-Principle revivals.
In plain terms: The free account of the grand attempt to make arithmetic pure logic — its fall and modern resurrections.
A System of Logic, Ratiocinative and Inductive
The empirical option's classic statement: arithmetic as very general natural fact — the third answer in the original.
In plain terms: The free primary text daring to say two plus two is known the way stones are — by experience.
Background: Logicism · Foundations of mathematics
30. What is the strongest challenge to a law of classical logic, and can it be resisted?
The strongest challenge stated by its champions: true contradictions from the paradoxes, against non-contradiction itself.
In plain terms: The free case for the unthinkable — that some contradictions are true — argued at full strength.
The formal vehicle: explosion rejected, and logics in which the challenge can be coherently housed.
In plain terms: The free entry on logics that survive contradiction — proof the challenge is at least statable.
The rival challenge for comparison: excluded middle rejected on constructive grounds — and the classical recapture results that resist both.
In plain terms: The free account of the other great heresy, so the 'strongest' claim can be judged between them.
Background: Paraconsistent logic · Law of excluded middle