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All the ideas for 'Function and Concept', 'The Tarskian Turn' and 'Principia Mathematica'

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100 ideas

1. Philosophy / D. Nature of Philosophy / 1. Philosophy
Philosophy must abstract from the senses [Newton]
1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
Philosophy is the most general intellectual discipline [Horsten]
2. Reason / D. Definition / 2. Aims of Definition
A definition should allow the defined term to be eliminated [Horsten]
3. Truth / A. Truth Problems / 1. Truth
Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles [Horsten]
Truth is a property, because the truth predicate has an extension [Horsten]
3. Truth / A. Truth Problems / 2. Defining Truth
Truth has no 'nature', but we should try to describe its behaviour in inferences [Horsten]
3. Truth / A. Truth Problems / 5. Truth Bearers
Propositions have sentence-like structures, so it matters little which bears the truth [Horsten]
3. Truth / C. Correspondence Truth / 2. Correspondence to Facts
Modern correspondence is said to be with the facts, not with true propositions [Horsten]
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
The correspondence 'theory' is too vague - about both 'correspondence' and 'facts' [Horsten]
3. Truth / D. Coherence Truth / 2. Coherence Truth Critique
The coherence theory allows multiple coherent wholes, which could contradict one another [Horsten]
3. Truth / E. Pragmatic Truth / 1. Pragmatic Truth
The pragmatic theory of truth is relative; useful for group A can be useless for group B [Horsten]
3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
Tarski's hierarchy lacks uniform truth, and depends on contingent factors [Horsten]
Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa [Horsten]
3. Truth / F. Semantic Truth / 1. Tarski's Truth / c. Meta-language for truth
Semantic theories have a regress problem in describing truth in the languages for the models [Horsten]
3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models [Horsten]
'Reflexive' truth theories allow iterations (it is T that it is T that p) [Horsten]
A good theory of truth must be compositional (as well as deriving biconditionals) [Horsten]
The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar [Horsten]
Axiomatic theories take truth as primitive, and propose some laws of truth as axioms [Horsten]
By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content! [Horsten]
Axiomatic approaches to truth avoid the regress problem of semantic theories [Horsten]
An axiomatic theory needs to be of maximal strength, while being natural and sound [Horsten]
3. Truth / G. Axiomatic Truth / 2. FS Truth Axioms
Friedman-Sheard theory keeps classical logic and aims for maximum strength [Horsten]
3. Truth / G. Axiomatic Truth / 3. KF Truth Axioms
Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength [Horsten]
3. Truth / H. Deflationary Truth / 2. Deflationary Truth
Inferential deflationism says truth has no essence because no unrestricted logic governs the concept [Horsten]
Deflationism skips definitions and models, and offers just accounts of basic laws of truth [Horsten]
Deflationism concerns the nature and role of truth, but not its laws [Horsten]
This deflationary account says truth has a role in generality, and in inference [Horsten]
Deflationism says truth isn't a topic on its own - it just concerns what is true [Horsten]
Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property [Horsten]
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
Frege thought traditional categories had psychological and linguistic impurities [Frege, by Rumfitt]
4. Formal Logic / E. Nonclassical Logics / 1. Nonclassical Logics
Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well [Horsten]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Doubt is thrown on classical logic by the way it so easily produces the liar paradox [Horsten]
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms [Horsten]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
First-level functions have objects as arguments; second-level functions take functions as arguments [Frege]
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
Relations are functions with two arguments [Frege]
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
A theory is 'non-conservative' if it facilitates new mathematical proofs [Horsten]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F) [Horsten]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
Satisfaction is a primitive notion, and very liable to semantical paradoxes [Horsten]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
The first incompleteness theorem means that consistency does not entail soundness [Horsten]
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated [Horsten]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
Newton developed a kinematic approach to geometry [Newton, by Kitcher]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
Quantities and ratios which continually converge will eventually become equal [Newton]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
ZFC showed that the concept of set is mathematical, not logical, because of its existence claims [Horsten]
Set theory is substantial over first-order arithmetic, because it enables new proofs [Horsten]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Predicativism says mathematical definitions must not include the thing being defined [Horsten]
7. Existence / A. Nature of Existence / 6. Criterion for Existence
Frege takes the existence of horses to be part of their concept [Frege, by Sommers]
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
We may believe in atomic facts, but surely not complex disjunctive ones? [Horsten]
7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
In the supervaluationist account, disjunctions are not determined by their disjuncts [Horsten]
If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't [Horsten]
8. Modes of Existence / B. Properties / 10. Properties as Predicates
Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver on Frege]
8. Modes of Existence / C. Powers and Dispositions / 2. Powers as Basic
I suspect that each particle of bodies has attractive or repelling forces [Newton]
9. Objects / A. Existence of Objects / 3. Objects in Thought
The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place [Frege]
9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
Particles mutually attract, and cohere at short distances [Newton]
9. Objects / C. Structure of Objects / 8. Parts of Objects / b. Sums of parts
The place of a thing is the sum of the places of its parts [Newton]
11. Knowledge Aims / A. Knowledge / 4. Belief / c. Aim of beliefs
Some claim that indicative conditionals are believed by people, even though they are not actually held true [Horsten]
14. Science / B. Scientific Theories / 6. Theory Holism
If you changed one of Newton's concepts you would destroy his whole system [Heisenberg on Newton]
14. Science / C. Induction / 1. Induction
Science deduces propositions from phenomena, and generalises them by induction [Newton]
14. Science / D. Explanation / 2. Types of Explanation / g. Causal explanations
We should admit only enough causes to explain a phenomenon, and no more [Newton]
Natural effects of the same kind should be assumed to have the same causes [Newton]
14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence
From the phenomena, I can't deduce the reason for the properties of gravity [Newton]
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
Concepts are the ontological counterparts of predicative expressions [Frege, by George/Velleman]
An assertion about the concept 'horse' must indirectly speak of an object [Frege, by Hale]
A concept is a function whose value is always a truth-value [Frege]
18. Thought / D. Concepts / 4. Structure of Concepts / a. Conceptual structure
Unlike objects, concepts are inherently incomplete [Frege, by George/Velleman]
19. Language / B. Reference / 5. Speaker's Reference
I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false [Frege]
19. Language / C. Assigning Meanings / 1. Syntax
A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding [Horsten]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / c. Ultimate substances
Newton's four fundamentals are: space, time, matter and force [Newton, by Russell]
26. Natural Theory / A. Speculations on Nature / 7. Later Matter Theories / a. Early Modern matter
Mass is central to matter [Newton, by Hart,WD]
26. Natural Theory / A. Speculations on Nature / 7. Later Matter Theories / b. Corpuscles
An attraction of a body is the sum of the forces of their particles [Newton]
26. Natural Theory / C. Causation / 1. Causation
Newtonian causation is changes of motion resulting from collisions [Newton, by Baron/Miller]
26. Natural Theory / D. Laws of Nature / 6. Laws as Numerical
You have discovered that elliptical orbits result just from gravitation and planetary movement [Newton, by Leibniz]
We have given up substantial forms, and now aim for mathematical laws [Newton]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / c. Essence and laws
I am not saying gravity is essential to bodies [Newton]
27. Natural Reality / A. Classical Physics / 1. Mechanics / a. Explaining movement
Newton reclassified vertical motion as violent, and unconstrained horizontal motion as natural [Newton, by Harré]
27. Natural Reality / A. Classical Physics / 1. Mechanics / b. Laws of motion
Inertia rejects the Aristotelian idea of things having natural states, to which they return [Newton, by Alexander,P]
1: Bodies rest, or move in straight lines, unless acted on by forces [Newton]
2: Change of motion is proportional to the force [Newton]
3: All actions of bodies have an equal and opposite reaction [Newton]
Newton's Third Law implies the conservation of momentum [Newton, by Papineau]
27. Natural Reality / A. Classical Physics / 1. Mechanics / c. Forces
Newton's idea of force acting over a long distance was very strange [Heisenberg on Newton]
Newton introduced forces other than by contact [Newton, by Papineau]
Newton's laws cover the effects of forces, but not their causes [Newton, by Papineau]
Newton's forces were accused of being the scholastics' real qualities [Pasnau on Newton]
I am studying the quantities and mathematics of forces, not their species or qualities [Newton]
The aim is to discover forces from motions, and use forces to demonstrate other phenomena [Newton]
27. Natural Reality / A. Classical Physics / 1. Mechanics / d. Gravity
Newton showed that falling to earth and orbiting the sun are essentially the same [Newton, by Ellis]
27. Natural Reality / A. Classical Physics / 2. Thermodynamics / c. Conservation of energy
Early Newtonians could not formulate conservation of energy, having no concept of potential energy [Newton, by Papineau]
27. Natural Reality / C. Space / 4. Substantival Space
Absolute space is independent, homogeneous and immovable [Newton]
27. Natural Reality / D. Time / 1. Nature of Time / a. Absolute time
Newton needs intervals of time, to define velocity and acceleration [Newton, by Le Poidevin]
Newton thought his laws of motion needed absolute time [Newton, by Bardon]
Time exists independently, and flows uniformly [Newton]
Absolute time, from its own nature, flows equably, without relation to anything external [Newton]
27. Natural Reality / D. Time / 2. Passage of Time / g. Time's arrow
Newtonian mechanics does not distinguish negative from positive values of time [Newton, by Coveney/Highfield]
27. Natural Reality / D. Time / 3. Parts of Time / d. Measuring time
If there is no uniform motion, we cannot exactly measure time [Newton]
28. God / A. Divine Nature / 3. Divine Perfections
If a perfect being does not rule the cosmos, it is not God [Newton]
28. God / B. Proving God / 2. Proofs of Reason / b. Ontological Proof critique
The Ontological Argument fallaciously treats existence as a first-level concept [Frege]
28. God / B. Proving God / 3. Proofs of Evidence / b. Teleological Proof
The elegance of the solar system requires a powerful intellect as designer [Newton]