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All the ideas for 'Defending the Axioms', 'Consciousness' and 'Logic for Philosophy'

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76 ideas

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
'Theorems' are formulas provable from no premises at all [Sider]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Truth tables assume truth functionality, and are just pictures of truth functions [Sider]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
Intuitively, deontic accessibility seems not to be reflexive, but to be serial [Sider]
In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary [Sider]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B
System B introduces iterated modalities [Sider]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid [Sider]
4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K) [Sider]
4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
We can treat modal worlds as different times [Sider]
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
Converse Barcan Formula: □∀αφ→∀α□φ [Sider]
The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic [Sider]
System B is needed to prove the Barcan Formula [Sider]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
You can employ intuitionist logic without intuitionism about mathematics [Sider]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Physicalism requires the naturalisation or rejection of set theory [Lycan]
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
The most popular account of logical consequence is the semantic or model-theoretic one [Sider]
Maybe logical consequence is more a matter of provability than of truth-preservation [Sider]
Maybe logical consequence is impossibility of the premises being true and the consequent false [Sider]
Maybe logical consequence is a primitive notion [Sider]
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
A 'theorem' is an axiom, or the last line of a legitimate proof [Sider]
5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity [Sider]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'total' function must always produce an output for a given domain [Sider]
5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
λ can treat 'is cold and hungry' as a single predicate [Sider]
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Good axioms should be indisputable logical truths [Sider]
No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider]
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [Sider]
Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Natural deduction helpfully allows reasoning with assumptions [Sider]
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
We can build proofs just from conclusions, rather than from plain formulae [Sider]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Valuations in PC assign truth values to formulas relative to variable assignments [Sider]
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
The semantical notion of a logical truth is validity, being true in all interpretations [Sider]
It is hard to say which are the logical truths in modal logic, especially for iterated modal operators [Sider]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation [Sider]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
If two mathematical themes coincide, that suggest a single deep truth [Maddy]
5. Theory of Logic / K. Features of Logics / 4. Completeness
In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider]
5. Theory of Logic / K. Features of Logics / 6. Compactness
Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
7. Existence / C. Structure of Existence / 2. Reduction
Institutions are not reducible as types, but they are as tokens [Lycan]
Types cannot be reduced, but levels of reduction are varied groupings of the same tokens [Lycan]
7. Existence / C. Structure of Existence / 3. Levels of Reality
One location may contain molecules, a metal strip, a key, an opener of doors, and a human tragedy [Lycan]
7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation [Sider]
Supervaluational logic is classical, except when it adds the 'Definitely' operator [Sider]
A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification [Sider]
We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings [Sider]
7. Existence / E. Categories / 3. Proposed Categories
I see the 'role'/'occupant' distinction as fundamental to metaphysics [Lycan]
8. Modes of Existence / A. Relations / 1. Nature of Relations
A relation is a feature of multiple objects taken together [Sider]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
The identity of indiscernibles is necessarily true, if being a member of some set counts as a property [Sider]
10. Modality / A. Necessity / 3. Types of Necessity
'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist [Sider]
10. Modality / A. Necessity / 5. Metaphysical Necessity
Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible [Sider]
10. Modality / A. Necessity / 6. Logical Necessity
Logical truths must be necessary if anything is [Sider]
10. Modality / B. Possibility / 8. Conditionals / b. Types of conditional
'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true [Sider]
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
Transworld identity is not a problem in de dicto sentences, which needn't identify an individual [Sider]
10. Modality / E. Possible worlds / 3. Transworld Objects / e. Possible Objects
Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost [Sider]
11. Knowledge Aims / C. Knowing Reality / 1. Perceptual Realism / b. Direct realism
I think greenness is a complex microphysical property of green objects [Lycan]
15. Nature of Minds / B. Features of Minds / 4. Intentionality / a. Nature of intentionality
Intentionality comes in degrees [Lycan]
15. Nature of Minds / B. Features of Minds / 4. Intentionality / b. Intentionality theories
Teleological views allow for false intentional content, unlike causal and nomological theories [Lycan]
15. Nature of Minds / B. Features of Minds / 5. Qualia / c. Explaining qualia
Pain is composed of urges, desires, impulses etc, at different levels of abstraction [Lycan]
The right 'level' for qualia is uncertain, though top (behaviourism) and bottom (particles) are false [Lycan]
17. Mind and Body / A. Mind-Body Dualism / 8. Dualism of Mind Critique
If energy in the brain disappears into thin air, this breaches physical conservation laws [Lycan]
In lower animals, psychology is continuous with chemistry, and humans are continuous with animals [Lycan]
17. Mind and Body / B. Behaviourism / 4. Behaviourism Critique
Two behaviourists meet. The first says,"You're fine; how am I?" [Lycan]
17. Mind and Body / C. Functionalism / 1. Functionalism
If functionalism focuses on folk psychology, it ignores lower levels of function [Lycan]
Functionalism must not be too abstract to allow inverted spectrum, or so structural that it becomes chauvinistic [Lycan]
17. Mind and Body / C. Functionalism / 2. Machine Functionalism
The distinction between software and hardware is not clear in computing [Lycan]
17. Mind and Body / C. Functionalism / 5. Teleological Functionalism
Mental types are a subclass of teleological types at a high level of functional abstraction [Lycan]
Teleological characterisations shade off smoothly into brutely physical ones [Lycan]
17. Mind and Body / E. Mind as Physical / 1. Physical Mind
Identity theory is functionalism, but located at the lowest level of abstraction [Lycan]
17. Mind and Body / E. Mind as Physical / 2. Reduction of Mind
We reduce the mind through homuncular groups, described abstractly by purpose [Lycan]
Teleological functionalism helps us to understand psycho-biological laws [Lycan]
17. Mind and Body / E. Mind as Physical / 7. Anti-Physicalism / b. Multiple realisability
A Martian may exhibit human-like behaviour while having very different sensations [Lycan]
26. Natural Theory / A. Speculations on Nature / 2. Natural Purpose / b. Limited purposes
We need a notion of teleology that comes in degrees [Lycan]
27. Natural Reality / B. Modern Physics / 4. Standard Model / a. Concept of matter
'Physical' means either figuring in physics descriptions, or just located in space-time [Lycan]