Combining Texts

All the ideas for 'The Evolution of Modern Metaphysics', 'First-Order Modal Logic' and 'Philosophies of Mathematics'

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87 ideas

1. Philosophy / E. Nature of Metaphysics / 1. Nature of Metaphysics
Metaphysics is the most general attempt to make sense of things [Moore,AW]
2. Reason / D. Definition / 7. Contextual Definition
Contextual definitions replace a complete sentence containing the expression [George/Velleman]
2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions quantify over the thing being defined [George/Velleman]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Each line of a truth table is a model [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / a. Symbols of ML
Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic [Fitting/Mendelsohn]
We let 'R' be the accessibility relation: xRy is read 'y is accessible from x' [Fitting/Mendelsohn]
The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ [Fitting/Mendelsohn]
The prefix σ names a possible world, and σ.n names a world accessible from that one [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / b. Terminology of ML
A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate [Fitting/Mendelsohn]
Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x' [Fitting/Mendelsohn]
A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- > [Fitting/Mendelsohn]
A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R > [Fitting/Mendelsohn]
Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual) [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / c. Derivation rules of ML
S5: a) if n ◊X then kX b) if n ¬□X then k ¬X c) if n □X then k X d) if n ¬◊X then k ¬X [Fitting/Mendelsohn]
Negation: if σ ¬¬X then σ X [Fitting/Mendelsohn]
Disj: a) if σ ¬(X∨Y) then σ ¬X and σ ¬Y b) if σ X∨Y then σ X or σ Y [Fitting/Mendelsohn]
Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new] [Fitting/Mendelsohn]
T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X [Fitting/Mendelsohn]
D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X [Fitting/Mendelsohn]
B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs] [Fitting/Mendelsohn]
4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs] [Fitting/Mendelsohn]
4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs] [Fitting/Mendelsohn]
If a proposition is possibly true in a world, it is true in some world accessible from that world [Fitting/Mendelsohn]
If a proposition is necessarily true in a world, it is true in all worlds accessible from that world [Fitting/Mendelsohn]
Conj: a) if σ X∧Y then σ X and σ Y b) if σ ¬(X∧Y) then σ ¬X or σ ¬Y [Fitting/Mendelsohn]
Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails] [Fitting/Mendelsohn]
Implic: a) if σ ¬(X→Y) then σ X and σ ¬Y b) if σ X→Y then σ ¬X or σ Y [Fitting/Mendelsohn]
Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists] [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / b. System K
The system K has no accessibility conditions [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
□P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed [Fitting/Mendelsohn]
The system D has the 'serial' conditon imposed on its accessibility relation [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / d. System T
The system T has the 'reflexive' conditon imposed on its accessibility relation [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / e. System K4
The system K4 has the 'transitive' condition on its accessibility relation [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B
The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / g. System S4
The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
Modality affects content, because P→◊P is valid, but ◊P→P isn't [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
In epistemic logic knowers are logically omniscient, so they know that they know [Fitting/Mendelsohn]
Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P' [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
F: will sometime, P: was sometime, G: will always, H: was always [Fitting/Mendelsohn]
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability [Fitting/Mendelsohn]
The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity [Fitting/Mendelsohn]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'power set' of A is all the subsets of A [George/Velleman]
The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
Grouping by property is common in mathematics, usually using equivalence [George/Velleman]
'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
The Axiom of Reducibility made impredicative definitions possible [George/Velleman]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions [Fitting/Mendelsohn]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Differences between isomorphic structures seem unimportant [George/Velleman]
5. Theory of Logic / K. Features of Logics / 2. Consistency
Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman]
A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman]
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman]
5. Theory of Logic / K. Features of Logics / 4. Completeness
A 'complete' theory contains either any sentence or its negation [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Rational numbers give answers to division problems with integers [George/Velleman]
The integers are answers to subtraction problems involving natural numbers [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers provide answers to square root problems [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
Logicists say mathematics is applicable because it is totally general [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
The classical mathematician believes the real numbers form an actual set [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
A successor is the union of a set with its singleton [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory can prove the Peano Postulates [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 8. Finitism
Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman]
Much infinite mathematics can still be justified finitely [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
The intuitionists are the idealists of mathematics [George/Velleman]
Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
The Indiscernibility of Identicals has been a big problem for modal logic [Fitting/Mendelsohn]
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
□ must be sensitive as to whether it picks out an object by essential or by contingent properties [Fitting/Mendelsohn]
Objects retain their possible properties across worlds, so a bundle theory of them seems best [Fitting/Mendelsohn]
10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts
Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them [Fitting/Mendelsohn]
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / b. Transcendental idealism
Appearances are nothing beyond representations, which is transcendental ideality [Moore,AW]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Corresponding to every concept there is a class (some of them sets) [George/Velleman]