Combining Texts

All the ideas for 'works', 'Higher-Order Logic' and 'Replies on 'Limits of Abstraction''

expand these ideas     |    start again     |     specify just one area for these texts


27 ideas

1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
Concern for rigour can get in the way of understanding phenomena [Fine,K]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The axiom of choice is controversial, but it could be replaced [Shapiro]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
There is no stage at which we can take all the sets to have been generated [Fine,K]
4. Formal Logic / G. Formal Mereology / 3. Axioms of Mereology
We might combine the axioms of set theory with the axioms of mereology [Fine,K]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Some say that second-order logic is mathematics, not logic [Shapiro]
If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence can be defined in terms of the logical terminology [Shapiro]
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
If you ask what F the second-order quantifier quantifies over, you treat it as first-order [Fine,K]
Second-order variables also range over properties, sets, relations or functions [Shapiro]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Assigning an entity to each predicate in semantics is largely a technical convenience [Fine,K]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
Why should a Dedekind cut correspond to a number? [Fine,K]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
Unless we know whether 0 is identical with the null set, we create confusions [Fine,K]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
Set-theoretic imperialists think sets can represent every mathematical object [Fine,K]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Logicists say mathematics can be derived from definitions, and can be known that way [Fine,K]
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / b. Levels of abstraction
A generative conception of abstracts proposes stages, based on concepts of previous objects [Fine,K]
8. Modes of Existence / B. Properties / 11. Properties as Sets
Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Abstraction-theoretic imperialists think Fregean abstracts can represent every mathematical object [Fine,K]
We can combine ZF sets with abstracts as urelements [Fine,K]
We can create objects from conditions, rather than from concepts [Fine,K]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
Unlike us, the early Greeks thought envy was a good thing, and hope a bad thing [Hesiod, by Nietzsche]