Combining Texts

All the ideas for 'works', 'Carnap and Logical Truth' and 'Higher-Order Logic'

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

23 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10301 The axiom of choice is controversial, but it could be replaced [Shapiro]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
13010 In order to select the logic justified by experience, we would need to use a lot of logic [Boghossian on Quine]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
9002 Elementary logic requires truth-functions, quantifiers (and variables), identity, and also sets of variables [Quine]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
10298 Some say that second-order logic is mathematics, not logic [Shapiro]
10299 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
13681 Logical consequence is marked by being preserved under all nonlogical substitutions [Quine, by Sider]
10300 Logical consequence can be defined in terms of the logical terminology [Shapiro]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13829 If logical truths essentially depend on logical constants, we had better define the latter [Hacking on Quine]
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
10290 Second-order variables also range over properties, sets, relations or functions [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
9003 Set theory was struggling with higher infinities, when new paradoxes made it baffling [Quine]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10294 Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9004 If set theory is not actually a branch of logic, then Frege's derivation of arithmetic would not be from logic [Quine]
8. Modes of Existence / B. Properties / 11. Properties as Sets
10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
8. Modes of Existence / E. Nominalism / 1. Nominalism / b. Nominalism about universals
9006 Commitment to universals is as arbitrary or pragmatic as the adoption of a new system of bookkeeping [Quine]
10. Modality / A. Necessity / 6. Logical Necessity
9001 Frege moved Kant's question about a priori synthetic to 'how is logical certainty possible?' [Quine]
12. Knowledge Sources / A. A Priori Knowledge / 7. A Priori from Convention
9005 Examination of convention in the a priori begins to blur the distinction with empirical knowledge [Quine]
29. Religion / B. Monotheistic Religion / 4. Christianity / d. Heresy
16713 Philosophers are the forefathers of heretics [Tertullian]
29. Religion / D. Religious Issues / 1. Religious Commitment / e. Fideism
6610 I believe because it is absurd [Tertullian]