Combining Texts

All the ideas for 'works', 'Higher-Order Logic' and 'Penses'

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28 ideas

2. Reason / A. Nature of Reason / 9. Limits of Reason
6675 The heart has its reasons of which reason knows nothing [Pascal]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
9565 Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara]
3339 For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn]
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 / 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 / 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
10300 Logical consequence can be defined in terms of the logical terminology [Shapiro]
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
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
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]
8. Modes of Existence / B. Properties / 11. Properties as Sets
10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
22011 The first principles of truth are not rational, but are known by the heart [Pascal]
19. Language / F. Communication / 1. Rhetoric
6681 We only want to know things so that we can talk about them [Pascal]
21. Aesthetics / C. Artistic Issues / 3. Artistic Representation
6676 Painting makes us admire things of which we do not admire the originals [Pascal]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention
6680 It is a funny sort of justice whose limits are marked by a river [Pascal]
22. Metaethics / B. Value / 1. Nature of Value / d. Subjective value
6677 Imagination creates beauty, justice and happiness, which is the supreme good [Pascal]
22. Metaethics / C. The Good / 2. Happiness / d. Routes to happiness
6678 We live for the past or future, and so are never happy in the present [Pascal]
23. Ethics / F. Existentialism / 3. Angst
20732 If man considers himself as lost and imprisoned in the universe, he will be terrified [Pascal]
24. Political Theory / D. Ideologies / 5. Democracy / a. Nature of democracy
6682 Majority opinion is visible and authoritative, although not very clever [Pascal]
25. Social Practice / A. Freedoms / 5. Freedom of lifestyle
6679 It is not good to be too free [Pascal]
28. God / B. Proving God / 2. Proofs of Reason / d. Pascal's Wager
7457 Pascal is right, but relies on the unsupported claim of a half as the chance of God's existence [Hacking on Pascal]
7455 Pascal knows you can't force belief, but you can make it much more probable [Pascal, by Hacking]
7456 The libertine would lose a life of enjoyable sin if he chose the cloisters [Hacking on Pascal]
6684 If you win the wager on God's existence you win everything, if you lose you lose nothing [Pascal]