Combining Texts

All the ideas for 'fragments/reports', 'Rationality in Action' and 'A Tour through Mathematical Logic'

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54 ideas

2. Reason / A. Nature of Reason / 1. On Reason
Entailment and validity are relations, but inference is a human activity [Searle]
Theory involves accepting conclusions, and so is a special case of practical reason [Searle]
2. Reason / A. Nature of Reason / 8. Naturalising Reason
Rationality is built into the intentionality of the mind, and its means of expression [Searle]
Rationality is the way we coordinate our intentionality [Searle]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
A 'tautology' must include connectives [Wolf,RS]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
If complex logic requires rules, then so does basic logic [Searle]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
In real reasoning semantics gives validity, not syntax [Searle]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
Model theory reveals the structures of mathematics [Wolf,RS]
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 4. Completeness
If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Modern mathematics has unified all of its objects within set theory [Wolf,RS]
7. Existence / C. Structure of Existence / 5. Supervenience / b. Types of supervenience
Users of 'supervenience' blur its causal and constitutive meanings [Searle]
11. Knowledge Aims / A. Knowledge / 4. Belief / c. Aim of beliefs
Our beliefs are about things, not propositions (which are the content of the belief) [Searle]
A belief is a commitment to truth [Searle]
We can't understand something as a lie if beliefs aren't commitment to truth [Searle]
11. Knowledge Aims / B. Certain Knowledge / 4. The Cogito
Thinking must involve a self, not just an "it" [Searle]
13. Knowledge Criteria / A. Justification Problems / 1. Justification / a. Justification issues
Reasons can either be facts in the world, or intentional states [Searle]
13. Knowledge Criteria / C. External Justification / 1. External Justification
In the past people had a reason not to smoke, but didn't realise it [Searle]
13. Knowledge Criteria / C. External Justification / 2. Causal Justification
Causes (usually events) are not the same as reasons (which are never events) [Searle]
13. Knowledge Criteria / E. Relativism / 2. Knowledge as Convention
By nature people are close to one another, but culture drives them apart [Hippias]
16. Persons / A. Concept of a Person / 2. Persons as Responsible
Being held responsible for past actions makes no sense without personal identity [Searle]
16. Persons / A. Concept of a Person / 3. Persons as Reasoners
Giving reasons for action requires reference to a self [Searle]
A 'self' must be capable of conscious reasonings about action [Searle]
An intentional, acting, rational being must have a self [Searle]
16. Persons / A. Concept of a Person / 4. Persons as Agents
Action requires a self, even though perception doesn't [Searle]
16. Persons / B. Nature of the Self / 1. Self and Consciousness
Selfs are conscious, enduring, reasonable, active, free, and responsible [Searle]
A self must at least be capable of consciousness [Searle]
16. Persons / B. Nature of the Self / 4. Presupposition of Self
The self is neither an experience nor a thing experienced [Searle]
16. Persons / B. Nature of the Self / 5. Self as Associations
The bundle must also have agency in order to act, and a self to act rationally [Searle]
16. Persons / F. Free Will / 4. For Free Will
Free will is most obvious when we choose between several reasons for an action [Searle]
Rational decision making presupposes free will [Searle]
We freely decide whether to make a reason for action effective [Searle]
20. Action / C. Motives for Action / 1. Acting on Desires
Preferences can result from deliberation, not just precede it [Searle]
20. Action / C. Motives for Action / 3. Acting on Reason / a. Practical reason
We don't accept practical reasoning if the conclusion is unpalatable [Searle]
20. Action / C. Motives for Action / 3. Acting on Reason / b. Intellectualism
The essence of humanity is desire-independent reasons for action [Searle]
Only an internal reason can actually motivate the agent to act [Searle]
22. Metaethics / B. Value / 1. Nature of Value / b. Fact and value
If it is true, you ought to believe it [Searle]
If this is a man, you ought to accept similar things as men [Searle]
23. Ethics / B. Contract Ethics / 3. Promise Keeping
Promises hold because I give myself a reason, not because it is an institution [Searle]
23. Ethics / D. Deontological Ethics / 2. Duty
'Ought' implies that there is a reason to do something [Searle]