54 ideas
3811 | Entailment and validity are relations, but inference is a human activity [Searle] |
3822 | Theory involves accepting conclusions, and so is a special case of practical reason [Searle] |
3806 | Rationality is built into the intentionality of the mind, and its means of expression [Searle] |
3812 | Rationality is the way we coordinate our intentionality [Searle] |
13520 | A 'tautology' must include connectives [Wolf,RS] |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
3809 | If complex logic requires rules, then so does basic logic [Searle] |
13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
3810 | In real reasoning semantics gives validity, not syntax [Searle] |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
3841 | Users of 'supervenience' blur its causal and constitutive meanings [Searle] |
3816 | Our beliefs are about things, not propositions (which are the content of the belief) [Searle] |
3833 | A belief is a commitment to truth [Searle] |
3837 | We can't understand something as a lie if beliefs aren't commitment to truth [Searle] |
3828 | Thinking must involve a self, not just an "it" [Searle] |
3831 | Reasons can either be facts in the world, or intentional states [Searle] |
3830 | In the past people had a reason not to smoke, but didn't realise it [Searle] |
3832 | Causes (usually events) are not the same as reasons (which are never events) [Searle] |
1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
3823 | Being held responsible for past actions makes no sense without personal identity [Searle] |
3821 | Giving reasons for action requires reference to a self [Searle] |
3824 | A 'self' must be capable of conscious reasonings about action [Searle] |
3834 | An intentional, acting, rational being must have a self [Searle] |
3825 | Action requires a self, even though perception doesn't [Searle] |
3829 | Selfs are conscious, enduring, reasonable, active, free, and responsible [Searle] |
3826 | A self must at least be capable of consciousness [Searle] |
3827 | The self is neither an experience nor a thing experienced [Searle] |
3820 | The bundle must also have agency in order to act, and a self to act rationally [Searle] |
3817 | Free will is most obvious when we choose between several reasons for an action [Searle] |
3808 | Rational decision making presupposes free will [Searle] |
3818 | We freely decide whether to make a reason for action effective [Searle] |
3814 | Preferences can result from deliberation, not just precede it [Searle] |
3840 | We don't accept practical reasoning if the conclusion is unpalatable [Searle] |
3815 | The essence of humanity is desire-independent reasons for action [Searle] |
3839 | Only an internal reason can actually motivate the agent to act [Searle] |
3835 | If it is true, you ought to believe it [Searle] |
3836 | If this is a man, you ought to accept similar things as men [Searle] |
3838 | Promises hold because I give myself a reason, not because it is an institution [Searle] |
3813 | 'Ought' implies that there is a reason to do something [Searle] |