78 ideas
10888 | Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo] |
10889 | The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo] |
10890 | A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo] |
10886 | Determinacy: an object is either in a set, or it isn't [Zalabardo] |
10887 | Specification: Determinate totals of objects always make a set [Zalabardo] |
10897 | A first-order 'sentence' is a formula with no free variables [Zalabardo] |
10893 | Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo] |
10899 | Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
10896 | Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo] |
6653 | Syntactical methods of proof need only structure, where semantic methods (truth-tables) need truth [Lowe] |
10898 | The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo] |
10902 | We can do semantics by looking at given propositions, or by building new ones [Zalabardo] |
10892 | We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo] |
10895 | 'Logically true' (|= φ) is true for every truth-assignment [Zalabardo] |
10900 | Logically true sentences are true in all structures [Zalabardo] |
10894 | A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true [Zalabardo] |
10901 | Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo] |
10903 | A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
10891 | If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
6618 | A 'substance' is a thing that remains the same when its properties change [Lowe] |
6635 | Causal theories of belief make all beliefs true, and can't explain belief about the future [Lowe] |
6619 | Perhaps 'I' no more refers than the 'it' in 'it is raining' [Lowe] |
6643 | 'Ecological' approaches say we don't infer information, but pick it up directly from reality [Lowe] |
6638 | One must be able to visually recognise a table, as well as knowing its form [Lowe] |
6644 | Computationalists object that the 'ecological' approach can't tell us how we get the information [Lowe] |
6647 | Comparing shapes is proportional in time to the angle of rotation [Lowe] |
6639 | The 'disjunctive' theory of perception says true perceptions and hallucinations need have nothing in common [Lowe] |
6640 | A causal theorist can be a direct realist, if all objects of perception are external [Lowe] |
6645 | If blindsight shows we don't need perceptual experiences, the causal theory is wrong [Lowe] |
6637 | How could one paraphrase very complex sense-data reports adverbially? [Lowe] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
6667 | There are memories of facts, memories of practical skills, and autobiographical memory [Lowe] |
6642 | Psychologists say illusions only occur in unnatural and passive situations [Lowe] |
6641 | Externalists say minds depend on environment for their very existence and identity [Lowe] |
6617 | The main questions are: is mind distinct from body, and does it have unique properties? [Lowe] |
6626 | 'Phenomenal' consciousness is of qualities; 'apperceptive' consciousness includes beliefs and desires [Lowe] |
6646 | The brain may have two systems for vision, with only the older one intact in blindsight [Lowe] |
6665 | Persons are selves - subjects of experience, with reflexive self-knowledge [Lowe] |
6670 | If my brain could survive on its own, I cannot be identical with my whole body [Lowe] |
6671 | It seems impossible to get generally applicable mental concepts from self-observation [Lowe] |
6666 | All human languages have an equivalent of the word 'I' [Lowe] |
6625 | If qualia are causally inert, how can we even know about them? [Lowe] |
6621 | You can only identify behaviour by ascribing belief, so the behaviour can't explain the belief [Lowe] |
6654 | A computer program is equivalent to the person AND the manual [Lowe] |
6629 | Functionalism commits us to bizarre possibilities, such as 'zombies' [Lowe] |
6623 | Functionalism can't distinguish our experiences in spectrum inversion [Lowe] |
6628 | Functionalism only discusses relational properties of mental states, not intrinsic properties [Lowe] |
6622 | Non-reductive physicalism accepts token-token identity (not type-type) and asserts 'supervenience' of mind and brain [Lowe] |
6634 | Physicalists must believe in narrow content (because thoughts are merely the brain states) [Lowe] |
6630 | Eliminativism is incoherent if it eliminates reason and truth as well as propositional attitudes [Lowe] |
6648 | Some behaviourists believe thought is just suppressed speech [Lowe] |
6651 | People are wildly inaccurate in estimating probabilities about an observed event [Lowe] |
6652 | 'Base rate neglect' makes people favour the evidence over its background [Lowe] |
6655 | The 'Frame Problem' is how to program the appropriate application of general knowledge [Lowe] |
6657 | Computers can't be rational, because they lack motivation and curiosity [Lowe] |
6656 | The Turing test is too behaviourist, and too verbal in its methods [Lowe] |
6636 | The naturalistic views of how content is created are the causal theory and the teleological theory [Lowe] |
6633 | Twin Earth cases imply that even beliefs about kinds of stuff are indexical [Lowe] |
6632 | The same proposition provides contents for the that-clause of an utterance and a belief [Lowe] |
6631 | If propositions are abstract entities, how can minds depend on their causal powers? [Lowe] |
6659 | The three main theories of action involve the will, or belief-plus-desire, or an agent [Lowe] |
6661 | Libet gives empirical support for the will, as a kind of 'executive' mental operation [Lowe] |
6662 | We feel belief and desire as reasons for choice, not causes of choice [Lowe] |
6663 | People's actions are explained either by their motives, or their reasons, or the causes [Lowe] |