100 ideas
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
| 13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
| 13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
| 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
| 13627 | There is no 'correct' logic for natural languages [Shapiro] |
| 13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
| 13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
| 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
| 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
| 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
| 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
| 13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
| 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
| 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
| 13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| 6653 | Syntactical methods of proof need only structure, where semantic methods (truth-tables) need truth [Lowe] |
| 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
| 13644 | Semantics for models uses set-theory [Shapiro] |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
| 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
| 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
| 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| 13628 | We can live well without completeness in logic [Shapiro] |
| 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
| 13646 | Compactness is derived from soundness and completeness [Shapiro] |
| 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
| 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
| 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
| 13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
| 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
| 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
| 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [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] |
| 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] |
| 13195 | To explain a house we must describe its use, as well as its parts [Leibniz] |
| 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] |
| 13193 | Active force is not just potential for action, since it involves a real effort or striving [Leibniz] |
| 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] |
| 13194 | God's laws would be meaningless without internal powers for following them [Leibniz] |
| 13196 | All qualities of bodies reduce to forces [Leibniz] |
| 13192 | Power is passive force, which is mass, and active force, which is entelechy or form [Leibniz] |