102 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] |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [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] |
| 13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
| 13627 | There is no 'correct' logic for natural languages [Shapiro] |
| 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [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] |
| 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] |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| 13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
| 4098 | The theory of descriptions supports internalism, since they are thinkable when the object is non-existent [Crane] |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| 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] |
| 4077 | Aesthetic properties of thing supervene on their physical properties [Crane] |
| 4078 | Constitution (as in a statue constituted by its marble) is supervenience without identity [Crane] |
| 4082 | The distinction between 'resultant' properties (weight) and 'emergent' properties is a bit vague [Crane] |
| 4083 | If mental properties are emergent they add a new type of causation, and physics is not complete [Crane] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 4079 | Properties are causes [Crane] |
| 4068 | Traditional substance is separate from properties and capable of independent existence [Crane] |
| 3756 | Perception, introspection, testimony, memory, reason, and inference can give us knowledge [Bernecker/Dretske] |
| 4096 | Maybe beliefs don't need to be conscious, if you are not conscious of the beliefs guiding your actions [Crane] |
| 4097 | Maybe there are two kinds of belief - 'de re' beliefs and 'de dicto' beliefs [Crane] |
| 4093 | Many cases of knowing how can be expressed in propositional terms (like how to get somewhere) [Crane] |
| 4108 | Phenol-thio-urea tastes bitter to three-quarters of people, but to the rest it is tasteless, so which is it? [Crane] |
| 4104 | One can taste that the wine is sour, and one can also taste the sourness of the wine [Crane] |
| 4105 | The traditional supports for the sense datum theory were seeing double and specks before one's eyes [Crane] |
| 4101 | If we smell something we are aware of the smell separately, but we don't perceive a 'look' when we see [Crane] |
| 4102 | The problems of perception disappear if it is a relation to an intentional state, not to an object or sense datum [Crane] |
| 4109 | If perception is much richer than our powers of description, this suggests that it is non-conceptual [Crane] |
| 3757 | Causal theory says true perceptions must be caused by the object perceived [Bernecker/Dretske] |
| 4103 | The adverbial theory of perceptions says it is the experiences which have properties, not the objects [Crane] |
| 3759 | You can acquire new knowledge by exploring memories [Bernecker/Dretske] |
| 3752 | Justification can be of the belief, or of the person holding the belief [Bernecker/Dretske] |
| 3753 | Foundationalism aims to avoid an infinite regress [Bernecker/Dretske] |
| 3754 | Infallible sensations can't be foundations if they are non-epistemic [Bernecker/Dretske] |
| 4065 | Is knowledge just a state of mind, or does it also involve the existence of external things? [Crane] |
| 3755 | Justification is normative, so it can't be reduced to cognitive psychology [Bernecker/Dretske] |
| 3761 | Modern arguments against the sceptic are epistemological and semantic externalism, and the focus on relevance [Bernecker/Dretske] |
| 3760 | Predictions are bound to be arbitrary if they depend on the language used [Bernecker/Dretske] |
| 4092 | The core of the consciousness problem is the case of Mary, zombies, and the Hard Question [Crane] |
| 4087 | Intentionalism does not require that all mental states be propositional attitudes [Crane] |
| 4095 | Object-directed attitudes like love are just as significant as propositional attitudes [Crane] |
| 4106 | If someone removes their glasses the content of experience remains, but the quality changes [Crane] |
| 4089 | Pains have a region of the body as their intentional content, not some pain object [Crane] |
| 4090 | Weak intentionalism says qualia are extra properties; strong intentionalism says they are intentional [Crane] |
| 4107 | With inverted qualia a person's experiences would change, but their beliefs remain the same [Crane] |
| 4069 | Descartes did not think of minds as made of a substance, because they are not divisible [Crane] |
| 4074 | Functionalism defines mental states by their causal properties, which rules out epiphenomenalism [Crane] |
| 4091 | The problems of misrepresentation and error have dogged physicalist reductions of intentionality [Crane] |
| 4070 | Properties dualism says mental properties are distinct from physical, despite a single underlying substance [Crane] |
| 4084 | Non-reductive physicalism seeks an explanation of supervenience, but emergentists accept it as basic [Crane] |
| 4080 | If mental supervenes on the physical, then every physical cause will be accompanied by a mental one [Crane] |
| 4075 | Identity theory is either of particular events, or of properties, depending on your theory of causation [Crane] |
| 4085 | Physicalism may be the source of the mind-body problem, rather than its solution [Crane] |
| 4073 | Overdetermination occurs if two events cause an effect, when each would have caused it alone [Crane] |
| 4072 | The completeness of physics must be an essential component of any physicalist view of mind [Crane] |
| 4094 | Experience teaches us propositions, because we can reason about our phenomenal experience [Crane] |
| 4100 | The Twin Earth argument depends on reference being determined by content, which may be false. [Crane] |
| 3758 | Semantic externalism ties content to the world, reducing error [Bernecker/Dretske] |
| 4067 | Broad content entails the existence of the object of the thought [Crane] |
| 4063 | In intensional contexts, truth depends on how extensions are conceived. [Crane] |
| 4071 | Causation can be seen in counterfactual terms, or as increased probability, or as energy flow [Crane] |
| 4076 | Causes are properties, not events, because properties are what make a difference in a situation [Crane] |
| 4066 | It seems that 'exists' could sometimes be a predicate [Crane] |