91 ideas
16227 | Philosophers are good at denying the obvious [Hawley] |
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] |
16216 | Part of the sense of a proper name is a criterion of the thing's identity [Hawley] |
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] |
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] |
13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [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] |
16211 | A homogeneous rotating disc should be undetectable according to Humean supervenience [Hawley] |
16219 | Non-linguistic things cannot be indeterminate, because they don't have truth-values at all [Hawley] |
16223 | Maybe for the world to be vague, it must be vague in its foundations? [Hawley] |
16226 | Epistemic vagueness seems right in the case of persons [Hawley] |
16208 | Supervaluation refers to one vaguely specified thing, through satisfaction by everything in some range [Hawley] |
16221 | Supervaluationism takes what the truth-value would have been if indecision was resolved [Hawley] |
16230 | Maybe the only properties are basic ones like charge, mass and spin [Hawley] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
16232 | An object is 'natural' if its stages are linked by certain non-supervenient relations [Hawley] |
16200 | Are sortals spatially maximal - so no cat part is allowed to be a cat? [Hawley] |
16237 | The modal features of statue and lump are disputed; when does it stop being that statue? [Hawley] |
16238 | Perdurantists can adopt counterpart theory, to explain modal differences of identical part-sums [Hawley] |
16220 | Vagueness is either in our knowledge, in our talk, or in reality [Hawley] |
16222 | Indeterminacy in objects and in properties are not distinct cases [Hawley] |
16228 | The constitution theory is endurantism plus more than one object in a place [Hawley] |
16229 | Constitution theory needs sortal properties like 'being a sweater' to distinguish it from its thread [Hawley] |
14492 | If the constitution view says thread and sweater are two things, why do we talk of one thing? [Hawley] |
16193 | 'Adverbialism' explains change by saying an object has-at-some-time a given property [Hawley] |
16195 | Presentism solves the change problem: the green banana ceases, so can't 'relate' to the yellow one [Hawley] |
16202 | The problem of change arises if there must be 'identity' of a thing over time [Hawley] |
16192 | Endurance theory can relate properties to times, or timed instantiations to properties [Hawley] |
16196 | Endurance is a sophisticated theory, covering properties, instantiation and time [Hawley] |
16197 | How does perdurance theory explain our concern for our own future selves? [Hawley] |
16191 | Perdurance needs an atemporal perspective, to say that the object 'has' different temporal parts [Hawley] |
16199 | If an object is the sum of all of its temporal parts, its mass is staggeringly large! [Hawley] |
16201 | Perdurance says things are sums of stages; Stage Theory says each stage is the thing [Hawley] |
16240 | If a life is essentially the sum of its temporal parts, it couldn't be shorter or longer than it was? [Hawley] |
16203 | Stage Theory seems to miss out the link between stages of the same object [Hawley] |
16204 | Stage Theory says every stage is a distinct object, which gives too many objects [Hawley] |
16212 | An isolated stage can't be a banana (which involves suitable relations to other stages) [Hawley] |
16213 | Stages of one thing are related by extrinsic counterfactual and causal relations [Hawley] |
16206 | Stages must be as fine-grained in length as change itself, so any change is a new stage [Hawley] |
16205 | The stages of Stage Theory seem too thin to populate the world, or to be referred to [Hawley] |
16225 | If two things might be identical, there can't be something true of one and false of the other [Hawley] |
16239 | To decide whether something is a counterpart, we need to specify a relevant sortal concept [Hawley] |
16218 | On any theory of self, it is hard to explain why we should care about our future selves [Hawley] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
16215 | Causation is nothing more than the counterfactuals it grounds? [Hawley] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
16207 | Time could be discrete (like integers) or dense (rationals) or continuous (reals) [Hawley] |