191 ideas
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| 10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
| 15186 | In the tenseless view, all times are equally real, so statements of the future have truth-values [Le Poidevin] |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 10206 | Modal operators are usually treated as quantifiers [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] |
| 10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
| 10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
| 10301 | The axiom of choice is controversial, but it could be replaced [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] |
| 10207 | Anti-realists reject set theory [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] |
| 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [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] |
| 10298 | Some say that second-order logic is mathematics, not logic [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] |
| 10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
| 10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
| 10259 | The two standard explanations of consequence are semantic (in models) and deductive [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] |
| 10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
| 10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
| 10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
| 10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| 10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
| 10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
| 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
| 10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
| 10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
| 10240 | Model theory deals with relations, reference and extensions [Shapiro] |
| 13644 | Semantics for models uses set-theory [Shapiro] |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [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] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [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] |
| 10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
| 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
| 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [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] |
| 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [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] |
| 10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
| 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [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] |
| 10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
| 18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
| 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
| 10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 10256 | For intuitionists, proof is inherently informal [Shapiro] |
| 13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
| 10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
| 10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
| 10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
| 10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
| 10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
| 10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
| 10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
| 10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
| 10276 | Mathematical structures are defined by axioms, or in set theory [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] |
| 10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
| 10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
| 10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
| 10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
| 10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
| 8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
| 10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
| 10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
| 10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
| 10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
| 10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
| 10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
| 10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [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] |
| 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] |
| 10254 | Can the ideal constructor also destroy objects? [Shapiro] |
| 10255 | Presumably nothing can block a possible dynamic operation? [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] |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
| 22919 | A thing which makes no difference seems unlikely to exist [Le Poidevin] |
| 10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
| 10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
| 10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
| 10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| 15207 | We want illuminating theories, rather than coherent theories [Le Poidevin] |
| 22926 | In addition to causal explanations, they can also be inferential, or definitional, or purposive [Le Poidevin] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| 9626 | A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro] |
| 10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
| 9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
| 10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
| 22932 | We don't just describe a time as 'now' from a private viewpoint, but as a fact about the world [Le Poidevin] |
| 6866 | It is disturbing if we become unreal when we die, but if time is unreal, then we remain real after death [Le Poidevin] |
| 15190 | Evil can't be an illusion, because then the illusion that there is evil would be evil [Le Poidevin] |
| 6867 | Existentialism focuses on freedom and self-making, and insertion into the world [Le Poidevin] |
| 22927 | The logical properties of causation are asymmetry, transitivity and irreflexivity [Le Poidevin] |
| 22922 | We can identify unoccupied points in space, so they must exist [Le Poidevin] |
| 22924 | If spatial points exist, then they must be stationary, by definition [Le Poidevin] |
| 22923 | Absolute space explains actual and potential positions, and geometrical truths [Le Poidevin] |
| 22928 | For relationists moving an object beyond the edge of space creates new space [Le Poidevin] |
| 22931 | We distinguish time from space, because it passes, and it has a unique present moment [Le Poidevin] |
| 22917 | Since nothing occurs in a temporal vacuum, there is no way to measure its length [Le Poidevin] |
| 22921 | Temporal vacuums would be unexperienced, unmeasured, and unending [Le Poidevin] |
| 15195 | If the future is not real, we don't seem to have any obligation to future individuals [Le Poidevin] |
| 15188 | If things don't persist through time, then change makes no sense [Le Poidevin] |
| 22934 | Time can't speed up or slow down, so it doesn't seem to be a 'process' [Le Poidevin] |
| 15191 | At the very least, minds themselves seem to be tensed [Le Poidevin] |
| 15197 | Fiction seems to lack a tensed perspective, and offers an example of tenseless language [Le Poidevin] |
| 15206 | It is the view of the future that really decides between tensed and tenseless views of time [Le Poidevin] |
| 15198 | In the B-series, time-positions are unchanging; in the A-series they change (from future to present to past) [Le Poidevin] |
| 15189 | Things which have ceased change their A-series position; things that persist change their B-series position [Le Poidevin] |
| 6865 | A-theory says past, present, future and flow exist; B-theory says this just reports our perspective [Le Poidevin] |
| 15187 | It is claimed that the tense view entails the unreality of both future and past [Le Poidevin] |
| 15205 | Tensed theorists typically try to reduce the tenseless to the tensed [Le Poidevin] |
| 15192 | We share a common now, but not a common here [Le Poidevin] |
| 22939 | The B-series doesn't seem to allow change [Le Poidevin] |
| 22940 | If the B-universe is eternal, why am I trapped in a changing moment of it? [Le Poidevin] |
| 15193 | The new tenseless theory offers indexical truth-conditions, instead of a reductive analysis [Le Poidevin] |
| 22938 | To say that the past causes the present needs them both to be equally real [Le Poidevin] |
| 22947 | An ordered series can be undirected, but time favours moving from earlier to later [Le Poidevin] |
| 22952 | If time's arrow is causal, how can there be non-simultaneous events that are causally unconnected? [Le Poidevin] |
| 22953 | Time's arrow is not causal if there is no temporal gap between cause and effect [Le Poidevin] |
| 22951 | If time's arrow is psychological then different minds can impose different orders on events [Le Poidevin] |
| 22948 | There are Thermodynamic, Psychological and Causal arrows of time [Le Poidevin] |
| 22949 | Presumably if time's arrow is thermodynamic then time ends when entropy is complete [Le Poidevin] |
| 22950 | If time is thermodynamic then entropy is necessary - but the theory says it is probable [Le Poidevin] |
| 22943 | Instantaneous motion is an intrinsic disposition to be elsewhere [Le Poidevin] |
| 22945 | The dynamic view of motion says it is primitive, and not reducible to objects, properties and times [Le Poidevin] |
| 22937 | If the present could have diverse pasts, then past truths can't have present truthmakers [Le Poidevin] |
| 22925 | The present is the past/future boundary, so the first moment of time was not present [Le Poidevin] |
| 22944 | The primitive parts of time are intervals, not instants [Le Poidevin] |
| 22942 | If time is infinitely divisible, then the present must be infinitely short [Le Poidevin] |
| 22946 | The multiverse is distinct time-series, as well as spaces [Le Poidevin] |
| 15196 | God being inside or outside of time both raise a group of difficult problems [Le Poidevin] |
| 22941 | How could a timeless God know what time it is? So could God be both timeless and omniscient? [Le Poidevin] |