190 ideas
21634 | Metaphysics is (supposedly) first the ontology, then in general what things are like [Hofweber] |
16415 | Esoteric metaphysics aims to be top science, investigating ultimate reality [Hofweber] |
16413 | Science has discovered properties of things, so there are properties - so who needs metaphysics? [Hofweber] |
21666 | 'Fundamentality' is either a superficial idea, or much too obscure [Hofweber] |
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] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
21640 | 'It's true that Fido is a dog' conjures up a contrast class, of 'it's false' or 'it's unlikely' [Hofweber] |
17990 | Instances of minimal truth miss out propositions inexpressible in current English [Hofweber] |
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] |
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] |
10207 | Anti-realists reject set theory [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] |
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] |
21657 | Since properties can have properties, some theorists rank them in 'types' [Hofweber] |
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] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [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] |
21653 | Maybe not even names are referential, but are just by used by speakers to refer [Hofweber] |
10001 | An adjective contributes semantically to a noun phrase [Hofweber] |
21636 | 'Singular terms' are not found in modern linguistics, and are not the same as noun phrases [Hofweber] |
21637 | If two processes are said to be identical, that doesn't make their terms refer to entities [Hofweber] |
16416 | The quantifier in logic is not like the ordinary English one (which has empty names, non-denoting terms etc) [Hofweber] |
21643 | The inferential quantifier focuses on truth; the domain quantifier focuses on reality [Hofweber] |
10007 | Quantifiers for domains and for inference come apart if there are no entities [Hofweber] |
13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
17988 | Quantification can't all be substitutional; some reference is obviously to objects [Hofweber] |
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] |
10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
13644 | Semantics for models uses set-theory [Shapiro] |
10240 | Model theory deals with relations, reference and extensions [Shapiro] |
13670 | Categoricity can't be reached in a first-order language [Shapiro] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [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] |
10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [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] |
10234 | Any theory with an infinite model has a model of every infinite cardinality [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] |
10002 | '2 + 2 = 4' can be read as either singular or plural [Hofweber] |
21644 | Numbers are used as singular terms, as adjectives, and as symbols [Hofweber] |
21646 | The Amazonian Piraha language is said to have no number words [Hofweber] |
9998 | What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber] |
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] |
21665 | The fundamental theorem of arithmetic is that all numbers are composed uniquely of primes [Hofweber] |
21649 | How can words be used for counting if they are objects? [Hofweber] |
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] |
10003 | Why is arithmetic hard to learn, but then becomes easy? [Hofweber] |
10008 | Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber] |
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] |
10005 | Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber] |
10000 | We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber] |
21647 | Logicism makes sense of our ability to know arithmetic just by thought [Hofweber] |
13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
21648 | Neo-Fregeans are dazzled by a technical result, and ignore practicalities [Hofweber] |
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] |
10006 | First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber] |
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] |
21664 | Supervenience offers little explanation for things which necessarily go together [Hofweber] |
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] |
21660 | Reality can be seen as the totality of facts, or as the totality of things [Hofweber] |
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] |
21661 | There are probably ineffable facts, systematically hidden from us [Hofweber] |
17989 | Since properties have properties, there can be a typed or a type-free theory of them [Hofweber] |
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] |
21652 | Our perceptual beliefs are about ordinary objects, not about simples arranged chair-wise [Hofweber] |
10275 | A blurry border is still a border [Shapiro] |
10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
21663 | Counterfactuals are essential for planning, and learning from mistakes [Hofweber] |
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] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
10004 | Our minds are at their best when reasoning about objects [Hofweber] |
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] |
21654 | The "Fido"-Fido theory of meaning says every expression in a language has a referent [Hofweber] |
21641 | Inferential role semantics is an alternative to semantics that connects to the world [Hofweber] |
21638 | Syntactic form concerns the focus of the sentence, as well as the truth-conditions [Hofweber] |
21658 | Properties can be expressed in a language despite the absence of a single word for them [Hofweber] |
21659 | 'Being taller than this' is a predicate which can express many different properties [Hofweber] |
21655 | Compositonality is a way to build up the truth-conditions of a sentence [Hofweber] |
21656 | Proposition have no content, because they are content [Hofweber] |
21635 | Without propositions there can be no beliefs or desires [Hofweber] |
21662 | Do there exist thoughts which we are incapable of thinking? [Hofweber] |
21645 | 'Semantic type coercion' is selecting the reading of a word to make the best sense [Hofweber] |
21639 | 'Background deletion' is appropriately omitting background from an answer [Hofweber] |
17991 | Holism says language can't be translated; the expressibility hypothesis says everything can [Hofweber] |
1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |