242 ideas
| 9593 | Progress in philosophy is incremental, not an immature seeking after drama [Williamson] |
| 9184 | We can't presume that all interesting concepts can be analysed [Williamson] |
| 6859 | Analytic philosophy has much higher standards of thinking than continental philosophy [Williamson] |
| 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] |
| 21616 | Truth and falsity apply to suppositions as well as to assertions [Williamson] |
| 21623 | True and false are not symmetrical; false is more complex, involving negation [Williamson] |
| 15134 | The truthmaker principle requires some specific named thing to make the difference [Williamson] |
| 15141 | Truthmaker is incompatible with modal semantics of varying domains [Williamson] |
| 15140 | The converse Barcan formula will not allow contingent truths to have truthmakers [Williamson] |
| 9594 | Correspondence to the facts is a bad account of analytic truth [Williamson] |
| 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] |
| 15131 | If metaphysical possibility is not a contingent matter, then S5 seems to suit it best [Williamson] |
| 14626 | In S5 matters of possibility and necessity are non-contingent [Williamson] |
| 15135 | If the domain of propositional quantification is constant, the Barcan formulas hold [Williamson] |
| 15130 | If a property is possible, there is something which can have it [Williamson] |
| 15139 | Converse Barcan: could something fail to meet a condition, if everything meets that condition? [Williamson] |
| 21602 | Many-valued logics don't solve vagueness; its presence at the meta-level is ignored [Williamson] |
| 6862 | Fuzzy logic uses a continuum of truth, but it implies contradictions [Williamson] |
| 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] |
| 6858 | Formal logic struck me as exactly the language I wanted to think in [Williamson] |
| 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] |
| 21611 | Formal semantics defines validity as truth preserved in every model [Williamson] |
| 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] |
| 21606 | 'Bivalence' is the meta-linguistic principle that 'A' in the object language is true or false [Williamson] |
| 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] |
| 21605 | Excluded Middle is 'A or not A' in the object language [Williamson] |
| 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] |
| 18492 | Not all quantification is either objectual or substitutional [Williamson] |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| 15136 | Substitutional quantification is metaphysical neutral, and equivalent to a disjunction of instances [Williamson] |
| 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] |
| 15138 | Not all quantification is objectual or substitutional [Williamson] |
| 21612 | Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B' [Williamson] |
| 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] |
| 10240 | Model theory deals with relations, reference and extensions [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] |
| 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] |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| 10234 | Any theory with an infinite model has a model of every infinite cardinality [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] |
| 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] |
| 21599 | A sorites stops when it collides with an opposite sorites [Williamson] |
| 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] |
| 9183 | Platonism claims that some true assertions have singular terms denoting abstractions, so abstractions exist [Williamson] |
| 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] |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| 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] |
| 9601 | The realist/anti-realist debate is notoriously obscure and fruitless [Williamson] |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| 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] |
| 15137 | If 'fact' is a noun, can we name the fact that dogs bark 'Mary'? [Williamson] |
| 21589 | When bivalence is rejected because of vagueness, we lose classical logic [Williamson] |
| 21596 | Vagueness undermines the stable references needed by logic [Williamson] |
| 21601 | A vague term can refer to very precise elements [Williamson] |
| 21629 | Equally fuzzy objects can be identical, so fuzziness doesn't entail vagueness [Williamson] |
| 9599 | There cannot be vague objects, so there may be no such thing as a mountain [Williamson] |
| 21591 | Vagueness is epistemic. Statements are true or false, but we often don't know which [Williamson] |
| 21619 | If a heap has a real boundary, omniscient speakers would agree where it is [Williamson] |
| 21620 | The epistemic view says that the essence of vagueness is ignorance [Williamson] |
| 21622 | If there is a true borderline of which we are ignorant, this drives a wedge between meaning and use [Williamson] |
| 9120 | Vagueness in a concept is its indiscriminability from other possible concepts [Williamson] |
| 6863 | Close to conceptual boundaries judgement is too unreliable to give knowledge [Williamson] |
| 21625 | The vagueness of 'heap' can remain even when the context is fixed [Williamson] |
| 21614 | The 'nihilist' view of vagueness says that 'heap' is not a legitimate concept [Williamson] |
| 21617 | We can say propositions are bivalent, but vague utterances don't express a proposition [Williamson] |
| 21618 | If the vague 'TW is thin' says nothing, what does 'TW is thin if his perfect twin is thin' say? [Williamson] |
| 21590 | Asking when someone is 'clearly' old is higher-order vagueness [Williamson] |
| 21592 | Supervaluation keeps classical logic, but changes the truth in classical semantics [Williamson] |
| 21603 | You can't give a precise description of a language which is intrinsically vague [Williamson] |
| 21604 | Supervaluation assigns truth when all the facts are respected [Williamson] |
| 21607 | Supervaluation has excluded middle but not bivalence; 'A or not-A' is true, even when A is undecided [Williamson] |
| 21608 | Truth-functionality for compound statements fails in supervaluation [Williamson] |
| 21609 | Supervaluationism defines 'supertruth', but neglects it when defining 'valid' [Williamson] |
| 21610 | Supervaluation adds a 'definitely' operator to classical logic [Williamson] |
| 21613 | Supervaluationism cannot eliminate higher-order vagueness [Williamson] |
| 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] |
| 21633 | Nominalists suspect that properties etc are our projections, and could have been different [Williamson] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 6861 | What sort of logic is needed for vague concepts, and what sort of concept of truth? [Williamson] |
| 9602 | Common sense and classical logic are often simultaneously abandoned in debates on vagueness [Williamson] |
| 21630 | If fuzzy edges are fine, then why not fuzzy temporal, modal or mereological boundaries? [Williamson] |
| 21632 | A river is not just event; it needs actual and counterfactual boundaries [Williamson] |
| 14625 | Necessity is counterfactually implied by its negation; possibility does not counterfactually imply its negation [Williamson] |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| 14623 | Strict conditionals imply counterfactual conditionals: □(A⊃B)⊃(A□→B) [Williamson] |
| 14624 | Counterfactual conditionals transmit possibility: (A□→B)⊃(◊A⊃◊B) [Williamson] |
| 14531 | Rather than define counterfactuals using necessity, maybe necessity is a special case of counterfactuals [Williamson, by Hale/Hoffmann,A] |
| 21621 | We can't infer metaphysical necessities to be a priori knowable - or indeed knowable in any way [Williamson] |
| 9598 | Modal thinking isn't a special intuition; it is part of ordinary counterfactual thinking [Williamson] |
| 16536 | Williamson can't base metaphysical necessity on the psychology of causal counterfactuals [Lowe on Williamson] |
| 9596 | We scorn imagination as a test of possibility, forgetting its role in counterfactuals [Williamson] |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| 15142 | Our ability to count objects across possibilities favours the Barcan formulas [Williamson] |
| 18925 | If talking donkeys are possible, something exists which could be a talking donkey [Williamson, by Cameron] |
| 21627 | We have inexact knowledge when we include margins of error [Williamson] |
| 4760 | Belief aims at knowledge (rather than truth), and mere believing is a kind of botched knowing [Williamson] |
| 19527 | We don't acquire evidence and then derive some knowledge, because evidence IS knowledge [Williamson] |
| 19512 | Don't analyse knowledge; use knowledge to analyse other concepts in epistemology [Williamson, by DeRose] |
| 19528 | Knowledge is prior to believing, just as doing is prior to trying to do [Williamson] |
| 19529 | Belief explains justification, and knowledge explains belief, so knowledge explains justification [Williamson] |
| 19530 | A neutral state of experience, between error and knowledge, is not basic; the successful state is basic [Williamson] |
| 19531 | Internalism about mind is an obsolete view, and knowledge-first epistemology develops externalism [Williamson] |
| 19536 | Knowledge-first says your total evidence IS your knowledge [Williamson] |
| 19526 | Surely I am acquainted with physical objects, not with appearances? [Williamson] |
| 9597 | There are 'armchair' truths which are not a priori, because experience was involved [Williamson] |
| 6860 | How can one discriminate yellow from red, but not the colours in between? [Williamson] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 9592 | Intuition is neither powerful nor vacuous, but reveals linguistic or conceptual competence [Williamson] |
| 20181 | When analytic philosophers run out of arguments, they present intuitions as their evidence [Williamson] |
| 21626 | Knowing you know (KK) is usually denied if the knowledge concept is missing, or not considered [Williamson] |
| 14628 | Imagination is important, in evaluating possibility and necessity, via counterfactuals [Williamson] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 21631 | To know, believe, hope or fear, one must grasp the thought, but not when you fail to do them [Williamson] |
| 21600 | 'Blue' is not a family resemblance, because all the blues resemble in some respect [Williamson] |
| 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] |
| 9595 | You might know that the word 'gob' meant 'mouth', but not be competent to use it [Williamson] |
| 21615 | References to the 'greatest prime number' have no reference, but are meaningful [Williamson] |
| 18038 | The 't' and 'f' of formal semantics has no philosophical interest, and may not refer to true and false [Williamson] |
| 19534 | How does inferentialism distinguish the patterns of inference that are essential to meaning? [Williamson] |
| 19535 | Internalist inferentialism has trouble explaining how meaning and reference relate [Williamson] |
| 19533 | Inferentialist semantics relies on internal inference relations, not on external references [Williamson] |
| 19532 | Truth-conditional referential semantics is externalist, referring to worldly items [Williamson] |
| 21624 | It is known that there is a cognitive loss in identifying propositions with possible worlds [Williamson] |
| 19216 | Propositions (such as 'that dog is barking') only exist if their items exist [Williamson] |
| 9600 | If languages are intertranslatable, and cognition is innate, then cultures are all similar [Williamson] |
| 15133 | A thing can't be the only necessary existent, because its singleton set would be as well [Williamson] |