208 ideas
| 16440 | I don't think Lewis's cost-benefit reflective equilibrium approach offers enough guidance [Stalnaker] |
| 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] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 10206 | Modal operators are usually treated as quantifiers [Shapiro] |
| 16468 | Non-S5 can talk of contingent or necessary necessities [Stalnaker] |
| 18823 | To say there could have been people who don't exist, but deny those possible things, rejects Barcan [Stalnaker, by Rumfitt] |
| 13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
| 16449 | In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker] |
| 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] |
| 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] |
| 12766 | Logical space is abstracted from the actual world [Stalnaker] |
| 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] |
| 16464 | We regiment to get semantic structure, for evaluating arguments, and understanding complexities [Stalnaker] |
| 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] |
| 16465 | In 'S was F or some other than S was F', the disjuncts need S, but the whole disjunction doesn't [Stalnaker] |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| 16405 | To understand a name (unlike a description) picking the thing out is sufficient? [Stalnaker] |
| 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] |
| 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] |
| 10240 | Model theory deals with relations, reference and extensions [Shapiro] |
| 10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
| 13644 | Semantics for models uses set-theory [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] |
| 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] |
| 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] |
| 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] |
| 16434 | Some say what exists must do so, and nothing else could possible exist [Stalnaker] |
| 16439 | A nominalist view says existence is having spatio-temporal location [Stalnaker] |
| 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
| 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] |
| 16443 | Properties are modal, involving possible situations where they are exemplified [Stalnaker] |
| 16471 | I accept a hierarchy of properties of properties of properties [Stalnaker] |
| 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] |
| 16452 | Dispositions have modal properties, of which properties things would have counterfactually [Stalnaker] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 14617 | Predicates can't apply to what doesn't exist [Stalnaker] |
| 10275 | A blurry border is still a border [Shapiro] |
| 12764 | For the bare particular view, properties must be features, not just groups of objects [Stalnaker] |
| 16407 | Possible worlds allow separating all the properties, without hitting a bare particular [Stalnaker] |
| 12761 | An essential property is one had in all the possible worlds where a thing exists [Stalnaker] |
| 16467 | 'Socrates is essentially human' seems to say nothing could be Socrates if it was not human [Stalnaker] |
| 12763 | Necessarily self-identical, or being what it is, or its world-indexed properties, aren't essential [Stalnaker] |
| 12762 | Bare particular anti-essentialism makes no sense within modal logic semantics [Stalnaker] |
| 16453 | The bundle theory makes the identity of indiscernibles a necessity, since the thing is the properties [Stalnaker] |
| 16466 | Strong necessity is always true; weak necessity is cannot be false [Stalnaker] |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| 14286 | In nearby worlds where A is true, 'if A,B' is true or false if B is true or false [Stalnaker] |
| 10994 | Conditionals are true if minimal revision of the antecedent verifies the consequent [Stalnaker, by Read] |
| 16438 | Necessity and possibility are fundamental, and there can be no reductive analysis of them [Stalnaker] |
| 16422 | The necessity of a proposition concerns reality, not our words or concepts [Stalnaker] |
| 16423 | Conceptual possibilities are metaphysical possibilities we can conceive of [Stalnaker] |
| 16436 | Modal concepts are central to the actual world, and shouldn't need extravagant metaphysics [Stalnaker] |
| 16421 | Critics say there are just an a priori necessary part, and an a posteriori contingent part [Stalnaker] |
| 16429 | A 'centred' world is an ordered triple of world, individual and time [Stalnaker] |
| 16397 | If it might be true, it might be true in particular ways, and possible worlds describe such ways [Stalnaker] |
| 16399 | Possible worlds are ontologically neutral, but a commitment to possibilities remains [Stalnaker] |
| 16398 | Possible worlds allow discussion of modality without controversial modal auxiliaries [Stalnaker] |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| 16433 | Given actualism, how can there be possible individuals, other than the actual ones? [Stalnaker] |
| 14285 | A possible world is the ontological analogue of hypothetical beliefs [Stalnaker] |
| 15793 | We can take 'ways things might have been' as irreducible elements in our ontology [Stalnaker, by Lycan] |
| 16396 | Kripke's possible worlds are methodological, not metaphysical [Stalnaker] |
| 16437 | Possible worlds are properties [Stalnaker] |
| 16444 | Possible worlds don't reduce modality, they regiment it to reveal its structure [Stalnaker] |
| 16445 | I think of worlds as cells (rather than points) in logical space [Stalnaker] |
| 12765 | Why imagine that Babe Ruth might be a billiard ball; nothing useful could be said about the ball [Stalnaker] |
| 16408 | Rigid designation seems to presuppose that differing worlds contain the same individuals [Stalnaker] |
| 16409 | Unlike Lewis, I defend an actualist version of counterpart theory [Stalnaker] |
| 16411 | If possible worlds really differ, I can't be in more than one at a time [Stalnaker] |
| 16412 | If counterparts exist strictly in one world only, this seems to be extreme invariant essentialism [Stalnaker] |
| 16454 | Modal properties depend on the choice of a counterpart, which is unconstrained by metaphysics [Stalnaker] |
| 16450 | Anti-haecceitism says there is no more to an individual than meeting some qualitative conditions [Stalnaker] |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 16428 | Meanings aren't in the head, but that is because they are abstract [Stalnaker] |
| 16474 | How can we know what we are thinking, if content depends on something we don't know? [Stalnaker] |
| 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] |
| 16406 | If you don't know what you say you can't mean it; what people say usually fits what they mean [Stalnaker] |
| 16404 | In the use of a name, many individuals are causally involved, but they aren't all the referent [Stalnaker] |
| 16432 | One view says the causal story is built into the description that is the name's content [Stalnaker] |
| 16403 | 'Descriptive' semantics gives a system for a language; 'foundational' semantics give underlying facts [Stalnaker] |
| 16461 | We still lack an agreed semantics for quantifiers in natural language [Stalnaker] |
| 16401 | To understand an utterance, you must understand what the world would be like if it is true [Stalnaker] |
| 16410 | Extensional semantics has individuals and sets; modal semantics has intensions, functions of world to extension [Stalnaker] |
| 16448 | Possible world semantics may not reduce modality, but it can explain it [Stalnaker] |
| 16430 | Two-D says that a posteriori is primary and contingent, and the necessity is the secondary intension [Stalnaker] |
| 16431 | In one view, the secondary intension is metasemantic, about how the thinker relates to the content [Stalnaker] |
| 16442 | I take propositions to be truth conditions [Stalnaker] |
| 16447 | A theory of propositions at least needs primitive properties of consistency and of truth [Stalnaker] |
| 14616 | A 'Russellian proposition' is an ordered sequence of individual, properties and relations [Stalnaker] |
| 16446 | Propositions presumably don't exist if the things they refer to don't exist [Stalnaker] |
| 18052 | An assertion aims to add to the content of a context [Stalnaker, by Magidor] |
| 14718 | An assertion is an attempt to rule out certain possibilities, narrowing things down for good planning [Stalnaker, by Schroeter] |
| 5064 | Rights are moral significance, or liberty, or right not to be restrained, or entitlement [Mawson] |