112 ideas
16325 | Analysis rests on natural language, but its ideal is a framework which revises language [Halbach] |
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
16292 | An explicit definition enables the elimination of what is defined [Halbach] |
10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
16307 | Don't trust analogies; they are no more than a guideline [Halbach] |
16330 | Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth [Halbach] |
16339 | Truth axioms prove objects exist, so truth doesn't seem to be a logical notion [Halbach] |
16324 | Any definition of truth requires a metalanguage [Halbach] |
16293 | Traditional definitions of truth often make it more obscure, rather than less [Halbach] |
16301 | If people have big doubts about truth, a definition might give it more credibility [Halbach] |
16297 | Semantic theories avoid Tarski's Theorem by sticking to a sublanguage [Halbach] |
16337 | Disquotational truth theories are short of deductive power [Halbach] |
16322 | CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA [Halbach] |
16294 | Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage [Halbach] |
16326 | The main semantic theories of truth are Kripke's theory, and revisions semantics [Halbach] |
16311 | To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction' [Halbach] |
16318 | Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach] |
16299 | Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory [Halbach] |
16340 | Truth axioms need a base theory, because that is where truth issues arise [Halbach] |
16305 | We know a complete axiomatisation of truth is not feasible [Halbach] |
16313 | A theory is 'conservative' if it adds no new theorems to its base theory [Halbach, by PG] |
16315 | The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals [Halbach] |
16314 | Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free' [Halbach] |
16327 | Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth [Halbach] |
16332 | The KF theory is useful, but it is not a theory containing its own truth predicate [Halbach] |
16331 | The KF is much stronger deductively than FS, which relies on classical truth [Halbach] |
16329 | Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts [Halbach] |
16319 | Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism [Halbach] |
16320 | Some say deflationism is axioms which are conservative over the base theory [Halbach] |
16338 | Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge [Halbach] |
16317 | The main problem for deflationists is they can express generalisations, but not prove them [Halbach] |
16316 | Deflationists say truth is just for expressing infinite conjunctions or generalisations [Halbach] |
10206 | Modal operators are usually treated as quantifiers [Shapiro] |
16335 | In Strong Kleene logic a disjunction just needs one disjunct to be true [Halbach] |
16334 | In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value [Halbach] |
16309 | Every attempt at formal rigour uses some set theory [Halbach] |
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] |
10207 | Anti-realists reject set theory [Shapiro] |
16333 | The underestimated costs of giving up classical logic are found in mathematical reasoning [Halbach] |
10259 | The two standard explanations of consequence are semantic (in models) and deductive [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] |
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] |
16310 | A theory is some formulae and all of their consequences [Halbach] |
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] |
10240 | Model theory deals with relations, reference and extensions [Shapiro] |
10239 | The central notion of model theory is the relation of 'satisfaction' [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] |
10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |
16341 | Normally we only endorse a theory if we believe it to be sound [Halbach] |
16344 | Soundness must involve truth; the soundness of PA certainly needs it [Halbach] |
16347 | Many new paradoxes may await us when we study interactions between frameworks [Halbach] |
16336 | The liar paradox applies truth to a negated truth (but the conditional will serve equally) [Halbach] |
10201 | Virtually all of mathematics can be modeled in set theory [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] |
18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
10256 | For intuitionists, proof is inherently informal [Shapiro] |
10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
16321 | The compactness theorem can prove nonstandard models of PA [Halbach] |
16343 | The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach] |
10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
16312 | To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach] |
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] |
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] |
16308 | Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach] |
10254 | Can the ideal constructor also destroy objects? [Shapiro] |
10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
16345 | That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction [Halbach] |
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] |
10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
10275 | A blurry border is still a border [Shapiro] |
16346 | Maybe necessity is a predicate, not the usual operator, to make it more like truth [Halbach] |
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] |
6457 | Sensations are mental, but sense-data could be mind-independent [Vesey] |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
10229 | Simple types can be apprehended through their tokens, via abstraction [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] |
16298 | We need propositions to ascribe the same beliefs to people with different languages [Halbach] |