115 ideas
16440 | I don't think Lewis's cost-benefit reflective equilibrium approach offers enough guidance [Stalnaker] |
9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
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] |
9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
9701 | A 'relation' is a set of ordered pairs [Enderton] |
9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
16449 | In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker] |
13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
12766 | Logical space is abstracted from the actual world [Stalnaker] |
16464 | We regiment to get semantic structure, for evaluating arguments, and understanding complexities [Stalnaker] |
16465 | In 'S was F or some other than S was F', the disjuncts need S, but the whole disjunction doesn't [Stalnaker] |
16405 | To understand a name (unlike a description) picking the thing out is sufficient? [Stalnaker] |
9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
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] |
16443 | Properties are modal, involving possible situations where they are exemplified [Stalnaker] |
16471 | I accept a hierarchy of properties of properties of properties [Stalnaker] |
16452 | Dispositions have modal properties, of which properties things would have counterfactually [Stalnaker] |
14617 | Predicates can't apply to what doesn't exist [Stalnaker] |
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] |
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] |
9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
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] |
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] |
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] |
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] |