101 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] |
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] |
9535 | 'Contradictory' propositions always differ in truth-value [Lemmon] |
9509 | That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon] |
9508 | The sign |- may be read as 'therefore' [Lemmon] |
9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon] |
9510 | That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon] |
9512 | We write the 'negation' of P (not-P) as ¬ [Lemmon] |
9513 | We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) [Lemmon] |
9514 | If A and B are 'interderivable' from one another we may write A -||- B [Lemmon] |
9516 | A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ [Lemmon] |
9517 | The 'scope' of a connective is the connective, the linked formulae, and the brackets [Lemmon] |
9519 | A 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon] |
9529 | A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon] |
9531 | 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon] |
9534 | Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon] |
9530 | A wff is 'contingent' if produces at least one T and at least one F [Lemmon] |
9532 | 'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon] |
9533 | A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon] |
9528 | A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon] |
9518 | A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon] |
9399 | ∧E: Given A∧B, we may derive either A or B separately [Lemmon] |
9396 | DN: Given A, we may derive ¬¬A [Lemmon] |
9393 | A: we may assume any proposition at any stage [Lemmon] |
9401 | ∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon] |
9398 | ∧I: Given A and B, we may derive A∧B [Lemmon] |
9397 | CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon] |
9394 | MPP: Given A and A→B, we may derive B [Lemmon] |
9402 | RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon] |
9395 | MTT: Given ¬B and A→B, we derive ¬A [Lemmon] |
9400 | ∨I: Given either A or B separately, we may derive A∨B [Lemmon] |
9521 | 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon] |
9522 | 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon] |
9525 | We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) [Lemmon] |
9524 | We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q [Lemmon] |
9523 | De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon] |
9527 | The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon] |
9526 | We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [Lemmon] |
9537 | Truth-tables are good for showing invalidity [Lemmon] |
9538 | A truth-table test is entirely mechanical, but this won't work for more complex logic [Lemmon] |
9536 | If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology [Lemmon] |
9539 | Propositional logic is complete, since all of its tautologous sequents are derivable [Lemmon] |
13909 | Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....' [Lemmon] |
13902 | 'Gm' says m has property G, and 'Pmn' says m has relation P to n [Lemmon] |
13911 | The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E [Lemmon] |
13910 | Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers' [Lemmon] |
13904 | Universal Elimination (UE) lets us infer that an object has F, from all things having F [Lemmon] |
13906 | With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro [Lemmon] |
13908 | UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one [Lemmon] |
13901 | Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules [Lemmon] |
13903 | Universal elimination if you start with the universal, introduction if you want to end with it [Lemmon] |
13905 | If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon] |
13900 | 'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional → [Lemmon] |
21657 | Since properties can have properties, some theorists rank them in 'types' [Hofweber] |
9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
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] |
17988 | Quantification can't all be substitutional; some reference is obviously to objects [Hofweber] |
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] |
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] |
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] |
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] |
21648 | Neo-Fregeans are dazzled by a technical result, and ignore practicalities [Hofweber] |
10006 | First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber] |
21664 | Supervenience offers little explanation for things which necessarily go together [Hofweber] |
21660 | Reality can be seen as the totality of facts, or as the totality of things [Hofweber] |
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] |
21652 | Our perceptual beliefs are about ordinary objects, not about simples arranged chair-wise [Hofweber] |
21663 | Counterfactuals are essential for planning, and learning from mistakes [Hofweber] |
10004 | Our minds are at their best when reasoning about objects [Hofweber] |
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] |
22595 | Liberty is the triumph of the individual, over both despotic government and enslaving majorities [Constant] |
22597 | Minority rights are everyone's rights, because we all have turns in the minority [Constant] |