138 ideas
| 19359 | Leibniz aims to give coherent rational support for empiricism [Leibniz, by Perkins] |
| 13086 | Metaphysics is a science of the intelligible nature of being [Leibniz, by Cover/O'Leary-Hawthorne] |
| 16710 | Leibniz tried to combine mechanistic physics with scholastic metaphysics [Leibniz, by Pasnau] |
| 16897 | Reason is the faculty for grasping apriori necessary truths [Leibniz, by Burge] |
| 3346 | For Leibniz rationality is based on non-contradiction and the principle of sufficient reason [Leibniz, by Benardete,JA] |
| 3347 | Leibniz said the principle of sufficient reason is synthetic a priori, since its denial is not illogical [Leibniz, by Benardete,JA] |
| 8627 | Leibniz is inclined to regard all truths as provable [Leibniz, by Frege] |
| 13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
| 13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
| 13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock] |
| 13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock] |
| 13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock] |
| 13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock] |
| 13350 | 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock] |
| 13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock] |
| 13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock] |
| 13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock] |
| 13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock] |
| 13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
| 13346 | Truth is the basic notion in classical logic [Bostock] |
| 13545 | Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock] |
| 13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock] |
| 13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
| 13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
| 13347 | Validity is a conclusion following for premises, even if there is no proof [Bostock] |
| 13348 | It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock] |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
| 13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock] |
| 13803 | If we are to express that there at least two things, we need identity [Bostock] |
| 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock] |
| 13357 | Truth-functors are usually held to be defined by their truth-tables [Bostock] |
| 13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
| 13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
| 13360 | In logic, a name is just any expression which refers to a particular single object [Bostock] |
| 13361 | An expression is only a name if it succeeds in referring to a real object [Bostock] |
| 13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
| 13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
| 13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock] |
| 13816 | Because of scope problems, definite descriptions are best treated as quantifiers [Bostock] |
| 13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock] |
| 13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock] |
| 13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock] |
| 13818 | If we allow empty domains, we must allow empty names [Bostock] |
| 13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock] |
| 13619 | Quantification adds two axiom-schemas and a new rule [Bostock] |
| 13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock] |
| 13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock] |
| 13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
| 13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock] |
| 13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock] |
| 13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock] |
| 13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock] |
| 13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock] |
| 13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock] |
| 13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
| 13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
| 13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
| 13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
| 13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
| 13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
| 13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock] |
| 13760 | A sequent calculus is good for comparing proof systems [Bostock] |
| 13364 | Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG] |
| 13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock] |
| 13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock] |
| 13618 | Compactness means an infinity of sequents on the left will add nothing new [Bostock] |
| 9147 | Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction [Fine,K on Leibniz] |
| 19375 | The continuum is not divided like sand, but folded like paper [Leibniz, by Arthur,R] |
| 18081 | Nature uses the infinite everywhere [Leibniz] |
| 18080 | A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz] |
| 13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock] |
| 13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock] |
| 7565 | Leibniz proposes monads, since there must be basic things, which are immaterial in order to have unity [Leibniz, by Jolley] |
| 10419 | If relations can be reduced to, or supervene on, monadic properties of relata, they are not real [Leibniz, by Swoyer] |
| 13078 | Relations aren't in any monad, so they are distributed, so they are not real [Leibniz] |
| 13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
| 13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
| 12713 | Forms have sensation and appetite, the latter being the ability to act on other bodies [Leibniz, by Garber] |
| 13087 | The essence of a thing is its real possibilities [Leibniz, by Cover/O'Leary-Hawthorne] |
| 12701 | Leibniz moved from individuation by whole entity to individuation by substantial form [Leibniz, by Garber] |
| 13105 | The laws-of-the-series plays a haecceitist role [Leibniz, by Cover/O'Leary-Hawthorne] |
| 16513 | Identity of a substance is the law of its persistence [Leibniz] |
| 12035 | Leibniz bases pure primitive entities on conjunctions of qualitative properties [Leibniz, by Adams,RM] |
| 13091 | Leibnizian substances add concept, law, force, form and soul [Leibniz, by Cover/O'Leary-Hawthorne] |
| 7561 | Substances are essentially active [Leibniz, by Jolley] |
| 12715 | Leibniz strengthened hylomorphism by connecting it to force in physics [Leibniz, by Garber] |
| 11878 | Leibniz's view (that all properties are essential) is extreme essentialism, not its denial [Leibniz, by Mackie,P] |
| 11862 | Leibniz was not an essentialist [Leibniz, by Wiggins] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 16504 | Two eggs can't be identical, because the same truths can't apply to both of them [Leibniz] |
| 8650 | Things are the same if one can be substituted for the other without loss of truth [Leibniz] |
| 13828 | Necessary truths are those provable from identities by pure logic in finite steps [Leibniz, by Hacking] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 13084 | How can things be incompatible, if all positive terms seem to be compatible? [Leibniz] |
| 4307 | A reason must be given why contingent beings should exist rather than not exist [Leibniz] |
| 15883 | Leibniz narrows down God's options to one, by non-contradiction, sufficient reason, indiscernibles, compossibility [Leibniz, by Harré] |
| 18822 | Each monad expresses all its compatible monads; a possible world is the resulting equivalence class [Leibniz, by Rumfitt] |
| 7837 | Leibniz proposed possible worlds, because they might be evil, where God would not create evil things [Leibniz, by Stewart,M] |
| 13080 | Leibniz has a counterpart view of de re counterfactuals [Leibniz, by Cover/O'Leary-Hawthorne] |
| 19332 | For Leibniz, divine understanding grasps every conceivable possibility [Leibniz, by Perkins] |
| 5509 | Leibniz said dualism of mind and body is illusion, and there is only mind [Leibniz, by Martin/Barresi] |
| 7568 | Leibniz is an idealist insofar as the basic components of his universe are all mental [Leibniz, by Jolley] |
| 13092 | The essence of substance is the law of its changes, as in the series of numbers [Leibniz] |
| 19354 | Leibniz introduced the idea of degrees of consciousness, essential for his monads [Leibniz, by Perkins] |
| 7841 | We think we are free because the causes of the will are unknown; determinism is a false problem [Leibniz] |
| 5510 | Leibniz has a panpsychist view that physical points are spiritual [Leibniz, by Martin/Barresi] |
| 7564 | Occasionalism give a false view of natural laws, miracles, and substances [Leibniz, by Jolley] |
| 19451 | When absorbed in deep reflection, is your reason in control, or is it you? [Feuerbach] |
| 19372 | Concepts are ordered, and show eternal possibilities, deriving from God [Leibniz, by Arthur,R] |
| 13467 | Leibniz was the first modern to focus on sentence-sized units (where empiricists preferred word-size) [Leibniz, by Hart,WD] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 19365 | Limited awareness leads to bad choices, and unconscious awareness makes us choose the bad [Leibniz, by Perkins] |
| 8110 | Leibniz identified beauty with intellectual perfection [Leibniz, by Gardner] |
| 7569 | Humans are moral, and capable of reward and punishment, because of memory and self-consciousness [Leibniz, by Jolley] |
| 19450 | Reason, love and will are the highest perfections and essence of man - the purpose of his life [Feuerbach] |
| 7574 | Natural law theory is found in Aquinas, in Leibniz, and at the Nuremberg trials [Leibniz, by Jolley] |
| 12728 | Leibniz rejected atoms, because they must be elastic, and hence have parts [Leibniz, by Garber] |
| 19374 | Microscopes and the continuum suggest that matter is endlessly divisible [Leibniz] |
| 7560 | Leibniz struggled to reconcile bodies with a reality of purely soul-like entities [Jolley on Leibniz] |
| 16683 | Leibniz eventually said resistance, rather than extension, was the essence of body [Leibniz, by Pasnau] |
| 12725 | Leibniz wanted to explain motion and its laws by the nature of body [Leibniz, by Garber] |
| 16507 | The law within something fixes its persistence, and accords with general laws of nature [Leibniz] |
| 7859 | Leibniz had an unusual commitment to the causal completeness of physics [Leibniz, by Papineau] |
| 15307 | Leibniz uses 'force' to mean both activity and potential [Leibniz] |
| 19448 | Consciousness is said to distinguish man from animals - consciousness of his own species [Feuerbach] |
| 19454 | A God needs justice, kindness and wisdom, but those concepts don't depend on the concept of God [Feuerbach] |
| 3889 | God's existence is either necessary or impossible [Leibniz, by Scruton] |
| 19452 | The nature of God is an expression of human nature [Feuerbach] |
| 7842 | Leibniz was closer than Spinoza to atheism [Leibniz, by Stewart,M] |
| 19453 | If love, goodness and personality are human, the God who is their source is anthropomorphic [Feuerbach] |
| 19449 | Religion is the consciousness of the infinite [Feuerbach] |
| 19455 | Today's atheism will tomorrow become a religion [Feuerbach] |