175 ideas
| 15209 | Like disastrous small errors in navigation, small misunderstandings can wreck intellectual life [Harré/Madden] |
| 15215 | Philosophy devises and assesses conceptual schemes in the service of worldviews [Harré/Madden] |
| 15212 | Analysis of concepts based neither on formalism nor psychology can arise from examining what we know [Harré/Madden] |
| 15210 | Humeans see analysis in terms of formal logic, because necessities are fundamentally logical relations [Harré/Madden] |
| 15236 | Positivism says science only refers to immediate experiences [Harré/Madden] |
| 15227 | Logically, definitions have a subject, and a set of necessary predicates [Harré/Madden] |
| 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] |
| 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] |
| 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] |
| 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] |
| 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] |
| 13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [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] |
| 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] |
| 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] |
| 13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
| 13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [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] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
| 13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
| 13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
| 13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
| 13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
| 13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
| 13762 | Tableau rules are all elimination rules, gradually shortening formulae [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] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [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] |
| 15273 | Points can be 'dense' by unending division, but must meet a tougher criterion to be 'continuous' [Harré/Madden] |
| 15274 | Points are 'continuous' if any 'cut' point participates in both halves of the cut [Harré/Madden] |
| 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] |
| 15211 | There is not an exclusive dichotomy between the formal and the logical [Harré/Madden] |
| 15261 | Humeans can only explain change with continuity as successive replacement [Harré/Madden] |
| 15268 | Humeans construct their objects from events, but we construct events from objects [Harré/Madden] |
| 15257 | The induction problem fades if you work with things, rather than with events [Harré/Madden] |
| 15300 | Fundamental particulars can't change [Harré/Madden] |
| 15319 | Hard individual blocks don't fix what 'things' are; fluids are no less material things [Harré/Madden] |
| 15320 | Magnetic and gravity fields can occupy the same place without merging [Harré/Madden] |
| 15318 | Gravitational and electrical fields are, for a materialist, distressingly empty of material [Harré/Madden] |
| 15267 | Events are changes in states of affairs (which consist of structured particulars, with powers and relations) [Harré/Madden] |
| 13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
| 13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
| 15281 | Humeans see predicates as independent, but science says they are connected [Harré/Madden] |
| 15279 | Energy was introduced to physics to refer to the 'store of potency' of a moving ball [Harré/Madden] |
| 15276 | Some powers need a stimulus, but others are just released [Harré/Madden] |
| 15305 | Some powers are variable, others cannot change (without destroying an identity) [Harré/Madden] |
| 15218 | Scientists define copper almost entirely (bar atomic number) in terms of its dispositions [Harré/Madden] |
| 15302 | We explain powers by the natures of things, but explanations end in inexplicable powers [Harré/Madden] |
| 15303 | Maybe a physical field qualifies as ultimate, if its nature is identical with its powers [Harré/Madden] |
| 15258 | Powers are not qualities; they just point to directions of empirical investigation [Harré/Madden] |
| 15315 | What is a field of potentials, if it only consists of possible events? [Harré/Madden] |
| 15272 | The good criticism of substance by Humeans also loses them the vital concept of a thing [Harré/Madden] |
| 15304 | We can escape substance and its properties, if we take fields of pure powers as ultimate [Harré/Madden] |
| 15309 | The assumption that shape and solidity are fundamental implies dubious 'substance' in bodies [Harré/Madden] |
| 15264 | The notorious substratum results from substance-with-qualities; individuals-with-powers solves this [Harré/Madden] |
| 15262 | In logic the nature of a kind, substance or individual is the essence which is inseparable from what it is [Harré/Madden] |
| 15297 | We can infer a new property of a thing from its other properties, via its essential nature [Harré/Madden] |
| 15266 | We say the essence of particles is energy, but only so we can tell a story about the nature of things [Harré/Madden] |
| 15220 | To say something remains the same but lacks its capacities and powers seems a contradiction [Harré/Madden] |
| 15222 | Some individuals can gain or lose capacities or powers, without losing their identity [Harré/Madden] |
| 15296 | A particular might change all of its characteristics, retaining mere numerical identity [Harré/Madden] |
| 15275 | 'Dense' time raises doubts about continuous objects, so they need 'continuous' time [Harré/Madden] |
| 15271 | If things are successive instantaneous events, nothing requires those events to resemble one another [Harré/Madden] |
| 15256 | Humeans cannot step in the same river twice, because they cannot strictly form the concept of 'river' [Harré/Madden] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 15290 | What reduces the field of the possible is a step towards necessity [Harré/Madden] |
| 15291 | There is 'absolute' necessity (implied by all propositions) and 'relative' necessity (from what is given) [Harré/Madden] |
| 15230 | Logical necessity is grounded in the logical form of a statement [Harré/Madden] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 15214 | Natural necessity is not logical necessity or empirical contingency in disguise [Harré/Madden] |
| 15221 | The relation between what a thing is and what it can do or undergo relate by natural necessity [Harré/Madden] |
| 15224 | A necessity corresponds to the nature of the actual [Harré/Madden] |
| 15232 | Natural necessity is when powerful particulars must produce certain results in a situation [Harré/Madden] |
| 15288 | People doubt science because if it isn't logically necessary it seems to be absolutely contingent [Harré/Madden] |
| 15289 | Property or event relations are naturally necessary if generated by essential mechanisms [Harré/Madden] |
| 15231 | Transcendental necessity is conditions of a world required for a rational being to know its nature [Harré/Madden] |
| 15234 | There is a transcendental necessity for each logical necessity, but the transcendental extends further [Harré/Madden] |
| 15260 | Counterfactuals are just right for analysing statements about the powers which things have [Harré/Madden] |
| 15233 | If natural necessity is used to include or exclude some predicate, the predicate is conceptually necessary [Harré/Madden] |
| 15242 | Having a child is contingent for a 'man', necessary for a 'father'; the latter reflects a necessity of nature [Harré/Madden] |
| 15216 | Is conceptual necessity just conventional, or does it mirror something about nature? [Harré/Madden] |
| 15235 | There is a conceptual necessity when properties become a standard part of a nominal essence [Harré/Madden] |
| 15228 | Necessity and contingency are separate from the a priori and the a posteriori [Harré/Madden] |
| 15252 | If Goldbach's Conjecture is true (and logically necessary), we may be able to conceive its opposite [Harré/Madden] |
| 15245 | It is silly to say that direct experience must be justified, either by reason, or by more experience [Harré/Madden] |
| 15244 | We experience qualities as of objects, not on their own [Harré/Madden] |
| 15248 | Inference in perception is unconvincingly defended as non-conscious and almost instantaneous [Harré/Madden] |
| 15269 | Humean impressions are too instantaneous and simple to have structure or relations [Harré/Madden] |
| 15286 | Clavius's Paradox: purely syntactic entailment theories won't explain, because they are too profuse [Harré/Madden] |
| 15283 | Simplicity can sort theories out, but still leaves an infinity of possibilities [Harré/Madden] |
| 15316 | The powers/natures approach has been so successful (for electricity, magnetism, gravity) it may be universal [Harré/Madden] |
| 15225 | Science investigates the nature and constitution of things or substances [Harré/Madden] |
| 15298 | We prefer the theory which explains and predicts the powers and capacities of particulars [Harré/Madden] |
| 15255 | Conjunctions explain nothing, and so do not give a reason for confidence in inductions [Harré/Madden] |
| 15270 | Hume's atomic events makes properties independent, and leads to problems with induction [Harré/Madden] |
| 15284 | Contraposition may be equivalent in truth, but not true in nature, because of irrelevant predicates [Harré/Madden] |
| 15285 | The items put forward by the contraposition belong within different natural clusters [Harré/Madden] |
| 15287 | The possibility that all ravens are black is a law depends on a mechanism producing the blackness [Harré/Madden] |
| 15306 | Only changes require explanation [Harré/Madden] |
| 15293 | If explanation is by entailment, that lacks a causal direction, unlike natural necessity [Harré/Madden] |
| 15294 | Powers can explain the direction of causality, and make it a natural necessity [Harré/Madden] |
| 15254 | If the nature of particulars explains their powers, it also explains their relations and behaviour [Harré/Madden] |
| 15317 | Powers and natures lead us to hypothesise underlying mechanisms, which may be real [Harré/Madden] |
| 15310 | Solidity comes from the power of repulsion, and shape from the power of attraction [Harré/Madden] |
| 15219 | Essence explains passive capacities as well as active powers [Harré/Madden] |
| 15301 | The very concepts of a particular power or nature imply the possibility of being generalised [Harré/Madden] |
| 15226 | What properties a thing must have to be a type of substance can be laid down a priori [Harré/Madden] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 15229 | We say there is 'no alternative' in all sorts of contexts, and there are many different grounds for it [Harré/Madden] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 15292 | We can base the idea of a natural kind on the mechanisms that produce natural necessity [Harré/Madden] |
| 15299 | Species do not have enough constancy to be natural kinds [Harré/Madden] |
| 15253 | If the concept of a cause includes its usual effects, we call it a 'power' [Harré/Madden] |
| 15278 | Humean accounts of causal direction by time fail, because cause and effect can occur together [Harré/Madden] |
| 15246 | Active causal power is just objects at work, not something existing in itself [Harré/Madden] |
| 15213 | Causation always involves particular productive things [Harré/Madden] |
| 15217 | Efficient causes combine stimulus to individuals, absence of contraints on activity [Harré/Madden] |
| 15277 | The cause (or part of it) is what stimulates or releases the powerful particular thing involved [Harré/Madden] |
| 15237 | Originally Humeans based lawlike statements on pure qualities, without particulars [Harré/Madden] |
| 15238 | Being lawlike seems to resist formal analysis, because there are always counter-examples [Harré/Madden] |
| 15223 | Necessary effects will follow from some general theory specifying powers and structure of a world [Harré/Madden] |
| 15241 | Humeans say there is no necessity in causation, because denying an effect is never self-contradictory [Harré/Madden] |
| 15240 | In lawful universal statements (unlike accidental ones) we see why the regularity holds [Harré/Madden] |
| 15239 | We could call any generalisation a law, if it had reasonable support and no counter-evidence [Harré/Madden] |
| 15243 | We perceive motion, and not just successive occupations of different positions [Harré/Madden] |
| 15265 | 'Energy' is a quasi-substance invented as the bearer of change during interactions [Harré/Madden] |
| 15280 | 'Kinetic energy' is used to explain the effects of moving things when they are stopped [Harré/Madden] |
| 15321 | Space can't be an individual (in space), but it is present in all places [Harré/Madden] |
| 15259 | Chemical atoms have two powers: to enter certain combinations, and to emit a particular spectrum [Harré/Madden] |
| 15263 | Chemistry is not purely structural; CO2 is not the same as SO2 [Harré/Madden] |
| 15295 | Theism is supposed to make the world more intelligible - and should offer results [Harré/Madden] |