163 ideas
| 2136 | Philosophers become as divine and orderly as possible, by studying divinity and order [Plato] |
| 23767 | The winds of the discussion should decide its destination [Plato] |
| 23682 | It would be absurd to be precise about the small things, but only vague about the big things [Plato] |
| 2151 | Dialectic is the only method of inquiry which uproots the things which it takes for granted [Plato] |
| 2154 | The ability to take an overview is the distinguishing mark of a dialectician [Plato] |
| 4011 | For Plato, rationality is a vision of and love of a cosmic rational order [Plato, by Taylor,C] |
| 2093 | You must never go against what you actually believe [Plato] |
| 2130 | People often merely practice eristic instead of dialectic, because they don't analyse the subject-matter [Plato] |
| 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] |
| 2145 | In mathematics certain things have to be accepted without further explanation [Plato] |
| 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] |
| 8726 | Geometry can lead the mind upwards to truth and philosophy [Plato] |
| 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] |
| 9863 | We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato] |
| 9864 | In pure numbers, all ones are equal, with no internal parts [Plato] |
| 8727 | Geometry is not an activity, but the study of unchanging knowledge [Plato] |
| 9861 | The same thing is both one and an unlimited number at the same time [Plato] |
| 9862 | To become rational, philosophers must rise from becoming into being [Plato] |
| 21818 | Being depends on the Good, which is not itself being, but superior to being [Plato] |
| 2061 | The best things (gods, healthy bodies, good souls) are least liable to change [Plato] |
| 6562 | Plato's reality has unchanging Parmenidean forms, and Heraclitean flux [Plato, by Fogelin] |
| 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] |
| 2142 | The plurality of beautiful things must belong to a single class, because they have a single particular character [Plato] |
| 5094 | Plato's Forms are said to have no location in space [Plato, by Aristotle] |
| 12043 | Forms are not universals, as they don't cover every general term [Plato, by Annas] |
| 2159 | Craftsmen making furniture refer to the form, but no one manufactures the form of furniture [Plato] |
| 17 | A Form applies to a set of particular things with the same name [Plato] |
| 12122 | Plato mistakenly thought forms were totally abstracted away from matter [Bacon on Plato] |
| 5574 | Plato's Forms not only do not come from the senses, but they are beyond possibility of sensing [Plato, by Kant] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 2133 | Knowledge must be of the permanent unchanging nature of things [Plato] |
| 2162 | If theory and practice conflict, the best part of the mind accepts theory, so the other part is of lower grade [Plato] |
| 2140 | True belief without knowledge is like blind people on the right road [Plato] |
| 2096 | Is the function of the mind management, authority and planning - or is it one's whole way of life? [Plato] |
| 6009 | Psychic conflict is clear if appetite is close to the body and reason fairly separate [Plato, by Modrak] |
| 6041 | There is a third element to the mind - spirit - lying between reason and appetite [Plato] |
| 2127 | The mind has parts, because we have inner conflicts [Plato] |
| 1737 | The soul seems to have an infinity of parts [Aristotle on Plato] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 5945 | The 'Republic' is a great work of rhetorical theory [Lawson-Tancred on Plato] |
| 23316 | For Plato and Aristotle there is no will; there is only rational desire for what is seen as good [Plato, by Frede,M] |
| 16 | We avoid evil either through a natural aversion, or because we have acquired knowledge [Plato] |
| 16565 | Without the surface decoration, poetry shows only appearances and nothing of what is real [Plato] |
| 2160 | Representation is two steps removed from the truth [Plato] |
| 2163 | Artists should be excluded from a law-abiding community, because they destroy the rational mind [Plato] |
| 2135 | Truth is closely related to proportion [Plato] |
| 2141 | I suggest that we forget about trying to define goodness itself for the time being [Plato] |
| 1869 | The good cannot be expressed in words, but imprints itself upon the soul [Plato, by Celsus] |
| 4115 | Plato found that he could only enforce rational moral justification by creating an authoritarian society [Williams,B on Plato] |
| 4547 | Plato measured the degree of reality by the degree of value [Nietzsche on Plato] |
| 2094 | A thing's function is what it alone can do, or what it does better than other things [Plato] |
| 2095 | If something has a function then it has a state of being good [Plato] |
| 2129 | Goodness is mental health, badness is mental sickness [Plato] |
| 12 | If we were invisible, would the just man become like the unjust? [Plato] |
| 2168 | Clever criminals do well at first, but not in the long run [Plato] |
| 2137 | The main aim is to understand goodness, which gives everything its value and advantage [Plato] |
| 2139 | Every person, and every activity, aims at the good [Plato] |
| 2143 | Good has the same role in the world of knowledge as the sun has in the physical world [Plato] |
| 2147 | The sight of goodness leads to all that is fine and true and right [Plato] |
| 4007 | For Plato we abandon honour and pleasure once we see the Good [Plato, by Taylor,C] |
| 2144 | Goodness makes truth and knowledge possible [Plato] |
| 2164 | Bad is always destructive, where good preserves and benefits [Plato] |
| 2138 | Pleasure is commonly thought to be the good, though the more ingenious prefer knowledge [Plato] |
| 2070 | Even people who think pleasure is the good admit that there are bad pleasures [Plato] |
| 2157 | Nice smells are intensive, have no preceding pain, and no bad after-effect [Plato] |
| 2134 | Philosophers are concerned with totally non-physical pleasures [Plato] |
| 2156 | There are three types of pleasure, for reason, for spirit and for appetite [Plato] |
| 2158 | Pleasure-seekers desperately seek illusory satisfaction, like filling a leaky vessel [Plato] |
| 2123 | Excessive pleasure deranges people, making the other virtues impossible [Plato] |
| 2166 | We should behave well even if invisible, for the health of the mind [Plato] |
| 2097 | Isn't it better to have a reputation for goodness than to actually be good? [Plato] |
| 19946 | Morality is a compromise, showing restraint, to avoid suffering wrong without compensation [Plato] |
| 5 | Justice is merely the interests of the stronger party [Plato] |
| 7 | Surely you don't return a borrowed weapon to a mad friend? [Plato] |
| 8 | Is right just the interests of the powerful? [Plato] |
| 15 | Sin first, then sacrifice to the gods from the proceeds [Plato] |
| 5944 | For Plato, virtue is its own reward [Lawson-Tancred on Plato] |
| 2155 | True goodness requires mental unity and harmony [Plato] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 2126 | A good community necessarily has wisdom, courage, self-discipline and morality [Plato] |
| 23562 | If the parts of our soul do their correct work, we will be just people, and will act justly [Plato] |
| 2092 | Simonides said morality is helping one's friends and harming one's enemies [Plato] |
| 19889 | People need society because the individual has too many needs [Plato] |
| 19890 | All exchanges in a community are for mutual benefit [Plato] |
| 10 | After a taste of mutual harm, men make a legal contract to avoid it [Plato] |
| 23561 | People doing their jobs properly is the fourth cardinal virtue for a city [Plato] |
| 2149 | Reluctant rulers make a better and more unified administration [Plato] |
| 2132 | Only rule by philosophers of integrity can keep a community healthy [Plato] |
| 2131 | Is there anything better for a community than to produce excellent people? [Plato] |
| 2152 | Dialectic is the highest and most important part of the curriculum [Plato] |
| 2148 | To gain knowledge, turn away from the world of change, and focus on true goodness [Plato] |
| 2153 | Compulsory intellectual work never remains in the mind [Plato] |
| 2630 | If Plato's God is immaterial, he will lack consciousness, wisdom, pleasure and movement, which are essential to him [Cicero on Plato] |
| 14 | If the gods are non-existent or indifferent, why bother to deceive them? [Plato] |
| 2165 | Something is unlikely to be immortal if it is imperfectly made from diverse parts [Plato] |
| 13 | Is the supreme reward for virtue to be drunk for eternity? [Plato] |
| 2120 | God is responsible for the good things, but we must look elsewhere for the cause of the bad things [Plato] |