232 ideas
22115 | Wise people should contemplate and discuss the truth, and fight against falsehood [Aquinas] |
22101 | Philosophy aims to know the truth about the way things are [Aquinas] |
1848 | We are coerced into assent to a truth by reason's violence [Aquinas] |
1858 | The mind is compelled by necessary truths, but not by contingent truths [Aquinas] |
21267 | Supposing many principles is superfluous if a few will do it [Aquinas] |
22102 | Arguing with opponents uncovers truths, and restrains falsehoods [Aquinas] |
13070 | If definitions must be general, and general terms can't individuate, then Socrates can't be defined [Aquinas, by Cover/O'Leary-Hawthorne] |
11197 | The definitions expressing identity are used to sort things [Aquinas] |
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
23176 | Truth is universal, but knowledge of it is not [Aquinas] |
20621 | Types of lying: Speak lies, intend lies, intend deception, aim at deceptive goal? [Aquinas, by Tuckness/Wolf] |
1852 | For the mind Good is one truth among many, and Truth is one good among many [Aquinas] |
21248 | If the existence of truth is denied, the 'Truth does not exist' must be true! [Aquinas] |
22104 | Truth is the conformity of being to intellect [Aquinas] |
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] |
18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
18114 | There is no single agreed structure for set theory [Bostock] |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
18109 | The completeness of first-order logic implies its compactness [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] |
23173 | If a syllogism admits one absurdity, others must follow [Aquinas] |
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] |
18123 | Substitutional quantification is just standard if all objects in the domain have a name [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] |
18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [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] |
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] |
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] |
18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
18097 | The Peano Axioms describe a unique structure [Bostock] |
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] |
18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
18149 | There are many criteria for the identity of numbers [Bostock] |
18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
18140 | The best version of conceptualism is predicativism [Bostock] |
18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
11195 | If affirmative propositions express being, we affirm about what is absent [Aquinas] |
22103 | Being is basic to thought, and all other concepts are additions to being [Aquinas] |
15812 | Being implies distinctness, which implies division, unity, and multitude [Aquinas] |
12695 | Epicurean atomists say body is sensible, to distinguish it from space. [Garber] |
21268 | Non-human things are explicable naturally, and voluntary things by the will, so God is not needed [Aquinas] |
16655 | Different genera are delimited by modes of predication, which rest on modes of being [Aquinas] |
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] |
16641 | Whiteness does not exist, but by it something can exist-as-white [Aquinas] |
11201 | Properties have an incomplete essence, with definitions referring to their subject [Aquinas] |
11205 | If the form of 'human' contains 'many', Socrates isn't human; if it contains 'one', Socrates is Plato [Aquinas] |
13090 | The principle of diversity for corporeal substances is their matter [Aquinas, by Cover/O'Leary-Hawthorne] |
17555 | 'One' can mean undivided and not a multitude, or it can add measurement, giving number [Aquinas] |
16765 | Humans only have a single substantial form, which contains the others and acts for them [Aquinas] |
16766 | One thing needs a single thing to unite it; if there were two forms, something must unite them [Aquinas] |
11202 | It is by having essence that things exist [Aquinas] |
11203 | Specific individual essence is defined by material, and generic essence is defined by form [Aquinas] |
11200 | The definition of a physical object must include the material as well as the form [Aquinas] |
11196 | Essence is something in common between the natures which sort things into categories [Aquinas] |
11208 | A simple substance is its own essence [Aquinas] |
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] |
22170 | Senses grasp external properties, but the understanding grasps the essential natures of things [Aquinas] |
23175 | The conclusions of speculative reason about necessities are certain [Aquinas] |
21337 | A knowing being possesses a further reality, the 'presence' of the thing known [Aquinas] |
21249 | Some things are self-evident to us; others are only self-evident in themselves [Aquinas] |
22169 | Initial universal truths are present within us as potential, to be drawn out by reason [Aquinas] |
21250 | A proposition is self-evident if the predicate is included in the essence of the subject [Aquinas] |
22168 | Minds take in a likeness of things, which activates an awaiting potential [Aquinas] |
20224 | Sensation prepares the way for intellectual knowledge, which needs the virtues of reason [Aquinas] |
1860 | Knowledge may be based on senses, but we needn't sense all our knowledge [Aquinas] |
22109 | The fullest knowledge places a conclusion within an accurate theory [Aquinas, by Kretzmann/Stump] |
11198 | Definition of essence makes things understandable [Aquinas] |
22107 | Sensations are transmitted to 'internal senses' in the brain, chiefly to 'phantasia' and 'imagination' [Aquinas, by Kretzmann/Stump] |
9098 | Mental activity combines what we sense with imagination of what is not present [Aquinas] |
9092 | Abstracting A from B generates truth, as long as the connection is not denied [Aquinas] |
9093 | We understand the general nature of things by ignoring individual peculiarities [Aquinas] |
9097 | The mind abstracts generalities from images, but also uses images for understanding [Aquinas] |
9095 | Very general ideas (being, oneness, potentiality) can be abstracted from thought matter in general [Aquinas] |
9099 | Particular instances come first, and (pace Plato) generalisations are abstracted from them [Aquinas] |
10508 | Species are abstracted from appearances by ignoring individual conditions [Aquinas] |
22111 | Aquinas attributes freedom to decisions and judgements, and not to the will alone [Aquinas, by Kretzmann/Stump] |
1855 | If we saw something as totally and utterly good, we would be compelled to will it [Aquinas] |
1856 | Nothing can be willed except what is good, but good is very varied, and so choices are unpredictable [Aquinas] |
1862 | However habituated you are, given time to ponder you can go against a habit [Aquinas] |
1849 | Since will is a reasoning power, it can entertain opposites, so it is not compelled to embrace one of them [Aquinas] |
1861 | The will is not compelled to move, even if pleasant things are set before it [Aquinas] |
1853 | Because the will moves by examining alternatives, it doesn't compel itself to will [Aquinas] |
1854 | We must admit that when the will is not willing something, the first movement to will must come from outside the will [Aquinas] |
22105 | The human intellectual soul is an incorporeal, subsistent principle [Aquinas] |
20700 | Without God's influence every operation would stop, so God causes everything [Aquinas] |
22108 | First grasp what it is, then its essential features; judgement is their compounding and division [Aquinas] |
10503 | We abstract forms from appearances, and acquire knowledge of immaterial things [Aquinas] |
10509 | Understanding consists entirely of grasping abstracted species [Aquinas] |
10506 | Mathematics can be abstracted from sensible matter, and from individual intelligible matter [Aquinas] |
9094 | Mathematical objects abstract both from perceived matter, and from particular substance [Aquinas] |
10505 | We can just think of an apple's colour, because the apple is not part of the colour's nature [Aquinas] |
10504 | Abstracting either treats something as separate, or thinks of it separately [Aquinas] |
10507 | Numbers and shapes are abstracted by ignoring their sensible qualities [Aquinas] |
9096 | The mind must produce by its own power an image of the individual species [Aquinas] |
11206 | The mind constructs complete attributions, based on the unified elements of the real world [Aquinas] |
13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
1847 | The will must aim at happiness, but can choose the means [Aquinas] |
1857 | We don't have to will even perfect good, because we can choose not to think of it [Aquinas] |
1846 | The will can only want what it thinks is good [Aquinas] |
23180 | The will is the rational appetite [Aquinas] |
1850 | Without free will not only is ethical action meaningless, but also planning, commanding, praising and blaming [Aquinas] |
22112 | For humans good is accordance with reason, and bad is contrary to reason [Aquinas] |
22494 | We must know the end, know that it is the end, and know how to attain it [Aquinas] |
1851 | Good applies to goals, just as truth applies to ideas in the mind [Aquinas] |
23181 | All acts of virtue relate to justice, which is directed towards the common good [Aquinas] |
8009 | Aquinas wanted, not to escape desire, but to transform it for moral ends [Aquinas, by MacIntyre] |
23182 | Legal justice is supreme, because it directs the other virtues to the common good [Aquinas] |
22399 | Temperance prevents our passions from acting against reason [Aquinas] |
23177 | Justice directs our relations with others, because it denotes a kind of equality [Aquinas] |
23179 | People differ in their social degrees, and a particular type of right applies to each [Aquinas] |
23174 | Natural law is a rational creature's participation in eternal law [Aquinas] |
22113 | Right and wrong actions pertain to natural law, as perceived by practical reason [Aquinas] |
22114 | Tyrannical laws are irrational, and so not really laws [Aquinas] |
7291 | For Aquinas a war must be in a just cause, have proper authority, and aim at good [Aquinas, by Grayling] |
5508 | Aquinas says a fertilized egg is not human, and has no immortal soul [Aquinas, by Martin/Barresi] |
12705 | Epicurean atoms are distinguished by their extreme hardness [Garber] |
16687 | Bodies are three-dimensional substances [Aquinas] |
11207 | A cause can exist without its effect, but the effect cannot exist without its cause [Aquinas] |
1859 | Even a sufficient cause doesn't compel its effect, because interference could interrupt the process [Aquinas] |
15202 | Eternity coexists with passing time, as the centre of a circle coexists with its circumference [Aquinas] |
23178 | Divine law commands some things because they are good, while others are good because commanded [Aquinas] |
21251 | We can't know God's essence, so his existence can't be self-evident for us [Aquinas] |
5614 | If you assume that there must be a necessary being, you can't say which being has this quality [Kant on Aquinas] |
21269 | Way 1: the infinite chain of potential-to-actual movement has to have a first mover [Aquinas] |
21270 | Way 2: no effect without a cause, and this cannot go back to infinity, so there is First Cause [Aquinas] |
21271 | Way 3: contingent beings eventually vanish, so continuity needs a necessary being [Aquinas] |
21272 | Way 4: the source of all qualities is their maximum, so something (God) causes all perfections [Aquinas] |
21273 | Way 5: mindless things act towards an obvious end, so there is an intelligent director [Aquinas] |
20211 | Life aims at the Beatific Vision - of perfect happiness, and revealed truth [Aquinas, by Zagzebski] |
22106 | Aquinas saw angels as separated forms, rather than as made of 'spiritual matter' [Aquinas, by Kretzmann/Stump] |
16711 | Heretics should be eradicated like wolves [Aquinas] |
1863 | If the soul achieves well-being in another life, it doesn't follow that I do [Aquinas] |
23306 | Humans have a non-physical faculty of reason, so they can be immortal [Aquinas, by Sorabji] |
4412 | Those in bliss have their happiness increased by seeing the damned punished [Aquinas] |
21266 | God does not exist, because He is infinite and good, and so no evil should be discoverable [Aquinas] |
21274 | It is part of God's supreme goodness that He brings good even out of evil [Aquinas] |