151 ideas
| 7170 | 'Wisdom' attempts to get beyond perspectives, making it hostile to life [Nietzsche] |
| 7167 | Words such as 'I' and 'do' and 'done to' are placed at the point where our ignorance begins [Nietzsche] |
| 7196 | Pessimism is laughable, because the world cannot be evaluated [Nietzsche] |
| 7137 | Is a 'philosopher' now impossible, because knowledge is too vast for an overview? [Nietzsche] |
| 7132 | Philosophers should create and fight for their concepts, not just clean and clarify them [Nietzsche] |
| 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] |
| 13354 | 'Conjunction' 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] |
| 13355 | 'Disjunction' 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] |
| 7188 | Logic tries to understand the world according to a man-made scheme [Nietzsche] |
| 7145 | Logic is not driven by truth, but desire for a simple single viewpoint [Nietzsche] |
| 7144 | Logic must falsely assume that identical cases exist [Nietzsche] |
| 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] |
| 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] |
| 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] |
| 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] |
| 7153 | We can't be realists, because we don't know what being is [Nietzsche] |
| 7174 | Categories are not metaphysical truths, but inventions in the service of needs [Nietzsche] |
| 7175 | Philosophers find it particularly hard to shake off belief in necessary categories [Nietzsche] |
| 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] |
| 7189 | Maybe there are only subjects, and 'objects' result from relations between subjects [Nietzsche] |
| 7207 | Counting needs unities, but that doesn't mean they exist; we borrowed it from the concept of 'I' [Nietzsche] |
| 7161 | The essence of a thing is only an opinion about the 'thing' [Nietzsche] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 7134 | Something can be irrefutable; that doesn't make it true [Nietzsche] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 7186 | There are no necessary truths, but something must be held to be true [Nietzsche] |
| 7154 | We can't use our own self to criticise our own capacity for knowledge! [Nietzsche] |
| 7146 | Belief in the body is better established than belief in the mind [Nietzsche] |
| 7156 | Sense perceptions contain values (useful, so pleasant) [Nietzsche] |
| 7181 | Pain shows the value of the damage, not what has been damaged [Nietzsche] |
| 7129 | Perception is unconscious, and we are only conscious of processed perceptions [Nietzsche] |
| 7149 | Comprehending everything is impossible, because it abolishes perspectives [Nietzsche] |
| 7169 | Is the perspectival part of the essence, or just a relation between beings? [Nietzsche] |
| 7182 | 'Perspectivism': the world has no meaning, but various interpretations give it countless meanings [Nietzsche] |
| 7183 | 'Subjectivity' is an interpretation, since subjects (and interpreters) are fictions [Nietzsche] |
| 7133 | There are different eyes, so different 'truths', so there is no truth [Nietzsche] |
| 7139 | Explanation is just showing the succession of things ever more clearly [Nietzsche] |
| 7131 | The intellect and senses are a simplifying apparatus [Nietzsche] |
| 7152 | With protoplasm ½+½=2, so the soul is not an indivisible monad [Nietzsche] |
| 7130 | Unity is not in the conscious 'I', but in the organism, which uses the self as a tool [Nietzsche] |
| 7155 | Consciousness exists to the extent that consciousness is useful [Nietzsche] |
| 7143 | Consciousness is a 'tool' - just as the stomach is a tool [Nietzsche] |
| 7157 | We think each thought causes the next, unaware of the hidden struggle beneath [Nietzsche] |
| 7148 | The 'I' is a conceptual synthesis, not the governor of our being [Nietzsche] |
| 7138 | The 'I' is a fiction used to make the world of becoming 'knowable' [Nietzsche] |
| 7135 | 'Freedom of will' is the feeling of having a dominating force [Nietzsche] |
| 7171 | Rationality is a scheme we cannot cast away [Nietzsche] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 7209 | There is no will; weakness of will is splitting of impulses, strong will is coordination under one impulse [Nietzsche] |
| 7194 | Experiencing a thing as beautiful is to experience it wrongly [Nietzsche] |
| 7136 | Morality is a system of values which accompanies a being's life [Nietzsche] |
| 7163 | Morality is merely interpretations, which are extra-moral in origin [Nietzsche] |
| 7147 | Values are innate and inherited [Nietzsche] |
| 7190 | Our values express an earlier era's conditions for survival and growth [Nietzsche] |
| 7201 | Knowledge, wisdom and goodness only have value relative to a goal [Nietzsche] |
| 7205 | Altruism is praised by the egoism of the weak, who want everyone to be looked after [Nietzsche] |
| 7141 | A living being is totally 'egoistic' [Nietzsche] |
| 7168 | Modest people express happiness as 'Not bad' [Nietzsche] |
| 7159 | The only happiness is happiness with illusion [Nietzsche] |
| 7197 | Pleasure needs dissatisfaction, boundaries and resistances [Nietzsche] |
| 467 | A virtue is a combination of intelligence, strength and luck [Ion] |
| 7165 | Virtue is wasteful, as it reduces us all to being one another's nurse [Nietzsche] |
| 7193 | Virtue for everyone removes its charm of being exceptional and aristocratic [Nietzsche] |
| 7191 | What does not kill us makes us stronger [Nietzsche] |
| 7151 | Courage, compassion, insight, solitude are the virtues, with courtesy a necessary vice [Nietzsche] |
| 7185 | Replace the categorical imperative by the natural imperative [Nietzsche] |
| 7164 | Not feeling harnessed to a system of 'ends' is a wonderful feeling of freedom [Nietzsche] |
| 7198 | Nihilism results from measuring the world by our categories which are purely invented [Nietzsche] |
| 7150 | By developing herd virtues man fixes what has up to now been the 'unfixed animal' [Nietzsche] |
| 7177 | Virtues from outside are dangerous, and they should come from within [Nietzsche] |
| 7172 | Existence without meaning or goal or end, eternally recurring, is a terrible thought [Nietzsche] |
| 7166 | Man is above all a judging animal [Nietzsche] |
| 7204 | The upholding of the military state is needed to maintain the strong human type [Nietzsche] |
| 7173 | Rights arise out of contracts, which need a balance of power [Nietzsche] |
| 7176 | 'Purpose' is like the sun, where most heat is wasted, and a tiny part has 'purpose' [Nietzsche] |
| 7195 | If the world aimed at an end, it would have reached it by now [Nietzsche] |
| 7206 | Things are strong or weak, and do not behave regularly or according to rules or compulsions [Nietzsche] |
| 7140 | Chemical 'laws' are merely the establishment of power relations between weaker and stronger [Nietzsche] |
| 7142 | All motions and 'laws' are symptoms of inner events, traceable to the will to power [Nietzsche] |
| 7178 | The utility of an organ does not explain its origin, on the contrary! [Nietzsche] |
| 7179 | Survival might undermine an individual's value, or prevent its evolution [Nietzsche] |
| 7180 | Darwin overestimates the influence of 'external circumstances' [Nietzsche] |
| 7192 | Remove goodness and wisdom from our concept of God. Being the highest power is enough! [Nietzsche] |
| 7158 | Morality kills religion, because a Christian-moral God is unbelievable [Nietzsche] |
| 7199 | It is dishonest to invent a being containing our greatest values, thus ignoring why they exist and are valuable [Nietzsche] |
| 7162 | Morality can only be upheld by belief in God and a 'hereafter' [Nietzsche] |
| 7208 | Paganism is a form of thanking and affirming life? [Nietzsche] |
| 7160 | Christian belief is kept alive because it is soothing - the proof based on pleasure [Nietzsche] |
| 7203 | In heaven all the interesting men are missing [Nietzsche] |
| 7200 | A combination of great power and goodness would mean the disastrous abolition of evil [Nietzsche] |