47 ideas
| 6319 | Wise people choose inaction and silence [Laozi (Lao Tzu)] |
| Full Idea: The sage keeps to the deed that consists in taking no action and practises the teaching that uses no words. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], I.II.6) | |
| A reaction: Notice that this is an active 'deed', and a positive 'practice'. He is not just recommending indifference and lethargy. Personally I don't find the advice very appealing, but it might be good if you live in 'interesting times'. |
| 6325 | One who knows does not speak; one who speaks does not know [Laozi (Lao Tzu)] |
| Full Idea: One who knows does not speak; one who speaks does not know. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.LVI.128) | |
| A reaction: A famous remark, which my western mind finds simply perplexing. It strikes me as wicked selfishness to keep your wisdom to yourself, and not try to persuade others to follow it. We are all in this together, I say. |
| 6321 | Vulgar people are alert; I alone am muddled [Laozi (Lao Tzu)] |
| Full Idea: Vulgar people are alert; I alone am muddled. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], I.XX.47) | |
| A reaction: Personally I think all human beings are deeply perplexed when they actually address their situation, but most people never spend more than a few minutes a year worrying about it. |
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| Full Idea: Until the 1960s standard truth-table semantics were the only ones that there were. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.10.1) | |
| A reaction: The 1960s presumably marked the advent of possible worlds. |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| Full Idea: 'dom R' indicates the 'domain' of a relation, that is, the set of all objects that are members of ordered pairs and that have that relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| Full Idea: 'fld R' indicates the 'field' of a relation, that is, the set of all objects that are members of ordered pairs on either side of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| Full Idea: 'ran R' indicates the 'range' of a relation, that is, the set of all objects that are members of ordered pairs and that are related to by the first objects. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| Full Idea: We write F : A → B to indicate that A maps into B, that is, the domain of relating things is set A, and the things related to are all in B. If we add that F = B, then A maps 'onto' B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
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Full Idea:
F(x) is a 'function', which indicates the unique value which y takes in |
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| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| Full Idea: A relation is 'symmetric' on a set if every ordered pair in the set has the relation in both directions. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| Full Idea: A relation is 'transitive' on a set if the relation can be carried over from two ordered pairs to a third. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| Full Idea: The 'powerset' of a set is all the subsets of a given set. Thus: PA = {x : x ⊆ A}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| Full Idea: Two sets are 'disjoint' iff their intersection is empty (i.e. they have no members in common). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| Full Idea: The 'domain' of a relation is the set of all objects that are members of ordered pairs that are members of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| Full Idea: A 'relation' is a set of ordered pairs. The ordering relation on the numbers 0-3 is captured by - in fact it is - the set of ordered pairs {<0,1>,<0,2>,<0,3>,<1,2>,<1,3>,<2,3>}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) | |
| A reaction: This can't quite be a definition of order among numbers, since it relies on the notion of a 'ordered' pair. |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| Full Idea: A 'function' is a relation which is single-valued. That is, for each object, there is only one object in the function set to which that object is related. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| Full Idea: A function 'maps A into B' if the domain of relating things is set A, and the things related to are all in B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| Full Idea: A function 'maps A onto B' if the domain of relating things is set A, and the things related to are set B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| Full Idea: A relation is 'reflexive' on a set if every member of the set bears the relation to itself. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| Full Idea: A relation satisfies 'trichotomy' on a set if every ordered pair is related (in either direction), or the objects are identical. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| Full Idea: A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| Full Idea: An 'equivalence relation' is a binary relation which is reflexive, and symmetric, and transitive. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| Full Idea: Equivalence classes will 'partition' a set. That is, it will divide it into distinct subsets, according to each relation on the set. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| Full Idea: The process is dubbed 'conversational implicature' when the inference is not from the content of what has been said, but from the fact that it has been said. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7.3) |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| Full Idea: The point of logic is to give an account of the notion of validity,..in two standard ways: the semantic way says that a valid inference preserves truth (symbol |=), and the proof-theoretic way is defined in terms of purely formal procedures (symbol |-). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..) | |
| A reaction: This division can be mirrored in mathematics, where it is either to do with counting or theorising about things in the physical world, or following sets of rules from axioms. Language can discuss reality, or play word-games. |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| Full Idea: A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.3.4) | |
| A reaction: So the final column of every line of the truth table will be T. |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| Full Idea: A truth assignment 'satisfies' a formula, or set of formulae, if it evaluates as True when all of its components have been assigned truth values. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.2) | |
| A reaction: [very roughly what Enderton says!] The concept becomes most significant when a large set of wff's is pronounced 'satisfied' after a truth assignment leads to them all being true. |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| Full Idea: If every proof-theoretically valid inference is semantically valid (so that |- entails |=), the proof theory is said to be 'sound'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| Full Idea: If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| Full Idea: If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: [Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that? |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| Full Idea: A set of expressions is 'decidable' iff there exists an effective procedure (qv) that, given some expression, will decide whether or not the expression is included in the set (i.e. doesn't contradict it). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7) | |
| A reaction: This is obviously a highly desirable feature for a really reliable system of expressions to possess. All finite sets are decidable, but some infinite sets are not. |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| Full Idea: The Enumerability Theorem says that for a reasonable language, the set of valid wff's can be effectively enumerated. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: There are criteria for what makes a 'reasonable' language (probably specified to ensure enumerability!). Predicates and functions must be decidable, and the language must be finite. |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| Full Idea: Not all sentences using 'if' are conditionals. Consider 'if you want a banana, there is one in the kitchen'. The rough test is that a conditional can be rewritten as 'that A implies that B'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.6.4) |
| 6328 | To know yet to think that one does not know is best [Laozi (Lao Tzu)] |
| Full Idea: To know yet to think that one does not know is best. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.LXXI.173) | |
| A reaction: Tricky. Self-deception doesn't sound like a virtue to me. There are epistemic virtues, and caution about one's own knowledge has to be one of them, but a totally false assessment sounds counter-productive. |
| 6323 | Pursuit of learning increases activity; the Way decreases it [Laozi (Lao Tzu)] |
| Full Idea: In the pursuit of learning one knows more every day; in the pursuit of the Way one does less every day. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.XLVII.108) | |
| A reaction: Everything in my culture has raised the status of the pursuit of learning, so that I can hardly comprehend what is proposed by the Way. I don't believe that the Way can be achieved without great learning, but one might move beyond learning. |
| 6331 | Truth is not beautiful; beautiful speech is not truthful [Laozi (Lao Tzu)] |
| Full Idea: Truthful words are not beautiful; beautiful words are not truthful. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.LXXXI.194) | |
| A reaction: A sharp disagreement with Keats ('Ode to a Grecian Urn'). A deep and important question, especially in relation to Plato's discussion of rhetoric (where he is very ambivalent). Great mathematics is beautiful. Truth can harsh. On the whole, I disagree. |
| 6330 | One with no use for life is wiser than one who values it [Laozi (Lao Tzu)] |
| Full Idea: It is just because one has no use for life that one is wiser than the man who values life. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.LXXV.181a) | |
| A reaction: To have no use for life certainly seems to put a person into a position of superiority, especially when the 'Titanic' is sinking. However, if our lives have no value, I don't know what does. A balance must clearly be struck. |
| 6327 | Do good to him who has done you an injury [Laozi (Lao Tzu)] |
| Full Idea: Do good to him who has done you an injury. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.LXIII.148) | |
| A reaction: Compare Idea 6288 (Jesus). People like this really mess up the social contract theory of morality. If they are going to return good for your evil, there doesn't seem much point in helping them, given how much effort is involved. Most peculiar… |
| 23402 | The highest virtue is achieved without effort [Laozi (Lao Tzu)] |
| Full Idea: Those of highest virtue do not strive for virtue, and so they have it. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], 38), quoted by Bryan van Norden - Intro to Classical Chinese Philosophy 8.II.2 | |
| A reaction: Every moralist's dream is the person to whom virtue comes so naturally that no thought is required. This says they don't even notice it; Aristotle says they simply enjoying behaving virtuously. |
| 6324 | To gain in goodness, treat as good those who are good, and those who are not [Laozi (Lao Tzu)] |
| Full Idea: Those who are good I treat as good; those who are not good I also treat as good; in doing so I gain in goodness. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.XLIX.111) | |
| A reaction: Socrates (idea 346) and Jesus (Idea 6288) had similar ideas. Who, though, is going to administer justice, and where is the idea that people 'deserve' good or ill treatment? Schoolteachers should treat all children as if they were good. |
| 6322 | There is no crime greater than having too many desires [Laozi (Lao Tzu)] |
| Full Idea: There is no crime greater than having too many desires. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.XLVI.104) | |
| A reaction: It seems harsh to call this a 'crime', given that no one is likely to choose to have 'too many' desires. The crime is in deciding to stimulate desire to excess, or deciding to show no sensible restraint. |
| 6320 | The best rulers are invisible, the next admired, the next feared, and the worst are exploited [Laozi (Lao Tzu)] |
| Full Idea: The best of all rulers is but a shadowy presence to his subjects; next comes the ruler they love and praise; next comes one they fear; next comes one with whom they take liberties. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], I.XVII.39) | |
| A reaction: This fits our understanding of football referees to perfection. It might apply to anyone doing a vital adminstrative job, such as compiling a school timetable. It is hard, though, to accept anonymity as a mark of success. |
| 6329 | People are hard to govern because authorities love to do things [Laozi (Lao Tzu)] |
| Full Idea: It is because those in authority are too fond of action that the people are difficult to govern. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.LXXV.181) | |
| A reaction: I love this. It should be on the wall of every human institution in our civilization. How the heart sinks at the prospect of a 'new initiative'. Not that I am against action; it is just important to recognise that inaction is sometimes the best option. |
| 6326 | The better known the law, the more criminals there are [Laozi (Lao Tzu)] |
| Full Idea: The better known the laws and edicts, the more thieves and robbers there are. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], II.LVII.132) | |
| A reaction: Nice. I link this with my favourite moral maxim from Democritus (Idea 519). The idea is that continual emphasis on what you should not do fills the mind with evil possibilities. Moral perfection must start by taking goodness for granted. |
| 23401 | A military victory is not a thing of beauty [Laozi (Lao Tzu)] |
| Full Idea: A military victory is not a thing of beauty. | |
| From: Laozi (Lao Tzu) (Daodejing (Tao Te Ching) [c.530 BCE], 31), quoted by Bryan van Norden - Intro to Classical Chinese Philosophy 8.II.1 | |
| A reaction: Should be written on the wall of every military academy and barracks. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |