32 ideas
| 10888 | Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo] |
| Full Idea: We can define a set by 'enumeration' (by listing the items, within curly brackets), or by 'abstraction' (by specifying the elements as instances of a property), pretending that they form a determinate totality. The latter is written {x | x is P}. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) |
| 10889 | The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo] |
|
Full Idea:
The 'Cartesian Product' of two sets, written A x B, is the relation which pairs every element of A with every element of B. So A x B = { |
|
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
| 10890 | A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo] |
| Full Idea: A binary relation in a set is a 'partial ordering' just in case it is reflexive, antisymmetric and transitive. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
| 10886 | Determinacy: an object is either in a set, or it isn't [Zalabardo] |
| Full Idea: Principle of Determinacy: For every object a and every set S, either a is an element of S or a is not an element of S. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.2) |
| 10887 | Specification: Determinate totals of objects always make a set [Zalabardo] |
| Full Idea: Principle of Specification: Whenever we can specify a determinate totality of objects, we shall say that there is a set whose elements are precisely the objects that we have specified. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) | |
| A reaction: Compare the Axiom of Specification. Zalabardo says we may wish to consider sets of which we cannot specify the members. |
| 10897 | A first-order 'sentence' is a formula with no free variables [Zalabardo] |
| Full Idea: A formula of a first-order language is a 'sentence' just in case it has no free variables. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.2) |
| 10899 | Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo] |
| Full Idea: A formula is the 'logical consequence' of a set of formulas (Γ |= φ) if for every structure in the language and every variable interpretation of the structure, if all the formulas within the set are true and the formula itself is true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10893 | Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo] |
| Full Idea: A propositional logic sentence is a 'logical consequence' of a set of sentences (written Γ |= φ) if for every admissible truth-assignment all the sentences in the set Γ are true, then φ is true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
| A reaction: The definition is similar for predicate logic. |
| 10896 | Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo] |
| Full Idea: In propositional logic, any set containing ¬ and at least one of ∧, ∨ and → is expressively complete. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.8) |
| 10898 | The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo] |
| Full Idea: The semantic pattern of a first-order language is the ways in which truth values depend on which individuals instantiate the properties and relations which figure in them. ..So we pair a truth value with each combination of individuals, sets etc. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.3) | |
| A reaction: So truth reduces to a combination of 'instantiations', which is rather like 'satisfaction'. |
| 10902 | We can do semantics by looking at given propositions, or by building new ones [Zalabardo] |
| Full Idea: We can look at semantics from the point of view of how truth values are determined by instantiations of properties and relations, or by asking how we can build, using the resources of the language, a proposition corresponding to a given semantic pattern. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) | |
| A reaction: The second version of semantics is model theory. |
| 10892 | We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo] |
| Full Idea: A truth assignment is a function from propositions to the set {T,F}. We will think of T and F as the truth values true and false, but for our purposes all we need to assume about the identity of these objects is that they are different from each other. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
| A reaction: Note that T and F are 'objects'. This remark is important in understanding modern logical semantics. T and F can be equated to 1 and 0 in the language of a computer. They just mean as much as you want them to mean. |
| 10895 | 'Logically true' (|= φ) is true for every truth-assignment [Zalabardo] |
| Full Idea: A propositional logic sentence is 'logically true', written |= φ, if it is true for every admissible truth-assignment. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
| 10900 | Logically true sentences are true in all structures [Zalabardo] |
| Full Idea: In first-order languages, logically true sentences are true in all structures. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10894 | A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true [Zalabardo] |
| Full Idea: A propositional logic set of sentences Γ is 'satisfiable' if there is at least one admissible truth-assignment that makes all of its sentences true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
| 10901 | Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo] |
| Full Idea: A set of formulas of a first-order language is 'satisfiable' if there is a structure and a variable interpretation in that structure such that all the formulas of the set are true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10903 | A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo] |
| Full Idea: A structure is a model of a sentence if the sentence is true in the model; a structure is a model of a set of sentences if they are all true in the structure. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) |
| 10891 | If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] |
| Full Idea: Defining a set by induction enables us to use the method of proof by induction to establish that all the elements of the set have a certain property. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.3) |
| 7352 | Jesus said learning was unnecessary, and only the spirit of the Law was needed [Jesus, by Johnson,P] |
| Full Idea: Jesus was a learned Jew who said that learning was not necessary, who took the spirit and not the letter as the essence of the Law. | |
| From: report of Jesus (reports [c.32]) by Paul Johnson - The History of the Jews Pt II | |
| A reaction: This seems to me the perfect opposite of Socrates's intellectualism, offering the essence of morality as 'purity of heart', rather than careful thought about virtue or principles. On the whole I am with Socrates, but the idea is interesting. |
| 6292 | Love thy neighbour as thyself [Jesus] |
| Full Idea: Love thy neighbour as thyself. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 19.19 | |
| A reaction: It would be stronger and better to say 'Love your neighbour, even if you don't love yourself'. Self-loathing and vicious hatred often go together. For once Jesus does not attach an instant heavenly reward to obedience of the command. |
| 6289 | Love your enemies [Jesus] |
| Full Idea: Love your enemies. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 05.44 | |
| A reaction: The germ of this idea had been around for several hundred years, but this very forceful statement is perhaps Jesus' most distinctive contribution to moral thought. It has the same clarion call as the Stoic demand for pure virtue. What about deserving? |
| 5356 | Treat others as you would have them treat you [Jesus] |
| Full Idea: All things whatsoever ye would that men should do to you, so ye even so to them: for this is the law and the prophets. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 07.12 | |
| A reaction: A problem which probably didn't occur to Jesus and the prophets is that of masochists. Personally I like buying philosophy books, but most people have no such desire. The Rule needs restricting to the basics of pleasure and pain. |
| 6286 | Blessed are the merciful: for they shall obtain mercy [Jesus] |
| Full Idea: Blessed are the merciful: for they shall obtain mercy. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 05.07 | |
| A reaction: This appears to be a straightforward application of social contract morality, with God playing the role of Hobbes' absolute monarch. It highlights the uncomfortable fact at the heart of Christian morality, that the motivation for altruism is selfish. |
| 6290 | Except ye become as little children, ye shall not enter heaven [Jesus] |
| Full Idea: Except ye be converted, and become as little children, ye shall not enter into the kingdom of heaven. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 18.03 | |
| A reaction: The appeal of such purity of heart is undeniable, but essentially I dislike this remark. It is the opponent of education, reason, autonomy, responsibility, democracy and maturity. It confirms the view that religion is the opium of the people. |
| 6287 | If you lust after a woman, you have committed adultery [Jesus] |
| Full Idea: Whosoever look on a woman to lust after her hath committed adultery with her already in his heart. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 05.28 | |
| A reaction: Compare Democritus, Idea 503. Literally this idea seems absurd, but it is also at the heart of Greek virtue theory. Aristotle (Idea 34) defines virtue as an activity 'of the soul', not an action in the world. Excellence has become purity of soul. |
| 6285 | Blessed are the meek; for they shall inherit the earth [Jesus] |
| Full Idea: Blessed are the meek; for they shall inherit the earth. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 05.05 | |
| A reaction: If they are truly meek, why would they want to inherit the earth? This is the classic statement of Nietzsche's 'inversion of values', where the qualities of a good slave are elevated above those of the greatest human beings. |
| 6288 | Don't resist evil, but turn the other cheek [Jesus] |
| Full Idea: Ye have heard it said, An eye for an eye, and a tooth for a tooth; But I say unto you, That ye resist not evil, but whosoever shall smite thee on thy right cheek, turn to him the other also. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 05.38-9 | |
| A reaction: Compare Socrates, Idea 346. The viciousness of many Hollywood movies is that they set up a character as thoroughly evil so that we can have the pleasure of watching him pulverised. On the whole, Jesus gives bad advice. 'Doormats' in game theory. |
| 6293 | It is almost impossible for the rich to go to heaven [Jesus] |
| Full Idea: It is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 19.24 | |
| A reaction: Aristotle and others (Margaret Thatcher) have observed that you cannot practise charity if you are poor. Jesus implies that the human race should remain in poverty. No wonder autocratic medieval rulers taught Christianity to peasants. Cf. Matt 25.30. |
| 8326 | Science has shown that causal relations are just transfers of energy or momentum [Fair, by Sosa/Tooley] |
| Full Idea: Basic causal relations can, as a consequence of our scientific knowledge, be identified with certain physicalistic [sic] relations between objects that can be characterized in terms of transference of either energy or momentum between objects. | |
| From: report of David Fair (Causation and the Flow of Energy [1979]) by E Sosa / M Tooley - Introduction to 'Causation' §1 | |
| A reaction: Presumably a transfer of momentum is a transfer of energy. If only anyone had the foggiest idea what energy actually is, we'd be doing well. What is energy made of? 'No identity without substance', I say. I like Fair's idea. |
| 10379 | Fair shifted his view to talk of counterfactuals about energy flow [Fair, by Schaffer,J] |
| Full Idea: Fair, who originated the energy flow view of causation, moved to a view that understands connection in terms of counterfactuals about energy flow. | |
| From: report of David Fair (Causation and the Flow of Energy [1979]) by Jonathan Schaffer - The Metaphysics of Causation 2.1.2 | |
| A reaction: David Fair was a pupil of David Lewis, the king of the counterfactual view. To me that sounds like a disappointing move, but it is hard to think that a mere flow of energy through space would amount to causation. Cause must work back from an effect. |
| 6291 | No one is good except God [Jesus] |
| Full Idea: Why callest thou me good? There is none good but one, that is, God. | |
| From: Jesus (reports [c.32]), quoted by St Matthew - 01: Gospel of St Matthew 19.17 | |
| A reaction: This remark raises the problem that if God is good, there must be some separate moral standard by which he can be judged good. What is that standard? It is related to the problem of whether Plato's Form of the Beautiful is itself beautiful. |
| 7351 | Jesus turned the ideas of Hillel into a theology reduced to its moral elements [Jesus, by Johnson,P] |
| Full Idea: Jesus was a member of the school of Hillel the Elder, and may have sat under him. He repeated some of the sayings of Hillel, ...and turned his ideas into a moral theology, stripping the law of all but its moral and ethical elements. | |
| From: report of Jesus (reports [c.32]) by Paul Johnson - The History of the Jews Pt II | |
| A reaction: The crucial move, it seems to me, is to strip Judaism of its complexity, and reduce it to very simple moral maxims, which means that ordinary illiterate people no longer need priests to understand and follow it. Jesus was, above all, a great teacher. |