31 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] |
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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 = { |
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| 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) |
| 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. |
| 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) |
| 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) |
| 20429 | Most of us are too close to our own motives to understand them [Fry] |
| Full Idea: The motives we actually experience are too close to us to enable us to feel them clearly. They are in a sense unintelligible. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.30) | |
| A reaction: Fry is defending the role of art in clarifying and highlighting such things, but I am not convinced by his claim. We can grasp most of our motives with a little introspection, and those we can't grasp are probably too subtle for art as well. |
| 20424 | Imaginative life requires no action, so new kinds of perception and values emerge in art [Fry] |
| Full Idea: In the imaginative life no action is necessary, so the whole consciousness may be focused upon the perceptive and the emotional aspects of the experience. Hence we get a different set of values, and a different kind of perception | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.24) | |
| A reaction: Good. A huge range of human activities are like scientific experiments, where you draw on our evolved faculties, but put them in controlled conditions, where the less convenient and stressful parts are absent. War and sport. Real and theatrical tragedy. |
| 20427 | Everyone reveals an aesthetic attitude, looking at something which only exists to be seen [Fry] |
| Full Idea: It is only when an object exists for no other purpose than to be seen that we really look at it, …and then even the most normal person adopts to some extent the artistic attitude of pure vision abstracted from necessity. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.29) | |
| A reaction: A painter of still life looks at things which exist for other purposes, with just the attitude which Fry attributes to the viewers of the paintings. We can encourage a child to look at a flower with just this attitude. |
| 20433 | 'Beauty' can either mean sensuous charm, or the aesthetic approval of art (which may be ugly) [Fry] |
| Full Idea: There is an apparent contradiction between two distinct uses of the word 'beauty', one for that which has sensuous charm, and one for the aesthetic approval of works of imaginative art where the objects presented to us are often of extreme ugliness. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.33) | |
| A reaction: The gouging of eyes in 'King Lear' was always the big problem case for aesthetics, just as nowadays it is Marcel Duchamp's wretched 'Fountain'. |
| 20430 | In life we neglect 'cosmic emotion', but it matters, and art brings it to the fore [Fry] |
| Full Idea: Those feelings unhappily named cosmic emotion find almost no place in life, but, since they seem to belong to certain very deep springs of our nature, do become of great importance in the arts. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.31) | |
| A reaction: Focus on the sublime was big in the romantic era, but Fry still sees its importance, and I don't think it ever goes away. Art styles which scorn the sublime are failing to perform their social duty, say I. |
| 20431 | Art needs a mixture of order and variety in its sensations [Fry] |
| Full Idea: The first quality that we demand in our [artistic] sensations will be order, without which our sensations will be troubled and perplexed, and the other will be variety, without which they will not be fully stimulated. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.32) | |
| A reaction: He makes good claims, but gives unconvincing reasons for them. Some of us rather like 'troubled and perplexed' sensations. And a very narrow range of sensations could still be highly stimulated. Is Fry a good aesthetician but a modest philosopher? |
| 20423 | If graphic arts only aim at imitation, their works are only trivial ingenious toys [Fry] |
| Full Idea: If imitation is the sole purpose of the graphic arts, it is surprising that the works of such arts are ever looked upon as more than curiosities, or ingenious toys, and are ever taken seriously by grown-up people. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.23) | |
| A reaction: But then you might say that same about fine wines. A mere nice taste is hardly worthy of grown ups, and yet lots of grown ups feeling quite passionately about it. What about Fabergé eggs? |
| 20428 | Popular opinion favours realism, yet most people never look closely at anything! [Fry] |
| Full Idea: Ordinary people have almost no idea of what things really look like, so that the one standard that popular criticism applies to painting (whether it is like nature or not) is the one which most people are prevented frm applying properly. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.29) | |
| A reaction: A nice remark, though there is a streak of Bloomsbury artistic snobbery running through Fry. Ordinary people recognise photographic realism, so they can study things closely either in the reality or the picture, should they so choose. |
| 20432 | When viewing art, rather than flowers, we are aware of purpose, and sympathy with its creator [Fry] |
| Full Idea: In our reaction to a work of art (rather than a flower) there is the consciousness of purpose, of a peculiar relation of sympathy with the man who made this thing in order to arouse precisely the sensations we experience. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.33) | |
| A reaction: I think this is entirely right. I like the mention of 'sympathy' as well as 'purpose'. |
| 20425 | In the cinema the emotions are weaker, but much clearer than in ordinary life [Fry] |
| Full Idea: One notices in the visions of the cinematograph that whatever emotions are aroused by them, though they are likely to be weaker than those of ordinary life, are presented more clearly to the conscious. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.25) | |
| A reaction: Fry had probably only seen very simple melodramas, but the general idea that artistic emotions are weaker than real life, but much clearer, is quite plausible. |
| 20426 | For pure moralists art must promote right action, and not just be harmless [Fry] |
| Full Idea: To the pure moralist, accepting nothing but ethical values, to be justified, the life of the imagination must be shown not only not to hinder but actually to forward right action, otherwise it is not only useless but, by absorbing energies, harmful. | |
| From: Roger Fry (An Essay in Aesthetics [1909], p.26) | |
| A reaction: I think this is the sort of attitude you find in Samuel Johnson. Puritans even reject light music, which seems pleasantly harmless to the rest of us. 'Absorbing energies' doesn't sound much of an objection, and may not be the actual objection. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |