30 ideas
| 1489 | Modern philosophy tends to be a theory-constructing extension of science, but there is also problem-solving [Nagel] |
| Full Idea: Philosophy is now dominated by a spirit of theory construction which sees philosophy as continuous with science, but the other problem-centred style is still in existence and it is important to keep it alive. | |
| From: Thomas Nagel (The Philosophical Culture [1995], §6) |
| 18543 | Do aesthetic reasons count as reasons, if they are rejectable without contradiction? [Scruton] |
| Full Idea: The judgement of beauty makes a claim about its object, and can be supported by reasons. But the reasons do not compel the judgement and can be rejected without contradiction. So are they reasons or aren't they? | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 1) | |
| A reaction: I suspect that what he is really referring to is evidence rather than reasons. |
| 18542 | Defining truth presupposes that there can be a true definition [Scruton] |
| Full Idea: How can you define truth, without already assuming the distinction between a true definition and a false one? | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 1) | |
| A reaction: Don't say we have to accept truth as yet another primitive! Philosophers are out of business if all the basic concepts are primitive. The axiomatic approach to truth is an alternative - by specifying how the primitive should be used. |
| 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) |
| 18546 | The pleasure taken in beauty also aims at understanding and valuing [Scruton] |
| Full Idea: Like the pleasure in friendship, the pleasure in beauty is curious: it aims to understand its object, and to value what it finds. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 1) | |
| A reaction: At least he is trying to pin down the way in which aesthetic pleasure is phenomenologically different from other kinds of pleasure. |
| 18550 | Art gives us imaginary worlds which we can view impartially [Scruton] |
| Full Idea: One aim of art is to present imaginary worlds, towards which we can adopt, as part of the integral aesthetic attitude, a posture of impartial concern. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 5) | |
| A reaction: It connects to the pleasure of watching people when they don't know they are being watched (such as watching the street from a restaurant window). Scruton's suggestion makes art resemble examples in philosophy. Cf the Frege-Geach problem in ethics. |
| 18544 | Maybe 'beauty' is too loaded, and we should talk of fittingness or harmony [Scruton] |
| Full Idea: Maybe we can understand the 'beauty' of a building better if we describe it in another and less loaded way, as a form of fittingness or harmony. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 1) | |
| A reaction: Almost everyone accepts the word 'beauty' for some things, such as a beautiful face, or goal, or steak. I remember a female interviewer writing that, reluctantly, the only appropriate word she could find for Nureyev's face was 'beautiful'. |
| 18553 | Beauty shows us what we should want in order to achieve human fulfilment [Scruton] |
| Full Idea: Beauty speaks to us of human fulfilment: not of things that we want, but of things that we ought to want, because human nature requires them. Such, at least, is my belief. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 7) | |
| A reaction: I'm not sure how this works with a beautiful natural landscape. And what should I see that I ought to desire after viewing a great Rembrandt self-portrait? That I don't want to end up looking as bleak as that? Hm. Lofty words. |
| 18556 | Beauty is rationally founded, inviting meaning, comparison and self-reflection [Scruton] |
| Full Idea: Beauty is rationally founded; it challenges us to find meaning in its object, to make critical comparisons, and to examine our own lives and emotions in the light of what we find. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 9) | |
| A reaction: This is the Kantian tradition, and I'm not finding it very persuasive. It seems to place the value of beauty in what we do with it afterwards, and he seems to make beauty a necessary stepping stone to virtue. I see beauty as more sui generis. |
| 18548 | Natural beauty reassures us that the world is where we belong [Scruton] |
| Full Idea: The experience of natural beauty is not a sense of 'how nice!' or 'how pleasant!' It contains a reassurance that this world is a right and fitting place to be - a home in which our human powers and prospects find confirmation. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 2) | |
| A reaction: To call it a 'reassurance' and 'confirmation' sounds like theism, anthropomorphism, or the pathetic fallacy. That said, this is certainly a heart-warming idea, and hence must contain a grain of truth. |
| 18551 | Croce says art makes inarticulate intuitions conscious; rival views say the audience is the main concern [Scruton] |
| Full Idea: The Croce model is of an inarticulate inner state (an 'intuition') becoming articulate and conscious through artistic expression. The rival model is fitting thing together so as to create links which resonate in the audience's feelings. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 5) | |
| A reaction: The first model tells you nothing about how the artist imagines the audience reacting. The second model tells you nothing about what matters personally to the artist. A good theory must do both! |
| 18541 | Beauty (unlike truth and goodness) is questionable as an ultimate value [Scruton] |
| Full Idea: The status of beauty as an ultimate value is questionable, in the way that the status of truth and goodness are not. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 1) | |
| A reaction: We suspect that a love of beauty may be a bit parochial, where it is hard to conceive of living creatures anywhere in the cosmos who don't value the other two. |
| 18554 | Prostitution is wrong because it hardens the soul, since soul and body are one [Scruton] |
| Full Idea: The condemnation of prostitution was not just puritan bigotry; it was a recognition of a profound truth, that you and your body are not two things but one, and by selling the body you harden your soul. | |
| From: Roger Scruton (Beauty: a very short introduction [2011], 7) | |
| A reaction: No one, I imagine, who condones or even enthuses about prostitution would hope that their own daughter followed the profession, so there is something wrong with it. But must an enthusiastic and cheerful prostitute necessarily have a hard soul? |