33 ideas
| 22864 | Philosophy is the study and criticsm of cultural beliefs, to achieve new possibilities [Dewey] |
| Full Idea: Philosophy is criticism of the influential beliefs that underlie culture, tracking them to their generating conditions and results, and considering their mutual compatibility. This terminates in a new perspective, which leads to new possibilities. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 6:19), quoted by David Hildebrand - Dewey Intro | |
| A reaction: [compressed] This would make quite a good manifesto for French thinkers of the 1960s. Foucault could hardly disagree. An excellent idea. |
| 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. |
| 22873 | Liberalism should improve the system, and not just ameliorate it [Dewey] |
| Full Idea: Liberalism must become radical in the sense that, instead of using social power to ameliorate the evil consequences of the existing system, it shall use social power to change the system. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 11:287), quoted by David Hildebrand - Dewey 4 'Dewey' | |
| A reaction: Conservative liberals ask what people want, and try to give it to them. Radical liberals ask what people actually need, and try to make it possible. The latter is bound to be a bit paternalistic, but will probably create a better world. |
| 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) |
| 13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
| Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2 | |
| A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths). |
| 9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara] |
| Full Idea: Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1 | |
| A reaction: Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries. |
| 18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew] |
| Full Idea: In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2 | |
| A reaction: The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy. |
| 18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H] |
| Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Hartry Field - Science without Numbers 3 | |
| A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field. |
| 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) |
| 22869 | Knowledge is either the product of competent enquiry, or it is meaningless [Dewey] |
| Full Idea: Knowledge, as an abstract term, is a name for the product of competent enquiries. Apart from this relation, its meaning is so empty that any content or filling may be arbitrarily poured into it. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 12:16), quoted by David Hildebrand - Dewey 2 'Knowledge' | |
| A reaction: What is the criterion of 'competent'? Danger of tautology, if competent enquiry is what produces knowledge. |
| 22867 | The quest for certainty aims for peace, and avoidance of the stress of action [Dewey] |
| Full Idea: The quest for certainty is a quest for a peace which is assured, an object which is unqualified by risk and the shadow of fear which action costs. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 4:7), quoted by David Hildebrand - Dewey 2 'Intro' | |
| A reaction: This is a characteristic pragmatist account. I think Dewey and Peirce offer us the correct attitude to certainty. It is just not available to us, and can only be a delusion. That doesn't mean we don't know anything, however! |
| 22870 | No belief can be so settled that it is not subject to further inquiry [Dewey] |
| Full Idea: The attainment of settled beliefs is a progressive matter; there is no belief so settled as not to be exposed to further inquiry. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 12:16), quoted by David Hildebrand - Dewey 2 'Knowledge' | |
| A reaction: A nice pragmatist mantra, but no scientists gets a research grant to prove facts which have been securely established for a very long time. It is neurotic to keep returning to check that you have locked your front door. Dewey introduced 'warranted'. |
| 22866 | Mind is never isolated, but only exists in its interactions [Dewey] |
| Full Idea: Mind is primarily a verb. ...Mind never denotes anything self-contained, isolated from the world of persons and things, but is always used with respect to situations, events, objects, persons and groups. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 10:267), quoted by David Hildebrand - Dewey 1 'emerge' | |
| A reaction: I strongly agree with the idea that mind is a process, not a thing. Certain types of solitary introspection don't seem to quite fit his account, but in general he is right. |
| 22872 | Liberals aim to allow individuals to realise their capacities [Dewey] |
| Full Idea: Liberalism is committed to …the liberation of individuals so that realisation of their capacities may be the law of their life. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 11:41), quoted by David Hildebrand - Dewey 4 'Dewey' | |
| A reaction: Capacity expression as the main aim of politics is precisely the idea developed more fully in modern times by Amartya Sen and Martha Nussbaum. It strikes me as an excellent proposal. Does it need liberalism, or socialism? |
| 22880 | The things in civilisation we prize are the products of other members of our community [Dewey] |
| Full Idea: The things in civilisation we most prize are not of ourselves. They exist by grace of the doings and sufferings of the continuous human community in which we are a link | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:57), quoted by David Hildebrand - Dewey 7 'Reconstruct' | |
| A reaction: Dewey defends liberalism, but he has strong communitarian tendencies. What is the significance of an enduring community losing touch with its own achievements? |
| 22879 | 'God' is an imaginative unity of ideal values [Dewey] |
| Full Idea: 'God' represents a unification of ideal values that is essentially imaginative in origin. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:29), quoted by David Hildebrand - Dewey 7 'Construct' | |
| A reaction: This seems to have happened when a flawed God like Zeus is elevated to be the only God, and is given supreme power and wisdom. |
| 22877 | We should try attaching the intensity of religious devotion to intelligent social action [Dewey] |
| Full Idea: One of the few experiments in the attachment of emotion to ends that mankind has not tried is that of devotion (so intense as to be religious) to intelligence as a force in social action. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:53), quoted by David Hildebrand - Dewey 7 'Intro' | |
| A reaction: An interesting thought that religious emotions such as devotion are so distinctive that they can be treated as valuable, even in the absence of belief. He seems to be advocating Technocracy. |
| 22878 | Religions are so shockingly diverse that they have no common element [Dewey] |
| Full Idea: There is only a multitude of religions …and the differences between them are so great and so shocking that any common element that can be extracted is meaningless. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:7), quoted by David Hildebrand - Dewey 7 'Construct' | |
| A reaction: Religion is for Dewey what a game was for Wittgenstein, as an anti-essentialist example. I would have thought that they all involved some commitment to a realm of transcendent existence. |