44 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) |
| 22355 | In the realist view, the real external world explains how it (and perceptions of it) are possible [Williams,B] |
| Full Idea: The substance of the absolute conception [of external reality] lies in the idea that it could nonvacuously explain how it itself, and the various perspectival views of the world, are possible. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], p.139), quoted by Reiss,J/Spreger,J - Scientific Objectivity 2.1 | |
| A reaction: I like this. Explanation and understanding strike me as more important than justified truths, and I am struck by the complete inability of subjectivists, relativists and anti-realists to give any kinds of good explanation. |
| 4244 | It is very confused to deduce a nonrelativist morality of universal toleration from relativism [Williams,B] |
| Full Idea: Some people believe a properly relativist view requires you to be equally well disposed to everybody's ethical beliefs, but this is seriously confused, as relativism has led to a nonrelativist morality of universal toleration. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Good point. This need not stop a relativist from passionately defending tolerance - it is only that the lack of rational support for the passion must be recognised. |
| 4243 | Our ability to react to an alien culture shows that ethical thought extends beyond cultural boundaries [Williams,B] |
| Full Idea: The fact that people can and must react when confronted with another culture, and do so by applying existing notions, seems to show that ethical thought of a given culture can always stretch beyond its boundaries. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Hardly conclusive, but it does seem to show that there is an element of universalising in values, no matter how local you may consider them to be. |
| 4114 | Philosophers try to produce ethical theories because they falsely assume that ethics can be simple [Williams,B] |
| Full Idea: If there is a truth about the subject matter of ethics, why should it be simple? ..I shall argue that philosophy should not try to produce ethical theories. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: Bizarrely defeatist - in parallel with Mysterians about the mind like McGinn. Is there any point in thinking at all? I suggest the aim of life as the best starting point. |
| 4128 | Intuitionism has been demolished by critics, and no longer looks interesting [Williams,B] |
| Full Idea: Intuitionism in ethics has been demolished by a succession of critics, and the ruins of it that remain above ground are not impressive enough to invite much history of what happened to it. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: Why does intuitionism have such appeal to beginners in moral philosophy? There is a truth buried in it somewhere. See 'Sources of the Self' by Charles Taylor. |
| 4132 | The category of person is a weak basis for ethics, because it is not fixed but comes in degrees [Williams,B] |
| Full Idea: The category of person is a poor foundation for ethical thought, because it looks like a sortal or classificatory notion while in fact it signals characteristics that almost all come in degrees (responsibility, self-reflection etc). | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: On the contrary, it must be the basis of moral theory, and its shifting character is strong support for Aristotle's approach to moral growth and responsibility. |
| 4134 | The weakness of prescriptivism is shown by "I simply don't like staying at good hotels" [Williams,B] |
| Full Idea: That "I simply don't like staying at good hotels" is intelligible brings out the basic weakness of prescriptive accounts of the evaluative. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 7) | |
| A reaction: This might be an elision of two different prescriptions, mine and most people's. In what sense do I think the hotel good, as opposed to other people? |
| 4135 | Some ethical ideas, such as 'treachery' and 'promise', seem to express a union of facts and values [Williams,B] |
| Full Idea: Some 'thicker' ethical notions, such as 'treachery', 'promise', 'brutality' and 'courage', seem to express a union of facts and values. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 8) | |
| A reaction: The onus does seem to be on the followers of Hume to disentangle what the rest of us have united. They may, of course, manage it. |
| 4120 | It is an error of consequentialism to think we just aim at certain states of affairs; we also want to act [Williams,B] |
| Full Idea: We do not merely want the world to contain certain states of affairs (it is a deep error of consequentialism to believe that this is all we want). Among the things we basically want is to act in certain ways. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 4) | |
| A reaction: A key objection. Does it matter whether Hiroshima is destroyed by earthquake or bombing? |
| 4252 | Promise keeping increases reliability, by making deliberation focus on something which would be overlooked [Williams,B] |
| Full Idea: The institution of promise keeping operates to provide portable reliability, by offering a formula that will confer high deliberative priority on what might not otherwise receive it. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: This is a bit pessimistic. We do not perceive promise keeping as a mere suggestion that we should bear something in mind when making a decision. 'May I rot in hell if I fail you'. |
| 4116 | A weakness of contractual theories is the position of a person of superior ability and power [Williams,B] |
| Full Idea: A particular weakness of the contractual theory is that it is unstable with respect to a superior agent, one more intelligent and resourceful and persuasive than the rest. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 3) | |
| A reaction: The very weak are equally a problem. Democratic societies produce fewer inequalities. Hierarchical societies are miserable (I expect..). |
| 4112 | A crucial feature of moral thought is second-order desire - the desire to have certain desires [Williams,B] |
| Full Idea: Recently there has been much emphasis on the importance of our capacity to have second-order desires - the desire to have certain desires - and its significance for ethical reflection and the practical consciousness. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: This is a crucial point if we are to defend a reasonably rational view of morality against (say) emotivism. I agree that it is crucial to morality. |
| 4113 | 'Deon' in Greek means what one must do; there was no word meaning 'duty' [Williams,B] |
| Full Idea: There is no ancient Greek word for duty; the word 'deon' (the basis of 'deontology') means what one must do. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: Presumably it covered compulsions which were not duties, such as the need to eat or drink. Greeks thought morally, but lacked a good moral vocabulary? |
| 4250 | The concept of a 'duty to myself' is fraudulent [Williams,B] |
| Full Idea: The concept of a 'duty to myself' is fraudulent. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: The only person who can offer a rebuttal of this is Aristotle. With the magnet of the Platonic Form of the Good, I can perceive the natural excellences of which I am capable, and feel a duty to pursue them. |
| 4110 | Obligation and duty look backwards (because of a promise or job), although the acts are in the future [Williams,B] |
| Full Idea: Obligation and duty look backwards; the acts they require lie in the future, but the reasons for those acts lie in the fact that I have already promised, the job I have undertaken, the position I am already in. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: Maybe the central issue in morality is forwards versus backwards. It reflects two types of human temperament. Tomorrow is another day. Spilt milk. |
| 4249 | "Ought implies can" is a famous formula in connection with moral obligation [Williams,B] |
| Full Idea: "Ought implies can" is a famous formula in connection with moral obligation. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: Williams says it is true in particular instances, but is not generally true of 'ought'. Maybe you 'ought' before you know whether you 'can'. |
| 4248 | Not all moral deliberations lead to obligations; some merely reveal what 'may' be done [Williams,B] |
| Full Idea: Not every conclusion of moral deliberation expresses an obligation; for example, some moral conclusions merely announce that you 'may' do something. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: An important point for any deontological ethics. It may be possible to translate what 'may' be done into some form of duty, but it will probably involve contortions. |
| 4121 | Why should I think of myself as both the legislator and the citizen who follows the laws? [Williams,B] |
| Full Idea: Why should I think of myself as a legislator and at the same time a citizen of a republic governed by some notional laws? | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 4) | |
| A reaction: Kant's answer is supposed to be 'because you are rational, and hence must want consistency'. If we were all rational, Kant would be right. |
| 4122 | If the self becomes completely impartial, it no longer has enough identity to worry about its interests [Williams,B] |
| Full Idea: How can an 'I' that has taken on the perspective of impartiality be left with enough identity to live a life that respects its own interests? | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 4) | |
| A reaction: Not a big problem. Thought constantly flips between objective and subjective, as Nagel has shown us. Compare Nagel in Idea 6446. |
| 4124 | Utilitarian benevolence involves no particular attachments, and is immune to the inverse square law [Williams,B] |
| Full Idea: Utilitarian benevolence involves no particular attachments, and it is immune to the inverse square law. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 5) | |
| A reaction: Nicely put. The point is that the theory is inhuman, but Mill says it tells us what we should do, not what we actually tend to do. |
| 4245 | Ethical conviction must be to some extent passive, and can't just depend on the will and decisions [Williams,B] |
| Full Idea: The view that the only alternative to the intellect is the will, and the source of ethical conviction is decisions about principles and ways of life, cannot be right; ethical conviction, like any conviction, must to some extent come to you passively. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Seems right. We cannot choose our factual beliefs (look at the sun and believe it is cloudy!). Could I 'decide' that it was right to betray my family just for fun? |
| 4246 | Taking responsibility won't cure ethical uncertainty by; we are uncertain what to decide [Williams,B] |
| Full Idea: If ethics is a matter of decision, and we must face the responsibility and burden of those decisions, this ignores the obvious point that if we are uncertain, then we are uncertain what to decide. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Good point. The defence would be that the decision itself contains the seeds of certainty. Do something rather than nothing, and the sense of it will emerge. Modify as you go along. |
| 4247 | It is a mark of our having ethical values that we aim to reproduce them in our children [Williams,B] |
| Full Idea: It is a mark of our having ethical values that we aim to reproduce them in our children. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Maybe beliefs imply education. A commitment to truth is an aspiration that others will agree, especially those over whom we have the greatest influence. |
| 4131 | Most women see an early miscarriage and a late stillbirth as being very different in character [Williams,B] |
| Full Idea: Few women see a spontaneous abortion or early miscarriage as the same thing as having a child who is stillborn or who dies very soon after birth. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: This implies a theory about the nature of what is lost. Everyone sees the difference between potential and actual. |
| 4133 | Speciesism isn't like racism, because the former implies a viewpoint which belongs to no one [Williams,B] |
| Full Idea: Speciesism is falsely modelled on racism and sexism, which really are prejudices; ..our arguments have to be founded on the human point of view; they cannot be derived from a point of view that is no one's point of view at all. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: This must be wrong. How else are we going to judge cruelty to animals as wrong? The 'point of view of the Universe' (Sidgwick) is not an empty concept. |
| 15877 | The aim of science is just to create a comprehensive, elegant language to describe brute facts [Poincaré, by Harré] |
| Full Idea: In Poincaré's view, we try to construct a language within which the brute facts of experience are expressed as comprehensively and as elegantly as possible. The job of science is the forging of a language precisely suited to that purpose. | |
| From: report of Henri Poincaré (The Value of Science [1906], Pt III) by Rom Harré - Laws of Nature 2 | |
| A reaction: I'm often struck by how obscure and difficult our accounts of self-evident facts can be. Chairs are easy, and the metaphysics of chairs is hideous. Why is that? I'm a robust realist, but I like Poincaré's idea. He permits facts. |