37 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) |
| 3772 | The will, in the beginning, is entirely produced by desire [Mill] |
| Full Idea: The will, in the beginning, is entirely produced by desire. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.4) | |
| A reaction: This is the sort of simplistic psychology that modern philosophers tend to avoid. Personally I am more Kantian. I will and desire that the answer to 3+2=? is 5, simply because it is true. Mill must realise we can will ourselves to desire something. |
| 3769 | With early training, any absurdity or evil may be given the power of conscience [Mill] |
| Full Idea: There is hardly anything so absurd or so mischievous that it may not, by means of early sanctions and influence, be made to act on the human mind with all the influence of conscience. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.3) | |
| A reaction: Like this! Think of all the people who have had weird upbringings, and end up feeling guilty about absurd things. Conscience just summarise upbringing and social conventions. |
| 3767 | Motive shows the worth of the agent, but not of the action [Mill] |
| Full Idea: The motive has nothing to do with the morality of the action, though much with the worth of the agent. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) | |
| A reaction: I think it is an error to try to separate these too sharply. Morality can't be purely consequential, because it would make earthquakes immoral. Actions indicate the worth of agents. |
| 3771 | Virtues only have value because they achieve some further end [Mill] |
| Full Idea: Utilitarians believe that actions and dispositions are only virtuous because they promote another end than virtue. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.4) |
| 3768 | Orthodox morality is the only one which feels obligatory [Mill] |
| Full Idea: The customary morality, that which education and opinion have consecrated, is the only one which presents itself to the mind with the feeling of being in itself obligatory. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.3) |
| 7202 | The English believe in the task of annihilating evil for the victory of good [Nietzsche on Mill] |
| Full Idea: One continues to believe in good and evil: in such a way that one feels the victory of good and the annihilation of evil to be a task (- this is English; a typical case is that shallow-headed John Stuart Mill). | |
| From: comment on John Stuart Mill (Utilitarianism [1861]) by Friedrich Nietzsche - Writings from Late Notebooks 11[148]e | |
| A reaction: The poor old English try very hard to be clear, sensible, practical and realistic, and get branded as 'shallow' for their pains. Nietzsche was a deeper thinker than Mill, but I would prefer Mill to Heidegger any day. |
| 5935 | Mill's qualities of pleasure is an admission that there are other good states of mind than pleasure [Ross on Mill] |
| Full Idea: Mill's introduction of quality of pleasures into the hedonistic calculus is an unconscious departure from hedonism and a half-hearted admission that there are other qualities than pleasantness in virtue of which states of mind are good. | |
| From: comment on John Stuart Mill (Utilitarianism [1861], Ch.2) by W. David Ross - The Right and the Good §VI | |
| A reaction: Mill argues that experienced people prefer some pleasures to others, but ducks the question of why they might prefer them. It can only be because they have some further desirable quality on top of the equal amount of pleasure. |
| 3764 | Actions are right if they promote pleasure, wrong if they promote pain [Mill] |
| Full Idea: The Greatest Happiness Principle holds that actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness. By happiness is intended pleasure, and the absence of pain. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) |
| 3776 | Utilitarianism only works if everybody has a totally equal right to happiness [Mill] |
| Full Idea: The Greatest Happiness Principle is a mere form of empty words unless one person's happiness, supposed equal in degree, is counted for exactly as much as another's (Bentham's "everybody to count for one, nobody for more than one"). | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |
| 3765 | Only pleasure and freedom from pain are desirable as ends [Mill] |
| Full Idea: Pleasure and freedom from pain are the only things desirable as ends. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) |
| 3763 | Ultimate goods such as pleasure can never be proved to be good [Mill] |
| Full Idea: What can be proved good must be so by being shown to be a means to something admitted to be good without proof. Music is good because it produces pleasure, but what proof is it possible to give that pleasure is good? | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.1) |
| 3766 | Better to be Socrates dissatisfied than a fool satisfied [Mill] |
| Full Idea: Better to be Socrates dissatisfied than a fool satisfied. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.2) |
| 3770 | General happiness is only desirable because individuals desire their own happiness [Mill] |
| Full Idea: No reason can be given why the general happiness is desirable, except that each person, so far as he believes it to be attainable, desires his own happiness. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.4) |
| 6697 | Moral rules protecting human welfare are more vital than local maxims [Mill] |
| Full Idea: Moral rules which forbid mankind to hurt one another are more vital to human well-being than any maxims about some department of human affairs; ..though in particular cases a social duty is so important, as to overrule any general maxim of justice. | |
| From: John Stuart Mill (Utilitarianism [1861]), quoted by Gordon Graham - Eight Theories of Ethics Ch.7 | |
| A reaction: The qualification is realistic, but raises the question of whether an 'act' calculation will always overrule any 'rule'. Maybe rule utilitirianism is just act utilitarianism, but ensuring that the calculations are long-term and extensive. (1871 edn) |
| 3774 | Rights are a matter of justice, not of benevolence [Mill] |
| Full Idea: Wherever there is a right, the case is one of justice, and not of the virtue of benevolence. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |
| 3773 | No individual has the right to receive our benevolence [Mill] |
| Full Idea: No one has a moral right to our generosity or beneficence, because we are not morally bound to practise those virtues towards any given individual. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |
| 3775 | A right is a valid claim to society's protection [Mill] |
| Full Idea: When we call anything a person's right, we mean that he has a valid claim on society to protect him in the possession of it. | |
| From: John Stuart Mill (Utilitarianism [1861], Ch.5) |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |