42 ideas
| 14970 | Normal system K has five axioms and rules [Cresswell] |
| Full Idea: Normal propositional modal logics derive from the minimal system K: wffs of PC are axioms; □(p⊃q)⊃(□p⊃□q); uniform substitution; modus ponens; necessitation (α→□α). | |
| From: Max J. Cresswell (Modal Logic [2001], 7.1) |
| 14971 | D is valid on every serial frame, but not where there are dead ends [Cresswell] |
| Full Idea: If a frame contains any dead end or blind world, then D is not valid on that frame, ...but D is valid on every serial frame. | |
| From: Max J. Cresswell (Modal Logic [2001], 7.1.1) |
| 14972 | S4 has 14 modalities, and always reduces to a maximum of three modal operators [Cresswell] |
| Full Idea: In S4 there are exactly 14 distinct modalities, and any modality may be reduced to one containing no more than three modal operators in sequence. | |
| From: Max J. Cresswell (Modal Logic [2001], 7.1.2) | |
| A reaction: The significance of this may be unclear, but it illustrates one of the rewards of using formal systems to think about modal problems. There is at least an appearance of precision, even if it is only conditional precision. |
| 14973 | In S5 all the long complex modalities reduce to just three, and their negations [Cresswell] |
| Full Idea: S5 contains the four main reduction laws, so the first of any pair of operators may be deleted. Hence all but the last modal operator may be deleted. This leaves six modalities: p, ◊p, □p, and their negations. | |
| From: Max J. Cresswell (Modal Logic [2001], 7.1.2) |
| 14976 | Reject the Barcan if quantifiers are confined to worlds, and different things exist in other worlds [Cresswell] |
| Full Idea: If one wants the quantifiers in each world to range only over the things that exist in that world, and one doesn't believe that the same things exist in every world, one would probably not want the Barcan formula. | |
| From: Max J. Cresswell (Modal Logic [2001], 7.2.2) | |
| A reaction: I haven't quite got this, but it sounds to me like I should reject the Barcan formula (but Idea 9449!). I like a metaphysics to rest on the actual world (with modal properties). I assume different things could have existed, but don't. |
| 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) |
| 17377 | All descriptive language is classificatory [Dupré] |
| Full Idea: Classification pervades any descriptive use of language whatever. | |
| From: John Dupré (The Disorder of Things [1993], 1) | |
| A reaction: This is because, as Aristotle well knew, language consists almost entirely of universals (apart from the proper names). Language just is classification. |
| 17376 | We should aim for a classification which tells us as much as possible about the object [Dupré] |
| Full Idea: The most important desideratum of a classificatory scheme is that assigning an object to a particular classification tell us as much as possible about that object. | |
| From: John Dupré (The Disorder of Things [1993], Ch 1) | |
| A reaction: We should probably say that the aim is a successful explanation, rather than a heap of information. If we are totally baffled by a particular type of object, it is presumably important to group the instances together, to focus the bafflement. |
| 14974 | A relation is 'Euclidean' if aRb and aRc imply bRc [Cresswell] |
| Full Idea: A relation is 'Euclidean' if aRb and aRc imply bRc. | |
| From: Max J. Cresswell (Modal Logic [2001], 7.1.2) | |
| A reaction: If a thing has a relation to two separate things, then those two things will also have that relation between them. If I am in the same family as Jim and as Jill, then Jim and Jill are in the same family. |
| 17390 | Natural kinds don't need essentialism to be explanatory [Dupré] |
| Full Idea: The importance of natural kinds for explanation does not depend on a doctrine of essences. | |
| From: John Dupré (The Disorder of Things [1993], 3) | |
| A reaction: He suggest as the alternative that laws do the explaining, employing natural kinds. He allows that individual essences might be explanatory. |
| 17389 | A species might have its essential genetic mechanism replaced by a new one [Dupré] |
| Full Idea: Contradicting one of the main points of essentialism, there is no reason in principle why a species should not survive the demise of its current genetic mechanisms (some other species coherence gradually taking over). | |
| From: John Dupré (The Disorder of Things [1993], 2) | |
| A reaction: I would say that this meant that the species had a new essence, because I don't take what is essential to be the same as what is necessary. The new genetics would replace the old as the basic explanation of the species. |
| 17388 | It seems that species lack essential properties, so they can't be natural kinds [Dupré] |
| Full Idea: It is widely agreed among biologists that no essential property can be found to demarcate species, so that if an essential property is necessary for a natural kind, species are not natural kinds. | |
| From: John Dupré (The Disorder of Things [1993], 2) | |
| A reaction: This uses 'essential' to mean 'necessary', but I would use 'essential' to mean 'deeply explanatory'. Biological species are, nevertheless, dubious members of an ontological system. Vegetables are the problem. |
| 14975 | A de dicto necessity is true in all worlds, but not necessarily of the same thing in each world [Cresswell] |
| Full Idea: A de dicto necessary truth says that something is φ, that this proposition is a necessary truth, i.e. that in every accessible world something (but not necessarily the same thing in each world) is φ. | |
| From: Max J. Cresswell (Modal Logic [2001], 7.2.1) | |
| A reaction: At last, a really clear and illuminating account of this term! The question is then invited of what is the truthmaker for a de dicto truth, assuming that the objects themselves are truthmakers for de re truths. |
| 17374 | The possibility of prediction rests on determinism [Dupré] |
| Full Idea: Determinism is the metaphysical underlay of the possibility of prediction. | |
| From: John Dupré (The Disorder of Things [1993], Intro) | |
| A reaction: Not convinced. There might be micro-indeterminacies which iron out into macro-regularities. |
| 17378 | Presumably molecular structure seems important because we never have the Twin Earth experience [Dupré] |
| Full Idea: It is surely the absence of experiences like the one Putnam describes that makes it reasonable to attach to molecular structure at least most of the importance that Putnam ascribes to it. | |
| From: John Dupré (The Disorder of Things [1993], 1) | |
| A reaction: That is, whenever we experience water-like stuff, it always turns out to have the same molecular structure. Twin Earth is a nice thought experiment, except that XZY is virtually inconceivable. |
| 17381 | Phylogenetics involves history, and cladism rests species on splits in lineage [Dupré] |
| Full Idea: The phylogenetic conception of classification reflects the facts of evolutionary history. Cladism insists that every taxonomic distinction should reflect an evolutionary event of lineage bifurcation. | |
| From: John Dupré (The Disorder of Things [1993], 1) | |
| A reaction: Devitt attacks cladism nicely. It rules out species change without bifurcation, and it insists on species change even in a line which remains unchanged after a split. |
| 17385 | Kinds don't do anything (including evolve) because they are abstract [Dupré] |
| Full Idea: A kind, being an abstract object, cannot do anything, including evolve. | |
| From: John Dupré (The Disorder of Things [1993], 2) | |
| A reaction: Maybe. We might have an extensional view of the kind, so that 'gold' is the set of extant gold atoms. But possible gold atoms are also gold, and defunct ones too. Virtually every word in English is abtract if you think about it long enough. |
| 17375 | Natural kinds are decided entirely by the intentions of our classification [Dupré] |
| Full Idea: The question of which natural kind a thing belongs to ....can be answered only in relation to some specification of the goal underlying the intent to classify the object. | |
| From: John Dupré (The Disorder of Things [1993], Intro) | |
| A reaction: I don't think I believe this. The situation is complex, and our intents are relevant, but to find an intent which no longer classifies tigers into the same category is wilful silliness. |
| 17379 | Borders between species are much less clear in vegetables than among animals [Dupré] |
| Full Idea: The richest source of illustrations is the vegetable kingdom, where specific differences tend to be much less clear than among animals, and considerable developmental plasticity is the rule. | |
| From: John Dupré (The Disorder of Things [1993], 1) | |
| A reaction: Nice. Just as the idea that laws of nature are mathematical suits physics, but founders on biology, so natural kinds founder in an area of biology to which we pay less attention. He cites prickly pears and lilies. I'm thinking oranges, satsumas etc. |
| 17384 | Even atoms of an element differ, in the energy levels of their electrons [Dupré] |
| Full Idea: Even if we claim that it is really isotopes not atoms that are the natural kinds (thus divorcing chemistry from ordinary language), atoms are said to differ with respect to such features as energy levels of the electrons. | |
| From: John Dupré (The Disorder of Things [1993], 2) | |
| A reaction: So we can't just pick out the features of one atom, and say that is the essence. Essence always involves some selection. I say the essence arises from the explanation of the atom's behaviour. |
| 17387 | Ecologists favour classifying by niche, even though that can clash with genealogy [Dupré] |
| Full Idea: To the extent that the occupants of a particular niche do not coincide with the members of a particular genealogical line, a possibility widely acknowledged to occur, ecologists must favour a method of classification lacking genealogical grounding. | |
| From: John Dupré (The Disorder of Things [1993], 2) | |
| A reaction: Zoo keepers probably classify by cages, or which zoo owns what, but that doesn't mean that they reject genealogy. Don't assume ecologists are rejecting any underlying classification that differs from theirs. Compare classification by economists. |
| 17380 | Wales may count as fish [Dupré] |
| Full Idea: The claim that whales are not fish is a debatable one | |
| From: John Dupré (The Disorder of Things [1993], 1) | |
| A reaction: A very nice challenge to an almost unquestioned orthodoxy. |
| 17382 | Cooks, unlike scientists, distinguish garlic from onions [Dupré] |
| Full Idea: It would be a severe culinary misfortune if no distinction were drawn between garlic and onions, a distinction that is not reflected in scientific taxonomy. | |
| From: John Dupré (The Disorder of Things [1993], 1) | |
| A reaction: Not every persuasive. We distinguish some cows from others because they taste better, but no one thinks that is a serious way in which to classify cows. |
| 17383 | Species are the lowest-level classification in biology [Dupré] |
| Full Idea: Species are, by definition, the lowest-level classificatory unit, or basal taxonomic unit, for biological organisms. | |
| From: John Dupré (The Disorder of Things [1993], 2) | |
| A reaction: I think this is the 'infima species' for Aristotelians. What about 'male' and 'female' in each species? |
| 17386 | The theory of evolution is mainly about species [Dupré] |
| Full Idea: Species are what the theory of evolution is centrally about. | |
| From: John Dupré (The Disorder of Things [1993], 2) |