30 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) |
| 17471 | Using mechanisms as explanatory schemes began in chemistry [Weisberg/Needham/Hendry] |
| Full Idea: The production of mechanisms as explanatory schemes finds its original home in chemistry. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 5.1) | |
| A reaction: This is as opposed to mechanisms in biology or neuroscience, which come later. |
| 17472 | Thick mechanisms map whole reactions, and thin mechanism chart the steps [Weisberg/Needham/Hendry] |
| Full Idea: In chemistry the 'thick' notion of a mechanism traces out positions of electrons and atomic cores, and correlates them with energies, showing the whole reaction. 'Thin' mechanisms focus on a discrete set of intermediate steps. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 5.1) |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 17465 | Lavoisier's elements included four types of earth [Weisberg/Needham/Hendry] |
| Full Idea: Four types of earth found a place on Lavoisier's list of elements. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 1.2) | |
| A reaction: A nice intermediate point between the ancient Greek and the modern view of earth. |
| 17468 | Over 100,000,000 compounds have been discovered or synthesised [Weisberg/Needham/Hendry] |
| Full Idea: There are well over 100,000,000 chemical compounds that have been discovered or synthesised, all of which have been formally characterised. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 4.3) |
| 17470 | Water molecules dissociate, and form large polymers, explaining its properties [Weisberg/Needham/Hendry] |
| Full Idea: Water's structure cannot simply be described as a collection of individual molecules. There is a continual dissociation of H2O molecules into hydrogen and hydroxide ions; they former larger polymeric species, explaining conductivity, melting and boiling. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 4.5) | |
| A reaction: [compressed] If philosophers try to state the 'essence of water', they had better not be too glib about it. |
| 17473 | It is unlikely that chemistry will ever be reduced to physics [Weisberg/Needham/Hendry] |
| Full Idea: Most philosophers believe chemistry has not been reduced to physics nor is it likely to be. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 6) | |
| A reaction: [Le Poidevin 2007 argues the opposite] That chemical features are actually metaphysically 'emergent' is a rare view, defended by Hendry. The general view is that the concepts are too different, and approximations render it hopeless. |
| 17474 | Quantum theory won't tell us which structure a set of atoms will form [Weisberg/Needham/Hendry] |
| Full Idea: Quantum mechanics cannot tell us why a given collection of atoms will adopt one molecular structure (and set of chemical properties) or the other. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 6.1) | |
| A reaction: Presumably it the 'chance' process of how the atoms are thrown together. |
| 17475 | For temperature to be mean kinetic energy, a state of equilibrium is also required [Weisberg/Needham/Hendry] |
| Full Idea: Having a particular average kinetic energy is only a necessary condition for having a given temperature, not a sufficient one, because only gases at equilibrium have a well-defined temperature. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 6.2) | |
| A reaction: If you try to pin it all down more precisely, the definition turns out to be circular. |
| 17469 | 'H2O' just gives the element proportions, not the microstructure [Weisberg/Needham/Hendry] |
| Full Idea: 'H2O' is not a description of any microstructure. It is a compositional formula, describing the combining proportions of hydrogen and oxygen to make water. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 4.5) |
| 17467 | Isotopes (such as those of hydrogen) can vary in their rates of chemical reaction [Weisberg/Needham/Hendry] |
| Full Idea: There are chemically salient differences among the isotopes, best illustrated by the three isotopes of hydrogen: protium, deuterium and tritium, which show different rates of reaction, making heavy water poisonous where ordinary water is not. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 1.4) | |
| A reaction: [They cite Paul Needham 2008] The point is that the isotopes are the natural kinds, rather than the traditional elements. The view is unorthodox, but clearly makes a good point. |
| 17466 | Mendeleev systematised the elements, and also gave an account of their nature [Weisberg/Needham/Hendry] |
| Full Idea: In addition to providing the systematization of the elements used in modern chemistry, Mendeleev also gave an account of the nature of the elements which informs contemporary philosophical understanding. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 1.3) |