24 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] |
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) |
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) |
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. |
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. |
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) |
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) |
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) |
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) |
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) |
22179 | Explanatory facts also predict, and predictive facts also explain [Hempel, by Okasha] |
Full Idea: Hempel said every scientific explanation is potentially a prediction - it would have predicted the phenomenon in question, had it not already been known. But also the information used to make a prediction is potentially an explanation. | |
From: report of Carl Hempel (Aspects of Scientific Explanation [1965]) by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 3 | |
A reaction: Sounds too neatly glib to be quite true. If you explain a single event there is nothing to predict. You might predict accurately from a repetitive pattern, with no understanding at all of the pattern. |
6755 | For Hempel, explanations are deductive-nomological or probabilistic-statistical [Hempel, by Bird] |
Full Idea: Hempel proposes that explanations involve covering laws and antecedent conditions; this view (the 'covering law' view) has two versions, the deductive-nomological model and the probabilistic-statistical model of explanation. | |
From: report of Carl Hempel (Aspects of Scientific Explanation [1965]) by Alexander Bird - Philosophy of Science Ch.2 | |
A reaction: The obvious problem with this approach, it seem to me, is that the laws themselves need explanation, and I don't see how a law can be foundational unless there is a divine law-giver. Are the laws arbitrary and axiomatic? |
17083 | The covering-law model is for scientific explanation; historical explanation is quite different [Hempel] |
Full Idea: To put forward the covering-law models of scientific explanation is not to deny that there are other contexts in which we speak of explanation. ….That it does not fit explaining the rules of Hanoverian succession is to miss the intent of our model. | |
From: Carl Hempel (Aspects of Scientific Explanation [1965], p. 412-3), quoted by David-Hillel Ruben - Explaining Explanation Ch 1 | |
A reaction: Important to get that clear. It then requires a clear demarcation between science and the rest, and it had better not rule out biology because it is having a love affair with physics. |
13052 | Hempel rejects causation as part of explanation [Hempel, by Salmon] |
Full Idea: Hempel explicitly rejects the idea that causality plays any essential explanatory role. | |
From: report of Carl Hempel (Aspects of Scientific Explanation [1965], p.352) by Wesley Salmon - Four Decades of Scientific Explanation 1.1 | |
A reaction: Hempel champions the 'covering-law' model of explanation. It strikes me that Hempel is so utterly wrong about this that his views aren't even a candidate for correctness, but then for a long time his views were orthodoxy. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |