28 ideas
| 2510 | Traditionally philosophy is an a priori enquiry into general truths about reality [Katz] |
| Full Idea: The traditional conception of philosophy is that it is an a priori enquiry into the most general facts about reality. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xi) | |
| A reaction: I think this still defines philosophy, though it also highlights the weakness of the subject, which is over-confidence about asserting necessary truths. How could the most god-like areas of human thought be about anything else? |
| 2516 | Most of philosophy begins where science leaves off [Katz] |
| Full Idea: Philosophy, or at least one large part of it, is subsequent to science; it begins where science leaves off. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxi) | |
| A reaction: In some sense this has to be true. Without metaphysics there couldn't be any science. Rationalists should not forget, though, the huge impact which Darwin's science has (or should have) on fairly abstract philosophy (e.g. epistemology). |
| 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) |
| 2521 | 'Real' maths objects have no causal role, no determinate reference, and no abstract/concrete distinction [Katz] |
| Full Idea: Three objections to realism in philosophy of mathematics: mathematical objects have no space/time location, and so no causal role; that such objects are determinate, but reference to numbers aren't; and that there is no abstract/concrete distinction. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxix) |
| 2513 | We don't have a clear enough sense of meaning to pronounce some sentences meaningless or just analytic [Katz] |
| Full Idea: Linguistic meaning is not rich enough to show either that all metaphysical sentences are meaningless or that all alleged synthetic a priori propositions are just analytic a priori propositions. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xx) |
| 2522 | Experience cannot teach us why maths and logic are necessary [Katz] |
| Full Idea: The Leibniz-Kant criticism of empiricism is that experience cannot teach us why mathematical and logical facts couldn't be otherwise than they are. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxxi) |
| 2517 | Structuralists see meaning behaviouristically, and Chomsky says nothing about it [Katz] |
| Full Idea: In linguistics there are two schools of thought: Bloomfieldian structuralism (favoured by Quine) conceives of sentences acoustically and meanings behaviouristically; and Chomskian generative grammar (which is silent about semantics). | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxiv) | |
| A reaction: They both appear to be wrong, so there is (or was) something rotten in the state of linguistics. Are the only options for meaning either behaviourist or eliminativist? |
| 2519 | It is generally accepted that sense is defined as the determiner of reference [Katz] |
| Full Idea: There is virtually universal acceptance of Frege's definition of sense as the determiner of reference. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxvi) | |
| A reaction: Not any more, since Kripke and Putnam. It is one thing to say sense determines reference, and quite another to say that this is the definition of sense. |
| 2520 | Sense determines meaning and synonymy, not referential properties like denotation and truth [Katz] |
| Full Idea: Pace Frege, sense determines sense properties and relations, like meaningfulness and synonymy, rather than determining referential properties, like denotation and truth. | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxvi) | |
| A reaction: This leaves room for Fregean 'sense', after Kripke has demolished the idea that sense determines reference. |
| 2518 | Sentences are abstract types (like musical scores), not individual tokens [Katz] |
| Full Idea: Sentences are types, not utterance tokens or mental/neural tokens, and hence sentences are abstract objects (like musical scores). | |
| From: Jerrold J. Katz (Realistic Rationalism [2000], Int.xxvi) | |
| A reaction: If sentences are abstract types, then two verbally indistinguishable sentences are the same sentence. But if I say 'I am happy', that isn't the same as you saying it. |
| 3284 | There is no one theory of how to act (or what to believe) [Nagel] |
| Full Idea: To look for a single general theory of how to decide the right thing to do is like looking for a single theory of how to decide what to believe. | |
| From: Thomas Nagel (The Fragmentation of Value [1977], p.135) | |
| A reaction: Depends on your level of generality. Values and virtues are general guides which should be brought to every action, with 'higher' values guiding choice of what is relevant. |