23 ideas
8138 | Beware lest any man spoil you through philosophy [Paul] |
Full Idea: Beware lest any man spoil you through philosophy. | |
From: St Paul (12: Colossians [c.55], 2.8) | |
A reaction: The same might be said of preaching. The two sorts of spoiling seem to be fanaticism and wickedness. While reason can lead to fanaticism, I believe (with Socrates) that it is unlikely to corrupt morally. |
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 = { |
|
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) |
22049 | Transcendental idealism aims to explain objectivity through subjectivity [Bowie] |
Full Idea: The aim of transcendental idealism is to give a basis for objectivity in terms of subjectivity. | |
From: Andrew Bowie (German Philosophy: a very short introduction [2010], 1) | |
A reaction: Hume used subjectivity to undermine the findings of objectivity. There was then no return to naive objectivity. Kant's aim then was to thwart global scepticism. Post-Kantians feared that he had failed. |
22055 | The Idealists saw the same unexplained spontaneity in Kant's judgements and choices [Bowie] |
Full Idea: The Idealist saw in Kant that knowledge, which depends on the spontaneity of judgement, and self-determined spontaneous action, can be seen as sharing the same source, which is not accessible to scientific investigation. | |
From: Andrew Bowie (German Philosophy: a very short introduction [2010]) | |
A reaction: This is the 'spontaneity' of judgements and choices which was seen as the main idea in Kant. It inspired romantic individualism. The judgements are the rule-based application of concepts. |
22054 | German Idealism tried to stop oppositions of appearances/things and receptivity/spontaneity [Bowie] |
Full Idea: A central aim of German Idealism is to overcome Kant's oppositions between appearances and thing in themselves, and between receptivity and spontaneity. | |
From: Andrew Bowie (German Philosophy: a very short introduction [2010], 2) | |
A reaction: I have the impression that there were two strategies: break down the opposition within the self (Fichte), or break down the opposition in the world (Spinozism). |
22056 | Crucial to Idealism is the idea of continuity between receptivity and spontaneous judgement [Bowie] |
Full Idea: A crucial idea for German Idealism (from Hamann) is that apparently passive receptivity and active spontaneity are in fact different degrees of the same 'activity, and the gap between subject and world can be closed. | |
From: Andrew Bowie (German Philosophy: a very short introduction [2010], 3) | |
A reaction: The 'passive' bit seems to be Hume's 'impressions', which are Kant's 'intuitions', which need 'spontaneous' interpretation to become experiences. Critics of Kant said this implied a dualism. |