Combining Texts

All the ideas for 'The Evolution of Modern Metaphysics', 'Introduction to the Theory of Logic' and 'Axiomatic Thought'

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26 ideas

1. Philosophy / E. Nature of Metaphysics / 1. Nature of Metaphysics
Metaphysics is the most general attempt to make sense of things [Moore,AW]
     Full Idea: Metaphysics is the most general attempt to make sense of things.
     From: A.W. Moore (The Evolution of Modern Metaphysics [2012], Intro)
     A reaction: This is the first sentence of Moore's book, and a touchstone idea all the way through. It stands up well, because it says enough without committing to too much. I have to agree with it. It implies explanation as the key. I like generality too.
4. Formal Logic / F. Set Theory ST / 1. Set Theory
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)
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
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 = { | x ∈ A and y ∈ B}.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6)
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)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
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.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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)
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
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.
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)
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
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)
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
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'.
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.
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
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.
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
'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)
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)
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
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)
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)
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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)
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
The facts of geometry, arithmetic or statics order themselves into theories [Hilbert]
     Full Idea: The facts of geometry order themselves into a geometry, the facts of arithmetic into a theory of numbers, the facts of statics, electrodynamics into a theory of statics, electrodynamics, or the facts of the physics of gases into a theory of gases.
     From: David Hilbert (Axiomatic Thought [1918], [03])
     A reaction: This is the confident (I would say 'essentialist') view of axioms, which received a bit of a setback with Gödel's Theorems. I certainly agree that the world proposes an order to us - we don't just randomly invent one that suits us.
Axioms must reveal their dependence (or not), and must be consistent [Hilbert]
     Full Idea: If a theory is to serve its purpose of orienting and ordering, it must first give us an overview of the independence and dependence of its propositions, and second give a guarantee of the consistency of all of the propositions.
     From: David Hilbert (Axiomatic Thought [1918], [09])
     A reaction: Gödel's Second theorem showed that the theory can never prove its own consistency, which made the second Hilbert requirement more difficult. It is generally assumed that each of the axioms must be independent of the others.
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
To decide some questions, we must study the essence of mathematical proof itself [Hilbert]
     Full Idea: It is necessary to study the essence of mathematical proof itself if one wishes to answer such questions as the one about decidability in a finite number of operations.
     From: David Hilbert (Axiomatic Thought [1918], [53])
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert]
     Full Idea: The linearity of the equation of the plane and of the orthogonal transformation of point-coordinates is completely adequate to produce the whole broad science of spatial Euclidean geometry purely by means of analysis.
     From: David Hilbert (Axiomatic Thought [1918], [05])
     A reaction: This remark comes from the man who succeeded in producing modern axioms for geometry (in 1897), so he knows what he is talking about. We should not be wholly pessimistic about Hilbert's ambitious projects. He had to dig deeper than this idea...
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
Number theory just needs calculation laws and rules for integers [Hilbert]
     Full Idea: The laws of calculation and the rules of integers suffice for the construction of number theory.
     From: David Hilbert (Axiomatic Thought [1918], [05])
     A reaction: This is the confident Hilbert view that the whole system can be fully spelled out. Gödel made this optimism more difficult.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
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)
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / b. Transcendental idealism
Appearances are nothing beyond representations, which is transcendental ideality [Moore,AW]
     Full Idea: Appearances in general are nothing outside our representations, which is just what we mean by transcendental ideality.
     From: A.W. Moore (The Evolution of Modern Metaphysics [2012], B535/A507)
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / d. Knowing essences
By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge [Hilbert]
     Full Idea: By pushing ahead to ever deeper layers of axioms ...we also win ever-deeper insights into the essence of scientific thought itself, and become ever more conscious of the unity of our knowledge.
     From: David Hilbert (Axiomatic Thought [1918], [56])
     A reaction: This is the less fashionable idea that scientific essentialism can also be applicable in the mathematic sciences, centring on the project of axiomatisation for logic, arithmetic, sets etc.