33 ideas
| 17713 | After 1903, Husserl avoids metaphysical commitments [Mares] |
| Full Idea: In Husserl's philosophy after 1903, he is unwilling to commit himself to any specific metaphysical views. | |
| From: Edwin D. Mares (A Priori [2011], 08.2) |
| 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) |
| 17715 | The truth of the axioms doesn't matter for pure mathematics, but it does for applied [Mares] |
| Full Idea: The epistemological burden of showing that the axioms are true is removed if we are only studying pure mathematics. If, however, we want to look at applied mathematics, then this burden returns. | |
| From: Edwin D. Mares (A Priori [2011], 11.4) | |
| A reaction: One of those really simple ideas that hits the spot. Nice. The most advanced applied mathematics must rest on counting and measuring. |
| 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) |
| 17716 | Mathematics is relations between properties we abstract from experience [Mares] |
| Full Idea: Aristotelians treat mathematical facts as relations between properties. These properties, moreover, are abstracted from our experience of things. ...This view finds a natural companion in structuralism. | |
| From: Edwin D. Mares (A Priori [2011], 11.7) | |
| A reaction: This is the view of mathematics that I personally favour. The view that we abstract 'five' from a group of five pebbles is too simplistic, but this is the right general approach. |
| 17703 | Light in straight lines is contingent a priori; stipulated as straight, because they happen to be so [Mares] |
| Full Idea: It seems natural to claim that light rays moving in straight lines is contingent but a priori. Scientists stipulate that they are the standard by which we measure straightness, but their appropriateness for this task is a contingent feature of the world. | |
| From: Edwin D. Mares (A Priori [2011], 02.9) | |
| A reaction: This resembles the metre rule in Paris. It is contingent that something is a certain way, so we make being that way a conventional truth, which can therefore be known via the convention, rather than via the contingent fact. |
| 17714 | Aristotelians dislike the idea of a priori judgements from pure reason [Mares] |
| Full Idea: Aristotelians tend to eschew talk about a special faculty of pure reason that is responsible for all of our a priori judgements. | |
| From: Edwin D. Mares (A Priori [2011], 08.9) | |
| A reaction: He is invoking Carrie Jenkins's idea that the a priori is knowledge of relations between concepts which have been derived from experience. Nice idea. We thus have an empirical a priori, integrated into the natural world. Abstraction must be involved. |
| 17705 | Empiricists say rationalists mistake imaginative powers for modal insights [Mares] |
| Full Idea: Empiricist critiques of rationalism often accuse rationalists of confusing the limits of their imaginations with real insight into what is necessarily true. | |
| From: Edwin D. Mares (A Priori [2011], 03.01) | |
| A reaction: See ideas on 'Conceivable as possible' for more on this. You shouldn't just claim to 'see' that something is true, but be willing to offer some sort of reason, truthmaker or grounding. Without that, you may be right, but you are on weak ground. |
| 17700 | The most popular view is that coherent beliefs explain one another [Mares] |
| Full Idea: In what is perhaps the most popular version of coherentism, a system of beliefs is a set of beliefs that explain one another. | |
| From: Edwin D. Mares (A Priori [2011], 01.5) | |
| A reaction: These seems too simple. My first response would be that explanations are what result from coherence sets of beliefs. I may have beliefs that explain nothing, but at least have the virtue of being coherent. |
| 17704 | Operationalism defines concepts by our ways of measuring them [Mares] |
| Full Idea: The central claim of Percy Bridgman's theory of operational definitions (1920s), is that definitions of certain scientific concepts are given by the ways that we have to measure them. For example, a straight line is 'the path of a light ray'. | |
| From: Edwin D. Mares (A Priori [2011], 02.9) | |
| A reaction: It is often observed that this captures the spirit of Special Relativity. |
| 17710 | Aristotelian justification uses concepts abstracted from experience [Mares] |
| Full Idea: Aristotelian justification is the process of reasoning using concepts that are abstracted from experience (rather than, say, concepts that are innate or those that we associate with the meanings of words). | |
| From: Edwin D. Mares (A Priori [2011], 08.1) | |
| A reaction: See Carrie Jenkins for a full theory along these lines (though she doesn't mention Aristotle). This is definitely my preferred view of concepts. |
| 17706 | The essence of a concept is either its definition or its conceptual relations? [Mares] |
| Full Idea: In the 'classical theory' a concept includes in it those concepts that define it. ...In the 'theory theory' view the content of a concept is determined by its relationship to other concepts. | |
| From: Edwin D. Mares (A Priori [2011], 03.10) | |
| A reaction: Neither of these seem to give an intrinsic account of a concept, or any account of how the whole business gets off the ground. |
| 17701 | Possible worlds semantics has a nice compositional account of modal statements [Mares] |
| Full Idea: Possible worlds semantics is appealing because it gives a compositional analysis of the truth conditions of statements about necessity and possibility. | |
| From: Edwin D. Mares (A Priori [2011], 02.2) | |
| A reaction: Not sure I get this. Is the meaning composed by the gradual addition of worlds? If not, how is meaning composed in the normal way, from component words and phrases? |
| 17702 | Unstructured propositions are sets of possible worlds; structured ones have components [Mares] |
| Full Idea: An unstructured proposition is a set of possible worlds. ....Structured propositions contain entities that correspond to various parts of the sentences or thoughts that express them. | |
| From: Edwin D. Mares (A Priori [2011], 02.3) | |
| A reaction: I am definitely in favour of structured propositions. It strikes me as so obvious as to be not worth discussion - so I am obviously missing something here. Mares says structured propositions are 'more convenient'. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 17708 | Maybe space has points, but processes always need regions with a size [Mares] |
| Full Idea: One theory is that space is made up of dimensionless points, but physical processes cannot take place in regions of less than a certain size. | |
| From: Edwin D. Mares (A Priori [2011], 06.7) | |
| A reaction: Thinkers in sympathy with verificationism presumably won't like this, and may prefer Feynman's view. |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |