30 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] |
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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) |
| 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) |
| 22121 | The concept of being has only one meaning, whether talking of universals or of God [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus was the first scholastic to hold that the concept of being and other transcendentals were univocal, not only in application to substance and accidents, but even to God and creatures. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.205 | |
| A reaction: So either it exists or it doesn't. No nonsense about 'subsisting'. Russell flirted with subsistence, but Quine agrees with Duns Scotus (and so do I). |
| 22122 | Being (not sensation or God) is the primary object of the intellect [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus said the primary object of the created intellect was being, rejecting Aquinas's Aristotelian view that it was limited to the quiddity of the sense particular, and Henry of Ghent's Augustinian view that it was God. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.205 | |
| A reaction: I suppose the 'primary object of the intellect' is the rationalist/empiricism disagreement. So (roughly) Aquinas was an empiricist, Duns Scotus was a rationalist, and Augustine was a transcendentalist? Augustine sounds like Spinoza. |
| 22125 | Duns Scotus was a realist about universals [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus was a realist on the issue of universals and one of the main adversaries of Ockham's programme of nominalism. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: The view of Scotus seems to be the minority view. It is hard to find thinkers who really believe that universals have an independent existence. My interest in Duns Scotus waned when I read this. How does he imagine universals? |
| 22127 | Scotus said a substantial principle of individuation [haecceitas] was needed for an essence [Duns Scotus, by Dumont] |
| Full Idea: Rejecting the standard views that essences are individuated by either actual existence, quantity or matter, Scotus said that the principle of individuation is a further substantial difference added to the species - the so-called haecceitas or 'thisness'. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: [Scotus seldom referred to 'haecceitas'] I suppose essences have prior existence, but are too generic, so something must fix an essence as pertaining to this particular object. Is the haecceitas part of the essence, or of the particular? |
| 22126 | Avicenna and Duns Scotus say essences have independent and prior existence [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus endorsed Avicenna's theory of the common nature, according to which the essences have an independence and priority to their existence as either universal in the mind or singular outside it. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: I occasionally meet this weird idea in modern discussions of essence (in Lowe?), and now see its origin. It makes little sense without a divine mind to support the independent essences. Scotus had to add a principle of individuation for essences. |
| 22129 | Certainty comes from the self-evident, from induction, and from self-awareness [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus grounded certitude in the knowledge of self-evident propositions, induction, and awareness of our own state. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: Induction looks like the weak link here. |
| 22130 | Scotus defended direct 'intuitive cognition', against the abstractive view [Duns Scotus, by Dumont] |
| Full Idea: Scotus allocated to the intellect a direct, existential awareness of the intelligible object, called 'intuitive cognition', in contrast to abstractive knowledge, which seized the object independently of its presence to the intellect in actual existence. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: Presumably if you see a thing, shut your eyes and then know it, that is 'abstractive'. Scotus says open your eyes for proper knowledge. |
| 22128 | Augustine's 'illumination' theory of knowledge leads to nothing but scepticism [Duns Scotus, by Dumont] |
| Full Idea: Scotus rejected Henry of Ghent's defence of Augustine's of knowledge by 'illumination', as leading to nothing but scepticism. ...After this, illumination never made a serious recovery. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 |
| 1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
| Full Idea: I regard you all as relatives - by nature, not by convention. By nature like is akin to like, but convention is a tyrant over humankind and often constrains people to act contrary to nature. | |
| From: Hippias (fragments/reports [c.430 BCE]), quoted by Plato - Protagoras 337c8 |
| 22131 | The will retains its power for opposites, even when it is acting [Duns Scotus, by Dumont] |
| Full Idea: Scotus said the will is a power for opposites, in the sense that even when actually willing one thing, it retains a real, active power to will the opposite. He detaches the idea of freedom from time and variability. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: In the sense that we can abandon an action when in the middle of it, this seems to be correct. Not just 'I could have done otherwise', but 'I don't have to be doing this'. This shows that the will has wide power, but not that it is 'free'. |
| 22123 | The concept of God is the unique first efficient cause, final cause, and most eminent being [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus establishes God as first efficient cause, as ultimate final cause, and as most eminent being - his so-called 'triple primacy' - and says there is a unique nature within these primacies. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: This is the first stage of Duns Scotus's unusually complex argument for God's existence. Asserting the actual infinity of this unique being concludes his argument. |
| 22124 | We can't infer the infinity of God from creation ex nihilo [Duns Scotus, by Dumont] |
| Full Idea: Duns Scotus rejected the traditional argument that the infinity of God can be inferred from creation ex nihilo. | |
| From: report of John Duns Scotus (works [1301]) by Stephen D. Dumont - Duns Scotus p.206 | |
| A reaction: He accepted the infinity of God, however, but not for this reason. I don't know why he rejected it. I suppose the rejected claim is that something has to be infinite, and if it isn't the Cosmos then that leaves God? |