39 ideas
| 5750 | Consistency is modal, saying propositions are consistent if they could be true together [Melia] |
| Full Idea: Consistency is a modal notion: a set of propositions is consistent iff all the members of the set could be true together. | |
| From: Joseph Melia (Modality [2003], Ch.6) | |
| A reaction: This shows why Kantian ethics, for example, needs a metaphysical underpinning. Maybe Kant should have believed in the reality of Leibnizian possible worlds? An account of reason requires an account of necessity and possibility. |
| 5737 | Predicate logic has connectives, quantifiers, variables, predicates, equality, names and brackets [Melia] |
| Full Idea: First-order predicate language has four connectives, two quantifiers, variables, predicates, equality, names, and brackets. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: Look up the reference for the details! The spirit of logic is seen in this basic framework, and the main interest is in the ontological commitment of the items on the list. The list is either known a priori, or it is merely conventional. |
| 5744 | First-order predicate calculus is extensional logic, but quantified modal logic is intensional (hence dubious) [Melia] |
| Full Idea: First-order predicate calculus is an extensional logic, while quantified modal logic is intensional (which has grave problems of interpretation, according to Quine). | |
| From: Joseph Melia (Modality [2003], Ch.3) | |
| A reaction: The battle is over ontology. Quine wants the ontology to stick with the values of the variables (i.e. the items in the real world that are quantified over in the extension). The rival view arises from attempts to explain necessity and counterfactuals. |
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| Full Idea: Set theory made a closer study of infinity possible. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 5740 | Second-order logic needs second-order variables and quantification into predicate position [Melia] |
| Full Idea: Permitting quantification into predicate position and adding second-order variables leads to second-order logic. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: Often expressed by saying that we now quantify over predicates and relations, rather than just objects. Depends on your metaphysical commitments. |
| 5741 | If every model that makes premises true also makes conclusion true, the argument is valid [Melia] |
| Full Idea: In first-order predicate calculus validity is defined thus: an argument is valid iff every model that makes the premises of the argument true also makes the conclusion of the argument true. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: See Melia Ch. 2 for an explanation of a 'model'. Traditional views of validity tend to say that if the premises are true the conclusion has to be true (necessarily), but this introduces the modal term 'necessarily', which is controversial. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 5735 | Maybe names and predicates can capture any fact [Melia] |
| Full Idea: Some philosophers think that any fact can be captured in a language containing only names and predicates. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: The problem case Melia is discussing is modal facts, such as 'x is possible'. It is hard to see how 'possible' could be an ordinary predicate, but then McGinn claims that 'existence' is, and that there are some predicates with unusual characters. |
| 5736 | No sort of plain language or levels of logic can express modal facts properly [Melia] |
| Full Idea: Some philosophers say that modal facts cannot be expressed either by name/predicate language, or by first-order predicate calculus, or even by second-order logic. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: If 'possible' were a predicate, none of this paraphernalia would be needed. If possible worlds are accepted, then the quantifiers of first-order predicate calculus will do the job. If neither of these will do, there seems to be a problem. |
| 5746 | The Identity of Indiscernibles is contentious for qualities, and trivial for non-qualities [Melia] |
| Full Idea: If the Identity of Indiscernibles is referring to qualitative properties, such as 'being red' or 'having mass', it is contentious; if it is referring to non-qualitative properties, such as 'member of set s' or 'brother of a', it is true but trivial. | |
| From: Joseph Melia (Modality [2003], Ch.3 n 11) | |
| A reaction: I would say 'false' rather than 'contentious'. No one has ever offered a way of distinguishing two electrons, but that doesn't mean there is just one (very busy) electron. The problem is that 'indiscernible' is only an epistemological concept. |
| 5738 | We may be sure that P is necessary, but is it necessarily necessary? [Melia] |
| Full Idea: We may have fairly firm beliefs as to whether or not P is necessary, but many of us find ourselves at a complete loss when wondering whether or not P is necessarily necessary. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: I think it is questions like this which are pushing philosophy back towards some sort of rationalism. See Idea 3651, for example. A regress of necessities would be mad, so necessity must be taken as self-evident (in itself, though maybe not to us). |
| 5732 | 'De re' modality is about things themselves, 'de dicto' modality is about propositions [Melia] |
| Full Idea: In cases of 'de re' modality, it is a particular thing that has the property essentially or accidentally; where the modality attaches to the proposition, it is 'de dicto' - it is the whole truth that all bachelors are unmarried that is necessary. | |
| From: Joseph Melia (Modality [2003], Ch.1) | |
| A reaction: This seems to me one of the most important distinctions in metaphysics (as practised by analytical philosophers, who like distinctions). The first type leads off into the ontology, the second type veers towards epistemology. |
| 5739 | Sometimes we want to specify in what ways a thing is possible [Melia] |
| Full Idea: Sometimes we want to count the ways in which something is possible, or say that there are many ways in which a certain thing is possible. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: This is a basic fact about talk of 'possibility'. It is not an all-or-nothing property of a situation. There can be 'faint' possibilities of things. The proximity of some possible worlds, especially those sharing our natural laws, is one answer. |
| 5734 | Possible worlds make it possible to define necessity and counterfactuals without new primitives [Melia] |
| Full Idea: In modal logic the concepts of necessity and counterfactuals are not interdefinable, so the language needs two primitives to represent them, but with the machinery of possible worlds they are defined by what is the case in all worlds, or close worlds. | |
| From: Joseph Melia (Modality [2003], Ch.1) | |
| A reaction: If your motivation is to reduce ontology to the barest of minimums (which it was for David Lewis) then it is paradoxical that the existence of possible worlds may be the way to achieve it. I doubt, though, whether a commitment to their reality is needed. |
| 5742 | In possible worlds semantics the modal operators are treated as quantifiers [Melia] |
| Full Idea: The central idea in possible worlds semantics is that the modal operators are treated as quantifiers. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: It seems an essential requirement of metaphysics that an account be given of possibility and necessity, and it is also a good dream to keep the ontology simple. Commitment to possible worlds is the bizarre outcome of this dream. |
| 5743 | If possible worlds semantics is not realist about possible worlds, logic becomes merely formal [Melia] |
| Full Idea: It has proved difficult to justify possible worlds semantics without accepting possible worlds. Without a secure metaphysical underpinning, the results in logic are in danger of having nothing more than a formal significance. | |
| From: Joseph Melia (Modality [2003], Ch.2) | |
| A reaction: This makes nicely clear why Lewis's controversial modal realism has to be taken seriously. It appears that the key problem is truth, because that is needed to define validity, but you can't have truth without some sort of metaphysics. |
| 5749 | Possible worlds could be real as mathematics, propositions, properties, or like books [Melia] |
| Full Idea: One can be a realist about possible worlds without adopting Lewis's extreme views; they might be abstract or mathematical entities; they might be sets of propositions or maximal uninstantiated properties; they might be like books or pictures. | |
| From: Joseph Melia (Modality [2003], Ch.6) | |
| A reaction: My intuition is that once you go down the road of realism about possible worlds, Lewis's full concrete realism looks at least as attractive as any of these options. You can discuss the 'average man' in an economic theory without realism. |
| 5751 | The truth of propositions at possible worlds are implied by the world, just as in books [Melia] |
| Full Idea: Propositions are true at possible worlds in much the same way as they are true at books: by being implied by the book. | |
| From: Joseph Melia (Modality [2003], Ch.7) | |
| A reaction: An intriguing way to introduce the view that possible worlds should be seen as like books. The truth-makers of propositions about the actual world are items in it, but the truth-makers in novels (say) are the conditions of the whole work as united. |
| 5748 | We accept unverifiable propositions because of simplicity, utility, explanation and plausibility [Melia] |
| Full Idea: Many philosophers now concede that it is rational to accept a proposition not because we can directly verify it but because it is supported by considerations of simplicity, theoretical utility, explanatory power and/or intuitive plausibility. | |
| From: Joseph Melia (Modality [2003], Ch.5) | |
| A reaction: This suggests how the weakness of logical positivism may have led us to the concept of epistemic virtues (such as those listed), which are, of course, largely a matter of community consensus, just as the moral virtues are. |
| 1868 | The world was made as much for animals as for man [Celsus] |
| Full Idea: The world was made as much for the irrational animals as for men. | |
| From: Celsus (On the True Doctrine (Against Christians) [c.178], §V) | |
| A reaction: A good remark. It seems to be a classic distortion of European Christianity that the world is made for us, and that animals only exist to fill our sandwiches. |
| 1867 | Christians presented Jesus as a new kind of logos to oppose that of the philosophers [Celsus] |
| Full Idea: Christians put forth this Jesus not only as the son of God, but as the very Logos - not the pure and holy Logos known to the philosophers, but a new kind of Logos. | |
| From: Celsus (On the True Doctrine (Against Christians) [c.178], III) |