30 ideas
| 17641 | Discoveries in mathematics can challenge philosophy, and offer it a new foundation [Russell] |
| Full Idea: Any new discovery as to mathematical method and principles is likely to upset a great deal of otherwise plausible philosophising, as well as to suggest a new philosophy which will be solid in proportion as its foundations in mathematics are securely laid. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.283) | |
| A reaction: This is a manifesto for modern analytic philosophy. I'm not convinced, especially if a fictionalist view of maths is plausible. What Russell wants is rigour, but there are other ways of getting that. Currently I favour artificial intelligence. |
| 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. |
| 17638 | If one proposition is deduced from another, they are more certain together than alone [Russell] |
| Full Idea: Two obvious propositions of which one can be deduced from the other both become more certain than either in isolation; thus in a complicated deductive system, many parts of which are obvious, the total probability may become all but absolute certainty. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.279) | |
| A reaction: Thagard picked this remark out, in support of his work on coherence. |
| 17632 | Non-contradiction was learned from instances, and then found to be indubitable [Russell] |
| Full Idea: The law of contradiction must have been originally discovered by generalising from instances, though, once discovered, it was found to be quite as indubitable as the instances. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.274) |
| 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. |
| 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. |
| 17629 | Which premises are ultimate varies with context [Russell] |
| Full Idea: Premises which are ultimate in one investigation may cease to be so in another. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.273) |
| 17630 | The sources of a proof are the reasons why we believe its conclusion [Russell] |
| Full Idea: In mathematics, except in the earliest parts, the propositions from which a given proposition is deduced generally give the reason why we believe the given proposition. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.273) |
| 17640 | Finding the axioms may be the only route to some new results [Russell] |
| Full Idea: The premises [of a science] ...are pretty certain to lead to a number of new results which could not otherwise have been known. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.282) | |
| A reaction: I identify this as the 'fruitfulness' that results when the essence of something is discovered. |
| 17627 | It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell] |
| Full Idea: It is an apparent absurdity in proceeding ...through many rather recondite propositions of symbolic logic, to the 'proof' of such truisms as 2+2=4: for it is plain that the conclusion is more certain than the premises, and the supposed proof seems futile. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.272) | |
| A reaction: Famously, 'Principia Mathematica' proved this fact at enormous length. I wonder if this thought led Moore to his common sense view of his own hand - the conclusion being better than the sceptical arguments? |
| 17628 | Arithmetic was probably inferred from relationships between physical objects [Russell] |
| Full Idea: When 2 + 2 =4 was first discovered, it was probably inferred from the case of sheep and other concrete cases. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.272) |
| 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. |
| 17637 | The most obvious beliefs are not infallible, as other obvious beliefs may conflict [Russell] |
| Full Idea: Even where there is the highest degree of obviousness, we cannot assume that we are infallible - a sufficient conflict with other obvious propositions may lead us to abandon our belief, as in the case of a hallucination afterwards recognised as such. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.279) | |
| A reaction: This approach to fallibilism seems to arise from the paradox that undermined Frege's rather obvious looking axioms. After Peirce and Russell, fallibilism has become a secure norm of modern thought. |
| 17639 | Believing a whole science is more than believing each of its propositions [Russell] |
| Full Idea: Although intrinsic obviousness is the basis of every science, it is never, in a fairly advanced science, the whole of our reason for believing any one proposition of the science. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.279) |
| 17631 | Induction is inferring premises from consequences [Russell] |
| Full Idea: The inferring of premises from consequences is the essence of induction. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.274) | |
| A reaction: So induction is just deduction in reverse? Induction is transcendental deduction? Do I deduce the premises from observing a lot of white swans? Hm. |
| 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. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 17633 | The law of gravity has many consequences beyond its grounding observations [Russell] |
| Full Idea: The law of gravitation leads to many consequences which could not be discovered merely from the apparent motions of the heavenly bodies. | |
| From: Bertrand Russell (Regressive Method for Premises in Mathematics [1907], p.275) |