47 ideas
| 14684 | A world is 'accessible' to another iff the first is possible according to the second [Salmon,N] |
| Full Idea: A world w' is accessible to a consistent world w if and only if w' is possible in w. Being 'inaccessible to' or 'possible relative to' a consistent world is simply being possible according to that world, nothing more and nothing less. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) | |
| A reaction: More illuminating than just saying that w can 'see' w'. Accessibility is internal to worIds. It gives some connection to why we spend time examining modal logic. There is no more important metaphysical notion than what is possible according to actuality. |
| 14669 | For metaphysics, T may be the only correct system of modal logic [Salmon,N] |
| Full Idea: Insofar as modal logic is concerned exclusively with the logic of metaphysical modality, ..T may well be the one and only (strongest) correct system of (first-order) propositional logic. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], Intro) | |
| A reaction: This contrasts sharply with the orthodox view, that S5 (or at the very least S4) is the correct system for metaphysics. |
| 14667 | System B has not been justified as fallacy-free for reasoning on what might have been [Salmon,N] |
| Full Idea: Even the conventionally accepted system B, which is weaker than S5 and independent of S4, has not been adequately justified as a fallacy-free system of reasoning about what might have been. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], Intro) |
| 14668 | In B it seems logically possible to have both p true and p is necessarily possibly false [Salmon,N] |
| Full Idea: The characteristic of B has the form φ⊃□◊φ. ...Even if these axioms are necessarily true, it seems logically possible for p to be true while the proposition that p is necessarily possible is at the same time false. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], Intro) |
| 14692 | System B implies that possibly-being-realized is an essential property of the world [Salmon,N] |
| Full Idea: Friends of B modal logic commit themselves to the loaded claim that it is logically true that the property of possibly being realized (or being a way things might have been) is an essential property of the world. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], V) | |
| A reaction: I think this 'loaded' formulation captures quite nicely the dispositional view I favour, that the possibilities of the actual world are built into the actual world, and define its nature just as much as the 'categorial' facts do. |
| 14671 | What is necessary is not always necessarily necessary, so S4 is fallacious [Salmon,N] |
| Full Idea: We can say of a wooden table that it would have been possible for it to have originated from some different matter, even though it is not actually possible. So what is necessary fails to be necessarily necessary, and S4 modal logic is fallacious. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], I) | |
| A reaction: [compressed] |
| 14686 | S5 modal logic ignores accessibility altogether [Salmon,N] |
| Full Idea: When we ignore accessibility altogether, we have finally zeroed in on S5 modal logic. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) |
| 14691 | S5 believers say that-things-might-have-been-that-way is essential to ways things might have been [Salmon,N] |
| Full Idea: Believers in S5 as a correct system of propositional reasoning about what might have been must claim that it is an essential property of any way things might have been that things might have been that way. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], V) | |
| A reaction: Salmon is working in a view where you are probably safe to substitute 'necessary' for 'essential' without loss of meaning. |
| 14693 | The unsatisfactory counterpart-theory allows the retention of S5 [Salmon,N] |
| Full Idea: Counterpart-theoretic modal semantics allows for the retention of S5 modal propositional logic, at a considerable cost. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], V n18) | |
| A reaction: See the other ideas in this paper by Salmon for his general attack on S5 as the appropriate system for metaphysical necessity. He favours the very modest System T. |
| 14670 | Metaphysical (alethic) modal logic concerns simple necessity and possibility (not physical, epistemic..) [Salmon,N] |
| Full Idea: Metaphysical modal logic concerns metaphysical (or alethic) necessity and metaphysical (alethic) possibility, or necessity and possibility tout court - as opposed to such other types of modality as physical necessity, epistemic necessity etc. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], Intro n2) |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 14678 | Any property is attached to anything in some possible world, so I am a radical anti-essentialist [Salmon,N] |
| Full Idea: By admitting possible worlds of unlimited variation and recombination, I simply abandon true metaphysical essentialism. By my lights, any property is attached to anything in some possible world or other. I am a closet radical anti-essentialist. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], II) | |
| A reaction: Salmon includes impossible worlds within his scheme of understanding. It strikes me that this is metaphysical system which tells us nothing about how things are: it is sort of 'logical idealist'. Later he talks of 'we essentialists'. |
| 14680 | Logical possibility contains metaphysical possibility, which contains nomological possibility [Salmon,N] |
| Full Idea: Just as nomological possibility is a special kind of metaphysical possibility, so metaphysical possibility is a special kind of logical possibility. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], III) | |
| A reaction: This is the standard view of how the three types of necessity are nested. He gives a possible counterexample in footnote 7. |
| 14690 | In the S5 account, nested modalities may be unseen, but they are still there [Salmon,N] |
| Full Idea: The S5 theorist's miscontrual of English (in the meaning of 'possibly possible') makes nested modality unseen, but it does not make nested modality vanish. Inaccessible worlds are still worlds. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) |
| 14677 | Metaphysical necessity is said to be unrestricted necessity, true in every world whatsoever [Salmon,N] |
| Full Idea: It is held that it is the hallmark of metaphysical necessity is that it is completely unrestricted, the limiting case of restricted necessity, with no restrictions whatever. A proposition is necessary only if it is true in absolutely every world whatever. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], II) | |
| A reaction: This is the standard picture which leads to the claim that S5 modal logic is appropriate for metaphysical necessity, because there are no restrictions on accessibility. Salmon raises objections to this conventional view. |
| 14679 | Bizarre identities are logically but not metaphysically possible, so metaphysical modality is restricted [Salmon,N] |
| Full Idea: Though there is a way things logically could be according to which I am a credit card account, there is no way things metaphysically might be according to which I am a credit card account. This illustrates the restricted nature of metaphysical modality. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], III) | |
| A reaction: His drift is that metaphyical modality is restricted, but expressing it in S5 modal logic (where all worlds see one another) makes it unrestricted, so S5 logic is wrong for metaphysics. I'm impressed by his arguments. |
| 14688 | Without impossible worlds, the unrestricted modality that is metaphysical has S5 logic [Salmon,N] |
| Full Idea: If one confines one's sights to genuinely possible worlds, disavowing the impossible worlds, then metaphysical modality emerges as the limiting case - the 'unrestricted' modality that takes account of 'every' world - and S5 emerges as its proper logic. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) | |
| A reaction: He observes that this makes metaphysical modality 'restricted' simply because you have restricted what 'all worlds' means. Could there be non-maximal worlds? Are logical and metaphysical modality coextensive? I think I like the S5 view. |
| 14685 | Metaphysical necessity is NOT truth in all (unrestricted) worlds; necessity comes first, and is restricted [Salmon,N] |
| Full Idea: A mythology gave us the idea that metaphysical necessity is truth in every world whatsoever, without restriction. But the notion of metaphysical modality comes first, and, like every notion of modality, it is restricted. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) |
| 14681 | Logical necessity is free of constraints, and may accommodate all of S5 logic [Salmon,N] |
| Full Idea: With its freedom from the constraint of metaphysical possibility, logical necessity may be construed as accommodating all the axioms and rules of S5. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], III) | |
| A reaction: He goes on to raise problems for this simple thought. The big question: what are the limits of what is actually possible? Compare: what are the limits of what is imaginable? what are the limits of what is meaningfully sayable? |
| 14676 | Nomological necessity is expressed with intransitive relations in modal semantics [Salmon,N] |
| Full Idea: Intransitive relations are introduced into modal semantics for the purposes of interpreting various 'real' or restricted types of modalities, such as nomological necessity. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], II) | |
| A reaction: The point here is that the (so-called) 'laws of nature' are held to change from world to world, so necessity in one could peter out in some more remote world, rather than being carried over everywhere. A very Humean view of such things. |
| 14689 | Necessity and possibility are not just necessity and possibility according to the actual world [Salmon,N] |
| Full Idea: The real meanings of the simple modal terms 'necessary' and 'possible' are not the same as the concepts of actual necessity and actual possibility, necessity and possibility according to the actual world. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) | |
| A reaction: If you were an 'actualist' (who denies everything except the actual world) then you are unlikely to agree with this. In unrestricted possible worlds, being true in one world makes it possible in all worlds. So actual necessity is possible everywhere. |
| 14674 | Impossible worlds are also ways for things to be [Salmon,N] |
| Full Idea: Total ways things cannot be are also 'worlds', or maximal ways for things to be. They are impossible worlds. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], I) | |
| A reaction: This unorthodox view doesn't sound too plausible to me. To think of a circular square as a 'way things could be' sounds pretty empty, and mere playing with words. The number 7 could be the Emperor of China? |
| 14682 | Denial of impossible worlds involves two different confusions [Salmon,N] |
| Full Idea: Every argument I am aware of against impossible worlds confuses ways for things to be with ways things might have been, or worse, confuses ways things cannot be with ways for things to be that cannot exist - or worse yet, commits both errors. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], III) | |
| A reaction: He is claiming that 'ways for things to be' allows impossible worlds, whereas 'ways things might have been' appears not to. (I think! Read the paragraph yourself!) |
| 14687 | Without impossible worlds, how things might have been is the only way for things to be [Salmon,N] |
| Full Idea: If one ignores impossible worlds, then ways things might have been are the only ways for things to be that are left. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) | |
| A reaction: Impossible worlds are included in 'ways for things to be', but excluded from 'ways things might have been'. I struggle with a circle being square as a 'way for circles to be'. I suppose being the greatest philosopher is a way for me to be. |
| 14683 | Possible worlds rely on what might have been, so they can' be used to define or analyse modality [Salmon,N] |
| Full Idea: On my conception, the notions of metaphysical necessity and possibility are not defined or analyzed in terms of the apparatus of possible worlds. The order of analysis is just the reverse: possible worlds rely on the notion of what might have been. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], IV) | |
| A reaction: This view seems to be becoming the new orthodoxy, and I certainly agree with it. I have no idea how you can begin to talk about possible worlds if you don't already have some idea of what 'possible' means. |
| 14672 | Possible worlds are maximal abstract ways that things might have been [Salmon,N] |
| Full Idea: I conceive of possible worlds as certain sorts of maximal abstract entities according to which certain things (facts, states of affairs) obtain and certain other things do not obtain. They are total ways things might have been. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], I) |
| 14675 | Possible worlds just have to be 'maximal', but they don't have to be consistent [Salmon,N] |
| Full Idea: As far as I can tell, worlds need not be logically consistent. The only restriction on worlds is that they must be (in some sense) 'maximal' ways for things to be. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], I) | |
| A reaction: The normal idea of a maximal model is that it must contain either p or ¬p, and not both, so I don't think I understand this thought, but I pass it on. |
| 14673 | You can't define worlds as sets of propositions, and then define propositions using worlds [Salmon,N] |
| Full Idea: It is not a good idea to think of possible worlds as sets of propositions, and at the same time to think of propositions as sets of possible worlds. | |
| From: Nathan Salmon (The Logic of What Might Have Been [1989], I n3) | |
| A reaction: Salmon favours thinking of worlds as sets of propositions, and hence rejects the account of propositions as sets of worlds. He favours the 'Russellian' view of propositions, which seem to me to be the same as 'facts'. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 22098 | Socrates neglects the gap between knowing what is good and doing good [Kierkegaard, by Carlisle] |
| Full Idea: There is a fundamental weakness in Socrates, that he does not take into account the gap between knowing what is good and actually putting this into action. | |
| From: report of Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: This rejects Socrates's intellectualism about weakness of will. It is perhaps a better criticism that Aristotle's view that desires sometimes overcome the will. It is also the problem of motivation in Kantian deontology. Or utilitarianism. |
| 22096 | Anxiety is not a passing mood, but a response to human freedom [Kierkegaard, by Carlisle] |
| Full Idea: For Kierkegaard anxiety is not simply a mood or an emotion that certain people experience at certain times, but a basic response to freedom that is part of the human condition. | |
| From: report of Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: Outside of Christianity, this may be Kierkegaard's most influential idea - since existential individualism is floating around in the romantic movement. But the Byronic hero experiences a sort of anxiety. If you can't face anxiety, become a monk or nun. |
| 22097 | The ultimate in life is learning to be anxious in the right way [Kierkegaard] |
| Full Idea: Every human being must learn to be anxious in order that he might not perish either by never having been in anxiety or by succumbing in anxiety. Whoever has learned to be anxious in the right way has learnt the ultimate. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.154), quoted by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: I think this is the most existentialist quotation I have found in Kierkegaard. It sounds circular. You must be in anxiety because otherwise you won't be able to cope with anxiety? I suppose anxiety is facing up to his concept of truth. |
| 21909 | Ultimate knowledge is being anxious in the right way [Kierkegaard] |
| Full Idea: Whoever learns to be anxious in the right way has learned the ultimate. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.187), quoted by Alastair Hannay - Kierkegaard 06 | |
| A reaction: This shows us that Kierkegaard had a rather bizarre mental life which the rest of us have little chance of penetrating. I'll have a go at cataloguing my types of anxiety, but I'm not hopeful. |
| 20758 | Anxiety is staring into the yawning abyss of freedom [Kierkegaard] |
| Full Idea: One may liken anxiety to dizziness. He whose eyes chance to look down into a yawning abyss becomes dizzy. Anxiety is the dizziness of freedom which is when freedom gazes down into its own possibility, grasping at finiteness to sustain itself. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.55), quoted by Kevin Aho - Existentialism: an introduction 6 'Moods' | |
| A reaction: Most of us rapidly retreat from the thought of the infinity of things we might choose. Choosing bizarrely merely to assert one's freedom is simple stupidity. |