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| 18835 | Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics |
| Full Idea: There is surely no metaphysical basis for logic, but equally there is no logical basis for metaphysics, if that implies that we can settle the choice of logic in advance of settling any seriously contested metaphysical-cum-semantic issues. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.5) | |||
| A reaction: Is this aimed at Tim Williamson's book on treating modal logic as metaphysics? I agree with the general idea that logic won't deliver a metaphysics. I might want to defend a good metaphysics giving rise to a good logic. |
| 18819 | The idea that there are unrecognised truths is basic to our concept of truth |
| Full Idea: The realist principle that a statement may be true even though no one is able to recognise its truth is so deeply embedded in our ordinary conception of truth that any account that flouts it is liable to engender confusion. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.1) |
| 18826 | 'True at a possibility' means necessarily true if what is said had obtained |
| Full Idea: A statement is 'true at a possibility' if, necessarily, things would have been as the statement (actually) says they are, had the possibility obtained. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.6) | |||
| A reaction: This is deliberately vague about what a 'possibility' is, but it is intended to be more than a property instantiation, and less than a possible world. |
| 18803 | Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables |
| Full Idea: The classical semantics of natural language propositions says 1) valid arguments preserve truth, 2) no statement is both true and false, 3) each statement is either true or false, 4) the familiar truth tables. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) |
| 18814 | 'Absolute necessity' would have to rest on S5 |
| Full Idea: If there is such a notion as 'absolute necessity', its logic is surely S5. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3) | |||
| A reaction: There are plenty of people (mainly in the strict empiricist tradition) who don't believe in 'absolute' necessity. |
| 18798 | It is the second-order part of intuitionistic logic which actually negates some classical theorems |
| Full Idea: Although intuitionistic propositional and first-order logics are sub-systems of the corresponding classical systems, intuitionistic second-order logic affirms the negations of some classical theorems. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) |
| 18799 | Intuitionists can accept Double Negation Elimination for decidable propositions |
| Full Idea: Double Negation Elimination is a rule of inference which the classicist accepts without restriction, but which the intuitionist accepts only for decidable propositions. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |||
| A reaction: This cures me of my simplistic understanding that intuitionists just reject the rules about double negation. |
| 18830 | Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic |
| Full Idea: Many set theorists doubt if the Generalised Continuum Hypothesis must be either true or false; certainly, its bivalence is far from obvious. All the same, almost all set theorists use classical logic in their proofs. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2) | |||
| A reaction: His point is that classical logic is usually taken to rest on bivalence. He offers the set theorists a helping hand, by defending classical logic without resorting to bivalence. |
| 18843 | The iterated conception of set requires continual increase in axiom strength |
| Full Idea: We are doomed to postulate an infinite sequence of successively stronger axiom systems as we try to spell out what is involved in iterating the power set operation 'as far as possible'. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.3) | |||
| A reaction: [W.W. Tait is behind this idea] The problem with set theory, then, especially as a foundation of mathematics, is that it doesn't just expand, but has to keep reinventing itself. The 'large cardinal axioms' are what is referred to. |
| 18836 | A set may well not consist of its members; the empty set, for example, is a problem |
| Full Idea: There seem strong grounds for rejecting the thesis that a set consists of its members. For one thing, the empty set is a perpetual embarrassment for the thesis. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4) | |||
| A reaction: Rumfitt also says that if 'red' has an extension, then membership of that set must be vague. Extensional sets are precise because their objects are decided in advance, but intensional (or logical) sets, decided by a predicate, can be vague. |
| 18837 | A set can be determinate, because of its concept, and still have vague membership |
| Full Idea: Vagueness in respect of membership is consistent with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of the concept A. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4) | |||
| A reaction: To be determinate, it must be presumed that there is some test which will decide what falls under the concept. The rule can say 'if it is vague, reject it' or 'if it is vague, accept it'. Without one of those, how could the set have a clear identity? |
| 18845 | If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom |
| Full Idea: Someone who is sympathetic to the thesis that the totality of sets is not well-defined ought to concede that we have no reason to think that the Power Set Axiom is true. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6) | |||
| A reaction: The point is that it is only this Axiom which generates the vast and expanding totality. In principle it is hard, though, to see what is intrinsically wrong with the operation of taking the power set of a set. Hence 'limitation of size'? |
| 18815 | Logic is higher-order laws which can expand the range of any sort of deduction |
| Full Idea: On the conception of logic recommended here, logical laws are higher-order laws that can be applied to expand the range of any deductive principles. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3) | |||
| A reaction: You need the concept of a 'deductive principle' to get this going, but I take it that might be directly known, rather than derived from a law. |
| 18805 | Classical logic rules cannot be proved, but various lines of attack can be repelled |
| Full Idea: There is not the slightest prospect of proving that the rules of classical logic are sound. ….All that the defender of classical logic can do is scrutinize particular attacks and try to repel them. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |||
| A reaction: This is the agenda for Rumfitt's book. |
| 18804 | The case for classical logic rests on its rules, much more than on the Principle of Bivalence |
| Full Idea: I think it is a strategic mistake to rest the case for classical logic on the Principle of Bivalence: the soundness of the classical logic rules is far more compelling than the truth of Bivalence. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |||
| A reaction: The 'rules' to which he is referring are those of 'natural deduction', which make very few assumptions, and are intended to be intuitively appealing. |
| 18827 | If truth-tables specify the connectives, classical logic must rely on Bivalence |
| Full Idea: If we specify the senses of the connectives by way of the standard truth-tables, then we must justify classical logic only by appeal to the Principle of Bivalence. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7) | |||
| A reaction: Rumfitt proposes to avoid the truth-tables, and hence not to rely on Bivalence for his support of classical logic. He accepts that Bivalence is doubtful, citing the undecidability of the Continuum Hypothesis as a problem instance. |
| 18813 | Logical consequence is a relation that can extended into further statements |
| Full Idea: Logical consequence, I argue, is distinguished from other implication relations by the fact that logical laws may be applied in extending any implication relation so that it applies among some complex statements involving logical connectives. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3) | |||
| A reaction: He offers implication in electronics as an example of a non-logical implication relation. This seems to indicate that logic must be monotonic, that consequence is transitive, and that the Cut Law always applies. |
| 18808 | Normal deduction presupposes the Cut Law |
| Full Idea: Our deductive practices seem to presuppose the Cut Law. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3) | |||
| A reaction: That is, if you don't believe that deductions can be transitive (and thus form a successful chain of implications), then you don't really believe in deduction. It remains a well known fact that you can live without the Cut Law. |
| 18840 | When faced with vague statements, Bivalence is not a compelling principle |
| Full Idea: I do not regard Bivalence, when applied to vague statements, as an intuitively compelling principle which we ought to try to preserve. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.7) | |||
| A reaction: The point of Rumfitt's book is to defend classical logic despite failures of bivalence. He also cites undecidable concepts such as the Continuum Hypothesis. |
| 18802 | In specifying a logical constant, use of that constant is quite unavoidable |
| Full Idea: There is no prospect whatever of giving the sense of a logical constant without using that very constant, and much else besides, in the metalinguistic principle that specifies that sense. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) |
| 18800 | Introduction rules give deduction conditions, and Elimination says what can be deduced |
| Full Idea: 'Introduction rules' state the conditions under which one may deduce a conclusion whose dominant logical operator is the connective. 'Elimination rules' state what may be deduced from some premises, where the major premise is dominated by the connective. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1) | |||
| A reaction: So Introduction gives conditions for deduction, and Elimination says what can actually be deduced. If my magic wand can turn you into a frog (introduction), and so I turn you into a frog, how does that 'eliminate' the wand? |
| 18809 | Logical truths are just the assumption-free by-products of logical rules |
| Full Idea: Gentzen's way of formalising logic has accustomed people to the idea that logical truths are simply the by-products of logical rules, that arise when all the assumptions on which a conclusion rests have been discharged. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.5) | |||
| A reaction: This is the key belief of those who favour the natural deduction account of logic. If you really believe in separate logic truths, then you can use them as axioms. |
| 18807 | Monotonicity means there is a guarantee, rather than mere inductive support |
| Full Idea: Monotonicity seems to mark the difference between cases in which a guarantee obtains and those where the premises merely provide inductive support for a conclusion. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3) | |||
| A reaction: Hence it is plausible to claim that 'non-monotonic logic' is a contradiction in terms. |
| 18842 | Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set |
| Full Idea: Menzel proposes that an ordinal is something isomorphic well-ordered sets have in common, so while an ordinal can be represented as a set, it is not itself a set, but a 'property' of well-ordered sets. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.2) | |||
| A reaction: [C.Menzel 1986] This is one of many manoeuvres available if you want to distance mathematics from set theory. |
| 18834 | Infinitesimals do not stand in a determinate order relation to zero |
| Full Idea: Infinitesimals do not stand in a determinate order relation to zero: we cannot say an infinitesimal is either less than zero, identical to zero, or greater than zero. ….Infinitesimals are so close to zero as to be theoretically indiscriminable from it. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.4) |
| 18846 | Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry) |
| Full Idea: One of the motivations behind Cantor's and Dedekind's pioneering explorations in the field was the ambition to give real analysis a new foundation in set theory - and hence a foundation independent of geometry. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6) | |||
| A reaction: Rumfitt is inclined to think that the project has failed, although a weaker set theory than ZF might do the job (within limits). |
| 18839 | An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases |
| Full Idea: A borderline red-orange object satisfies the disjunctive predicate 'red or orange', even though it satisfies neither 'red' or 'orange'. When applied to adjacent bands of colour, the disjunction 'sweeps up' objects which are reddish-orange. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5) | |||
| A reaction: Rumfitt offers a formal principle in support of this. There may be a problem with 'adjacent'. Different colour systems will place different colours adjacent to red. In other examples the idea of 'adjacent' may make no sense. Rumfitt knows this! |
| 18838 | The extension of a colour is decided by a concept's place in a network of contraries |
| Full Idea: On Sainsbury's picture, a colour has an extension that it has by virtue of its place in a network of contrary colour classifications. Something is determined to be 'red' by being a colour incompatible with orange, yellow, green, blue, indigo and violet. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5) | |||
| A reaction: Along with Idea 18839, this gives quite a nice account of vagueness, by requiring a foil to the vague predicate, and using the disjunction of the predicate and its foil to handle anything caught in between them. |
| 18816 | Metaphysical modalities respect the actual identities of things |
| Full Idea: The central characteristic mark of metaphysical necessity is that a metaphysical possibility respects the actual identities of things - in a capacious sense of 'thing'. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.4) | |||
| A reaction: He contrast this with logical necessity, and concludes that some truths are metaphysically but not logically necessary, such as 'Hesperus is identical with Phosphorus'. Personally I like the idea of a 'necessity-maker', so that fits. |
| 18825 | S5 is the logic of logical necessity |
| Full Idea: I accept the widely held thesis that S5 is the logic of logical necessity. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4 n16) | |||
| A reaction: It seems plausible that S5 is also the logic of metaphysical necessity, but that does not make them the same thing. The two types of necessity have two different grounds. |
| 18824 | Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right |
| Full Idea: Some philosophers describe the colour scarlet as a determination of the determinable red; since the ways the world might be are naturally taken to be properties of the world, it helps to bear this analogy in mind. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4) | |||
| A reaction: This fits nicely with the disposition accounts of modality which I favour. Hence being 'coloured' is a real property of objects, even in the absence of the name of its specific colour. |
| 18828 | If two possibilities can't share a determiner, they are incompatible |
| Full Idea: Two possibilities are incompatible when no possibility determines both. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1) | |||
| A reaction: This strikes me as just the right sort of language for building up a decent metaphysical picture of the world, which needs to incorporate possibilities as well as actualities. |
| 18821 | Possibilities are like possible worlds, but not fully determinate or complete |
| Full Idea: Possibilities are things of the same general character as possible worlds, on one popular conception of the latter. They differ from worlds, though, in that they are not required to be fully determinate or complete. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6) | |||
| A reaction: A rather promising approach to such things, even though a possibility is fairly determinate at its core, but very vague at the edges. It is possible that the UK parliament might be located in Birmingham, for example. Is this world 'complete'? |
| 18831 | Medieval logicians said understanding A also involved understanding not-A |
| Full Idea: Mediaeval logicians had a principle, 'Eadem est scientia oppositorum': in order to attain a clear conception of what it is for A to be the case, one needs to attain a conception of what it is for A not to be the case. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2) | |||
| A reaction: Presumably 'understanding' has to be a fairly comprehensive grasp of the matter, so understanding the negation sounds like a reasonable requirement for the real thing. |
| 18820 | In English 'evidence' is a mass term, qualified by 'little' and 'more' |
| Full Idea: In English, the word 'evidence' behaves as a mass term: we speak of someone's having little evidence for an assertion, and of one thinker's having more evidence than another for a claim. One the other hand, we also speak of 'pieces' of evidence. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.2) | |||
| A reaction: And having 'more' evidence does not mean having a larger number of pieces of evidence, so it really is like an accumulated mass. |
| 18817 | We understand conditionals, but disagree over their truth-conditions |
| Full Idea: It is striking that our understanding of conditionals is not greatly impeded by widespread disagreement about their truth-conditions. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 4.2) | |||
| A reaction: Compare 'if you dig there you might find gold' with 'if you dig there you will definitely find gold'. The second but not the first invites 'how do you know that?', implying truth. Two different ifs. |
| 18829 | The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A |
| Full Idea: The truth-grounds of '¬A' are precisely those possibilities that are incompatible with any truth-ground of A. | |||
| From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1) | |||
| A reaction: This is Rumfitt's proposal for the semantics of 'not', based on the central idea of a possibility, rather than a possible world. The incompatibility tracks back to an absence of shared grounding. |