95 ideas
| 21616 | Truth and falsity apply to suppositions as well as to assertions [Williamson] |
| Full Idea: The notion of truth and falsity apply to suppositions as well as to assertions. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This may not be obvious to those who emphasise pragmatics and ordinary language, but it is self-evident to anyone who emphasises logic. |
| 21623 | True and false are not symmetrical; false is more complex, involving negation [Williamson] |
| Full Idea: The concepts of truth and falsity are not symmetrical. The asymmetry is visible in the fundamental principles governing them, for F is essentially more complex than T, by its use of negation. | |
| From: Timothy Williamson (Vagueness [1994], 7.5) | |
| A reaction: If T and F are primitives, controlled by axioms, then they might be symmetrical in nature, but asymmetrical in use. However, if forced to choose just one primitive, I presume it would be T. |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| Full Idea: In a sense, satisfaction is the notion of 'truth in a model', and (as Hodes 1984 elegantly puts it) 'truth in a model' is a model of 'truth'. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |
| A reaction: So we can say that Tarski doesn't offer a definition of truth itself, but replaces it with a 'model' of truth. |
| 13643 | Aristotelian logic is complete [Shapiro] |
| Full Idea: Aristotelian logic is complete. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5) | |
| A reaction: [He cites Corcoran 1972] |
| 21602 | Many-valued logics don't solve vagueness; its presence at the meta-level is ignored [Williamson] |
| Full Idea: It is an illusion that many-valued logic constitutes a well-motivated and rigorously worked out theory of vagueness. ...[top] There has been a reluctance to acknowledge higher-order vagueness, or to abandon classical logic in the meta-language. | |
| From: Timothy Williamson (Vagueness [1994], 4.12) |
| 13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
| Full Idea: If, for every b∈d, a∈b entails that a∈d, the d is said to be 'transitive'. In other words, d is transitive if it contains every member of each of its members. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.2) | |
| A reaction: The alternative would be that the members of the set are subsets, but the members of those subsets are not themselves members of the higher-level set. |
| 13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
| Full Idea: The axiom of choice is essential for proving the downward Löwenheim-Skolem Theorem. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) |
| 13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
| Full Idea: Is there a notion of set in the jurisdiction of logic, or does it belong to mathematics proper? | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: It immediately strikes me that they might be neither. I don't see that relations between well-defined groups of things must involve number, and I don't see that mapping the relations must intrinsically involve logical consequence or inference. |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
| Full Idea: In set theory it is central to the iterative conception that the membership relation is well-founded, ...which means there are no infinite descending chains from any relation. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.4) |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| Full Idea: The argument behind Russell's paradox shows that in set theory there are logical sets (i.e. classes) that are not iterative sets. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3) | |
| A reaction: In his preface, Shapiro expresses doubts about the idea of a 'logical set'. Hence the theorists like the iterative hierarchy because it is well-founded and under control, not because it is comprehensive in scope. See all of pp.19-20. |
| 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
| Full Idea: Iterative sets do not exhibit a Boolean structure, because the complement of an iterative set is not itself an iterative set. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) |
| 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
| Full Idea: A 'well-ordering' of a set X is an irreflexive, transitive, and binary relation on X in which every non-empty subset of X has a least element. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.3) | |
| A reaction: So there is a beginning, an ongoing sequence, and no retracing of steps. |
| 13627 | There is no 'correct' logic for natural languages [Shapiro] |
| Full Idea: There is no question of finding the 'correct' or 'true' logic underlying a part of natural language. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: One needs the context of Shapiro's defence of second-order logic to see his reasons for this. Call me romantic, but I retain faith that there is one true logic. The Kennedy Assassination problem - can't see the truth because drowning in evidence. |
| 13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
| Full Idea: A logic can be seen as the ideal of what may be called 'relative justification', the process of coming to know some propositions on the basis of others. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.3.1) | |
| A reaction: This seems to be the modern idea of logic, as opposed to identification of a set of 'logical truths' from which eternal necessities (such as mathematics) can be derived. 'Know' implies that they are true - which conclusions may not be. |
| 13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
| Full Idea: Bernays (1918) formulated and proved the completeness of propositional logic, the first precise solution as part of the Hilbert programme. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.1) |
| 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
| Full Idea: In 1910 Weyl observed that set theory seemed to presuppose natural numbers, and he regarded numbers as more fundamental than sets, as did Fraenkel. Dedekind had developed set theory independently, and used it to formulate numbers. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.2) |
| 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
| Full Idea: Skolem and Gödel were the main proponents of first-order languages. The higher-order language 'opposition' was championed by Zermelo, Hilbert, and Bernays. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2) |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| Full Idea: Almost all the systems developed in the first part of the twentieth century are higher-order; first-order logic was an afterthought. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) |
| 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
| Full Idea: The 'triumph' of first-order logic may be related to the remnants of failed foundationalist programmes early this century - logicism and the Hilbert programme. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: Being complete must also be one of its attractions, and Quine seems to like it because of its minimal ontological commitment. |
| 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
| Full Idea: Tharp (1975) suggested that compactness, semantic effectiveness, and the Löwenheim-Skolem properties are consequences of features one would want a logic to have. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |
| A reaction: I like this proposal, though Shapiro is strongly against. We keep extending our logic so that we can prove new things, but why should we assume that we can prove everything? That's just what Gödel suggests that we should give up on. |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| Full Idea: The notion of finitude is explicitly 'built in' to the systems of first-order languages in one way or another. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1) | |
| A reaction: Personally I am inclined to think that they are none the worse for that. No one had even thought of all these lovely infinities before 1870, and now we are supposed to change our logic (our actual logic!) to accommodate them. Cf quantum logic. |
| 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
| Full Idea: Shapiro preferred second-order logic to set theory because second-order logic refers only to the relations and operations in a domain, and not to the other things that set-theory brings with it - other domains, higher-order relations, and so forth. | |
| From: report of Stewart Shapiro (Foundations without Foundationalism [1991]) by Shaughan Lavine - Understanding the Infinite VII.4 |
| 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
| Full Idea: Three systems of semantics for second-order languages: 'standard semantics' (variables cover all relations and functions), 'Henkin semantics' (relations and functions are a subclass) and 'first-order semantics' (many-sorted domains for variable-types). | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: [my summary] |
| 13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
|
Full Idea:
In 'Henkin' semantics, in a given model |
|
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3) |
| 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
| Full Idea: In the standard semantics of second-order logic, by fixing a domain one thereby fixes the range of both the first-order variables and the second-order variables. There is no further 'interpreting' to be done. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3) | |
| A reaction: This contrasts with 'Henkin' semantics (Idea 13650), or first-order semantics, which involve more than one domain of quantification. |
| 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
| Full Idea: The counterparts of Completeness, Compactness and the Löwenheim-Skolem theorems all fail for second-order languages with standard semantics, but hold for Henkin or first-order semantics. Hence such logics are much like first-order logic. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |
| A reaction: Shapiro votes for the standard semantics, because he wants the greater expressive power, especially for the characterization of infinite structures. |
| 21611 | Formal semantics defines validity as truth preserved in every model [Williamson] |
| Full Idea: An aim of formal semantics is to define in mathematical terms a set of models such that an argument is valid if and only if it preserves truth in every model in the set, for that will provide us with a precise standard of validity. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) |
| 13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
| Full Idea: It follows from Gödel's incompleteness theorem that the semantic consequence relation of second-order logic is not effective. For example, the set of logical truths of any second-order logic is not recursively enumerable. It is not even arithmetic. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: I don't fully understand this, but it sounds rather major, and a good reason to avoid second-order logic (despite Shapiro's proselytising). See Peter Smith on 'effectively enumerable'. |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| Full Idea: Second-order logic is inherently incomplete, so its semantic consequence relation is not effective. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1) |
| 21606 | 'Bivalence' is the meta-linguistic principle that 'A' in the object language is true or false [Williamson] |
| Full Idea: The meta-logical law of excluded middle is the meta-linguistic principle that any statement 'A' in the object language is either truth or false; it is now known as the principle of 'bivalence'. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: [He cites Henryk Mehlberg 1958] See also Idea 21605. Without this way of distinguishing bivalence from excluded middle, most discussions of them strikes me as shockingly lacking in clarity. Personally I would cut the normativity from this one. |
| 21605 | Excluded Middle is 'A or not A' in the object language [Williamson] |
| Full Idea: The logical law of excluded middle (now the standard one) is the schema 'A or not A' in the object-language. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: [He cites Henryk Mehlberg 1958] See Idea 21606. The only sensible way to keep Excluded Middle and Bivalence distinct. I would say: (meta-) only T and F are available, and (object) each proposition must have one of them. Are they both normative? |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
| Full Idea: It is sometimes difficult to find a formula that is a suitable counterpart of a particular sentence of natural language, and there is no acclaimed criterion for what counts as a good, or even acceptable, 'translation'. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| Full Idea: The main role of substitutional semantics is to reduce ontology. As an alternative to model-theoretic semantics for formal languages, the idea is to replace the 'satisfaction' relation of formulas (by objects) with the 'truth' of sentences (using terms). | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) | |
| A reaction: I find this very appealing, and Ruth Barcan Marcus is the person to look at. My intuition is that logic should have no ontology at all, as it is just about how inference works, not about how things are. Shapiro offers a compromise. |
| 21612 | Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B' [Williamson] |
| Full Idea: Argument by Cases (or or-elimination) is the standard way of using disjunctive premises. If one can argue from A and some premises to C, and from B and some premises to C, one can argue from 'A or B' and the combined premises to C. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) |
| 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
| Full Idea: The 'satisfaction' relation may be thought of as a function from models, assignments, and formulas to the truth values {true,false}. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |
| A reaction: This at least makes clear that satisfaction is not the same as truth. Now you have to understand how Tarski can define truth in terms of satisfaction. |
| 13644 | Semantics for models uses set-theory [Shapiro] |
| Full Idea: Typically, model-theoretic semantics is formulated in set theory. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5.1) |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| Full Idea: An axiomatization is 'categorical' if all its models are isomorphic to one another; ..hence it has 'essentially only one' interpretation [Veblen 1904]. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1) |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| Full Idea: Categoricity cannot be attained in a first-order language. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.3) |
| 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
| Full Idea: A language has the Downward Löwenheim-Skolem property if each satisfiable countable set of sentences has a model whose domain is at most countable. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |
| A reaction: This means you can't employ an infinite model to represent a fact about a countable set. |
| 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
| Full Idea: A language has the Upward Löwenheim-Skolem property if for each set of sentences whose model has an infinite domain, then it has a model at least as big as each infinite cardinal. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |
| A reaction: This means you can't have a countable model to represent a fact about infinite sets. |
| 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
| Full Idea: The Löwenheim-Skolem theorems mean that no first-order theory with an infinite model is categorical. If Γ has an infinite model, then it has a model of every infinite cardinality. So first-order languages cannot characterize infinite structures. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |
| A reaction: So much of the debate about different logics hinges on characterizing 'infinite structures' - whatever they are! Shapiro is a leading structuralist in mathematics, so he wants second-order logic to help with his project. |
| 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
| Full Idea: The Upward Löwenheim-Skolem theorem fails (trivially) with substitutional semantics. If there are only countably many terms of the language, then there are no uncountable substitution models. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) | |
| A reaction: Better and better. See Idea 13674. Why postulate more objects than you can possibly name? I'm even suspicious of all real numbers, because you can't properly define them in finite terms. Shapiro objects that the uncountable can't be characterized. |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| Full Idea: A logic is 'weakly sound' if every theorem is a logical truth, and 'strongly sound', or simply 'sound', if every deduction from Γ is a semantic consequence of Γ. Soundness indicates that the deductive system is faithful to the semantics. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1) | |
| A reaction: Similarly, 'weakly complete' is when every logical truth is a theorem. |
| 13628 | We can live well without completeness in logic [Shapiro] |
| Full Idea: We can live without completeness in logic, and live well. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: This is the kind of heady suggestion that American philosophers love to make. Sounds OK to me, though. Our ability to draw good inferences should be expected to outrun our ability to actually prove them. Completeness is for wimps. |
| 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
| Full Idea: It is sometimes said that non-compactness is a defect of second-order logic, but it is a consequence of a crucial strength - its ability to give categorical characterisations of infinite structures. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: The dispute between fans of first- and second-order may hinge on their attitude to the infinite. I note that Skolem, who was not keen on the infinite, stuck to first-order. Should we launch a new Skolemite Crusade? |
| 13646 | Compactness is derived from soundness and completeness [Shapiro] |
| Full Idea: Compactness is a corollary of soundness and completeness. If Γ is not satisfiable, then, by completeness, Γ is not consistent. But the deductions contain only finite premises. So a finite subset shows the inconsistency. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1) | |
| A reaction: [this is abbreviated, but a proof of compactness] Since all worthwhile logics are sound, this effectively means that completeness entails compactness. |
| 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
| Full Idea: A logical language is 'semantically effective' if the collection of logically true sentences is a recursively enumerable set of strings. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) |
| 21599 | A sorites stops when it collides with an opposite sorites [Williamson] |
| Full Idea: A sorites paradox is stopped when it collides with a sorites paradox going in the opposite direction. That account will not strike a logician as solving the sorites paradox. | |
| From: Timothy Williamson (Vagueness [1994], 3.3) |
| 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
| Full Idea: 'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.1) | |
| A reaction: On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before. |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| Full Idea: The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) |
| 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
| Full Idea: By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.3) |
| 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
| Full Idea: The 'continuum' is the cardinality of the powerset of a denumerably infinite set. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2) |
| 13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
| Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28) | |
| A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is. |
| 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
| Full Idea: There are sets of natural numbers definable in set-theory but not in arithmetic. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3) |
| 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
| Full Idea: It is claimed that aiming at a universal language for all contexts, and the thesis that logic does not involve a process of abstraction, separates the logicists from algebraists and mathematicians, and also from modern model theory. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) | |
| A reaction: I am intuitively drawn to the idea that logic is essentially the result of a series of abstractions, so this gives me a further reason not to be a logicist. Shapiro cites Goldfarb 1979 and van Heijenoort 1967. Logicists reduce abstraction to logic. |
| 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
| Full Idea: I extend Quinean holism to logic itself; there is no sharp border between mathematics and logic, especially the logic of mathematics. One cannot expect to do logic without incorporating some mathematics and accepting at least some of its ontology. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: I have strong sales resistance to this proposal. Mathematics may have hijacked logic and warped it for its own evil purposes, but if logic is just the study of inferences then it must be more general than to apply specifically to mathematics. |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
| Full Idea: Some authors (Poincaré and Russell, for example) were disposed to reject properties that are not definable, or are definable only impredicatively. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) | |
| A reaction: I take Quine to be the culmination of this line of thought, with his general rejection of 'attributes' in logic and in metaphysics. |
| 21589 | When bivalence is rejected because of vagueness, we lose classical logic [Williamson] |
| Full Idea: The principle of bivalence (that every statement is either true or false) has been rejected for vague languages. To reject bivalence is to reject classical logic or semantics. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: His example is specifying a moment when Rembrandt became 'old'. This is the number one reason why the problem of vagueness is seen as important. Is the rejection of classical logic a loss of our grip on the world? |
| 21596 | Vagueness undermines the stable references needed by logic [Williamson] |
| Full Idea: Logic requires expressions to have the same referents wherever they occur; vague natural languages violate this contraint. | |
| From: Timothy Williamson (Vagueness [1994], 2.2) | |
| A reaction: This doesn't mean that logic has to win. Maybe it is important for philosophers who see logic as central to be always aware of vagueness as the gulf between their precision and the mess of reality. Precision is worth trying for, though. |
| 21601 | A vague term can refer to very precise elements [Williamson] |
| Full Idea: Both 30° and 60° are clearly acute angles. 'Acute' is precise in all relevant respects. Nevertheless, 30° is acuter than 60°. | |
| From: Timothy Williamson (Vagueness [1994], 4.11) | |
| A reaction: A very nice example of something which is vague, despite involving precise ingredients. But then 'bald' is vague, while 'this is a hair on his head' is fairly precise. |
| 21629 | Equally fuzzy objects can be identical, so fuzziness doesn't entail vagueness [Williamson] |
| Full Idea: Fuzzy boundaries do not in any way require vague identity. Objects are identical only if their boundaries have exactly the same fuzziness. | |
| From: Timothy Williamson (Vagueness [1994], 9.2) | |
| A reaction: This all rests on the Fregean idea that determinate existence requires the ability to participate in an identity statement. |
| 21591 | Vagueness is epistemic. Statements are true or false, but we often don't know which [Williamson] |
| Full Idea: My thesis is that vagueness is an epistemic phenomenon. In cases of unclarity, statements remain true or false, but speakers of the language have no way of knowing which. Higher-order vagueness consists in ignorance about ignorance. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: He has plumped for the intuitively least plausible theory. It means that a hair dropping out of someone's head triggers a situation where they are 'bald', but none of us know when that was. And Rembrandt became 'old' in an instant. |
| 21619 | If a heap has a real boundary, omniscient speakers would agree where it is [Williamson] |
| Full Idea: If, in judging a heap as grains are removed, omniscient speakers all stop at the same point, it must does mark some sort of previously hidden boundary. ...If there is no hidden boundary, then different omniscient speakers would stop at different points. | |
| From: Timothy Williamson (Vagueness [1994], 7.3) | |
| A reaction: A very nice thought experiment, which obviously won't settle anything, but brings out nicely the view the vagueness is a sort of ignorance. God is never vague in the application of terms (though God might withhold the application if there is no boundary). |
| 21620 | The epistemic view says that the essence of vagueness is ignorance [Williamson] |
| Full Idea: The epistemic view is that ignorance is the real essence of the phenomenon ostensively identified as vagueness. ...[203] According to the epistemic view, I am either thin or not thin, ...and we have no idea how to find out out which. | |
| From: Timothy Williamson (Vagueness [1994], 7.4) | |
| A reaction: Presumably this implies that there is often a real border (of which we may be ignorant), but it doesn't seem to rule out cases where there just is no border. Where does the east Atlantic meet the west Atlantic? |
| 21622 | If there is a true borderline of which we are ignorant, this drives a wedge between meaning and use [Williamson] |
| Full Idea: A common complaint against the epistemic view is that to postulate a matter of fact in borderline cases is to suppose, incoherently, that the meanings of our words draw a line where our use of them does not. | |
| From: Timothy Williamson (Vagueness [1994], 7.5) | |
| A reaction: This doesn't necessarily seem to require the view that the meaning of words is their usage. Just that if there is one consensus on usage, it seems unlikely that there is a different underlying reality about the true meaning. Externalist meanings? |
| 9120 | Vagueness in a concept is its indiscriminability from other possible concepts [Williamson] |
| Full Idea: Vagueness in a concept is its indiscriminability from other possible concepts; this can be reconciled with our knowledge of vague terms. | |
| From: Timothy Williamson (Vagueness [1994], 8.1) | |
| A reaction: Sorensen objects that this makes vagueness too relative to members of a speech community. He prefers 'absolute borderline cases'. If you like the epistemic view, then Williamson seems more plausible. My 'vague' might differ from yours. |
| 21625 | The vagueness of 'heap' can remain even when the context is fixed [Williamson] |
| Full Idea: Vagueness remains even when the context is fixed. In principle, a vague word might exhibit no context dependence whatsoever. ...For example, a dispute over whether someone has left a 'heap' of sand on the floor. | |
| From: Timothy Williamson (Vagueness [1994], 7.7) | |
| A reaction: A fairly devastating rebuttal of what seems to be David Lewis's view. He talks of something being 'smooth' depending on context. |
| 21614 | The 'nihilist' view of vagueness says that 'heap' is not a legitimate concept [Williamson] |
| Full Idea: The 'nihilist' view is that no genuine distinction can be vaguely drawn; since vague expressions are not properly meaningful, there is nothing for sorites reasoning to betray; they are empty. | |
| From: Timothy Williamson (Vagueness [1994], 6.1) | |
| A reaction: He cites Frege as holding this view. The thought is that 'heap' is not a legitimate concept, so fussing over what qualifies as one is pointless. This seems to be a semantic view of vagueness, of which the main rival is the contextual view. |
| 21617 | We can say propositions are bivalent, but vague utterances don't express a proposition [Williamson] |
| Full Idea: A philosopher might endorse bivalence for propositions, while treating vagueness as the failure of an utterance to express a unique proposition. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This idea jumps at out me as an extremely promising approach to vagueness, because I am a fan of propositions (and have written a paper on them). The whole point of propositions is that they are not ambiguous (and probably not vague). |
| 21618 | If the vague 'TW is thin' says nothing, what does 'TW is thin if his perfect twin is thin' say? [Williamson] |
| Full Idea: If vague utterances in borderline cases fail to say anything, then if 'TW is thin' is vague, and TW has a twin of identical dimensions, it still seems that 'If TW is thin then his twin is thin' must be true, and so it must have said something. | |
| From: Timothy Williamson (Vagueness [1994], 7.2 (d)) | |
| A reaction: This an objection to the Fregean 'nihilistic' view of Idea 21614. I am inclined to a solution based on the proposition expressed, rather than the sentence. The first question is whether you are willing to assert 'TW is thin'. |
| 21590 | Asking when someone is 'clearly' old is higher-order vagueness [Williamson] |
| Full Idea: Difficulties of vagueness are presented by the question 'When did Rembrandt become clearly old?', and the iterating question 'When did he become clearly clearly old?'. This is the phenomenon of higher-order vagueness. The language of vagueness is vague. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: [compressed] I presume the bottom level is a question about Rembrandt, the second level is about this use of the word 'old', and the third level is about this particular application of the word 'clearly'. Meta-languages. |
| 21592 | Supervaluation keeps classical logic, but changes the truth in classical semantics [Williamson] |
| Full Idea: Supervaluationism preserves almost all of classical logic, at the expense of classical semantics, but giving a non-standard account of truth. I argue that its treatment of higher-order vagueness undermines the non-standard account of truth. | |
| From: Timothy Williamson (Vagueness [1994], Intro) |
| 21603 | You can't give a precise description of a language which is intrinsically vague [Williamson] |
| Full Idea: If a vague language is made precise, its expressions change in meaning, so an accurate semantic description of the precise language is inaccurate as a description of the vague one. | |
| From: Timothy Williamson (Vagueness [1994], 5.1) | |
| A reaction: Kind of obvious, really, but it clarifies the nature of any project (starting with Leibniz) to produce a wholly precise language. That is usually seen as a specialist language for science. |
| 21604 | Supervaluation assigns truth when all the facts are respected [Williamson] |
| Full Idea: 'Admissible' interpretations respect all the theoretical and ostensive connections. ...'Supervaluation' is the assignment of truth to the statements true on all admissible valuations, falsity to the false one, and neither to the rest. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: So 'he is bald' is true if when faced with all observations and definitions it is acceptable. Prima facie, that doesn't sound like a solution to the problem. Supervaluation started in philosophy of science. [p.156 'Admissible seems vague'] |
| 21607 | Supervaluation has excluded middle but not bivalence; 'A or not-A' is true, even when A is undecided [Williamson] |
| Full Idea: The supervaluationist denies bivalence but accepts excluded middle. The statement 'A or not-A' is true on each admissible interpretation, and therefore true, even if 'A' (and hence 'not-A') are true and some and false on others, so neither T nor F. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: See Ideas 21605 and 21606 for the distinction being used here. Denying bivalence allows 'A' to be neither true nor false. It seems common sense that 'he is either bald or not-bald' is true, without being sure about the disjuncts. |
| 21608 | Truth-functionality for compound statements fails in supervaluation [Williamson] |
| Full Idea: A striking fearure of supervaluations is the failure of truth-functionality for compound statements. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: Supervaluations has the initial appearance of enhancing classical logic, but turns out to somewhat undermine it. Hence Williamson's lack of sympathy. But see Idea 21610. |
| 21609 | Supervaluationism defines 'supertruth', but neglects it when defining 'valid' [Williamson] |
| Full Idea: Supervaluationists identify truth with 'supertruth'; since validity is necessary preservation of truth, they should identify it with necessary preservation of supertruth. But it plays no role in their definition of 'local' validity. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: [See text for 'local'] Generally Williamson's main concern with attempts to sort out vagueness is that higher-order and meta-language issues are neglected. |
| 21610 | Supervaluation adds a 'definitely' operator to classical logic [Williamson] |
| Full Idea: Supervaluation seems to inherit the power of classical logic, ...but also enables it to be extended. It makes room for a new operator 'definitely' to express supertruth in the object-language. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: Once you mention higher-order vagueness you can see a regress looming over the horizon. 'He is definitely definitely definitely bald'. [p.164 he says 'definitely' has no analysis, and is an uninteresting primitive] |
| 21613 | Supervaluationism cannot eliminate higher-order vagueness [Williamson] |
| Full Idea: Supervaluationism cannot eliminate higher-order vagueness. It must conduct its business in a vague meta-language. ...[162] All truth is at least disquotational, and supertruth is not. | |
| From: Timothy Williamson (Vagueness [1994], 5.6) | |
| A reaction: This is Williamson's final verdict on the supervaluation strategy for vagueness. Intuitively, it looks as if merely narrowing down the vagueness (by some sort of consensus) is no solution to the problem of vagueness. |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| Full Idea: Properties are often taken to be intensional; equiangular and equilateral are thought to be different properties of triangles, even though any triangle is equilateral if and only if it is equiangular. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3) | |
| A reaction: Many logicians seem to want to treat properties as sets of objects (red being just the set of red things), but this looks like a desperate desire to say everything in first-order logic, where only objects are available to quantify over. |
| 21633 | Nominalists suspect that properties etc are our projections, and could have been different [Williamson] |
| Full Idea: The nominalist suspects that properties, relations and states of affairs are mere projections onto the world of our forms of speech. One source of the suspicion is a sense that we could just as well have classified things differently. | |
| From: Timothy Williamson (Vagueness [1994], 9.3) | |
| A reaction: I know it is very wicked to say so, but I'm afraid I have some sympathy with this view. But I like the primary/secondary distinction, so there is more 'projection' in the latter case. Classification is not random; it is a response to reality. |
| 21630 | If fuzzy edges are fine, then why not fuzzy temporal, modal or mereological boundaries? [Williamson] |
| Full Idea: If objects can have fuzzy spatial boundaries, surely they can have fuzzy temporal, modal or mereological boundaries too. | |
| From: Timothy Williamson (Vagueness [1994], 9.2) | |
| A reaction: Fair point. I think there is a distinction between parts of the thing, such as its edges, being fuzzy, and the whole thing being fuzzy, in the temporal case. |
| 23647 | Objects have an essential constitution, producing its qualities, which we are too ignorant to define [Reid] |
| Full Idea: Individuals and objects have a real essence, or constitution of nature, from which all their qualities flow: but this essence our faculties do not comprehend. They are therefore incapable of definition. | |
| From: Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], 1) | |
| A reaction: Aha - he's one of us! I prefer the phrase 'essential nature' of an object, which is understood, I think, by everyone. I especially like the last bit, directed at those who mistakenly think that Aristotle identified the essence with the definition. |
| 21632 | A river is not just event; it needs actual and counterfactual boundaries [Williamson] |
| Full Idea: A river is not just an event. One would need to specify counterfactual as well as actual boundaries. | |
| From: Timothy Williamson (Vagueness [1994], 9.3) | |
| A reaction: In other words the same river can change its course a bit, but it can't head off in the opposite direction. |
| 21621 | We can't infer metaphysical necessities to be a priori knowable - or indeed knowable in any way [Williamson] |
| Full Idea: The inference from metaphysical necessity to a priori knowlability is, as Kripke has emphasized, fallacious. Indeed, metaphysical necessities cannot be assumed knowable in any way at all. | |
| From: Timothy Williamson (Vagueness [1994], 7.4) | |
| A reaction: The second sentence sounds like common sense. He cites Goldbach's Conjecture. A nice case of the procedural rule of keeping your ontology firmly separated from your epistemology. How is it? is not How do we know it? |
| 11958 | Impossibilites are easily conceived in mathematics and geometry [Reid, by Molnar] |
| Full Idea: Reid pointed out how easily conceivable mathematical and geometric impossibilities are. | |
| From: report of Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], IV.III) by George Molnar - Powers 11.3 | |
| A reaction: The defence would be that you have to really really conceive them, and the only way the impossible can be conceived is by blurring it at the crucial point, or by claiming to conceive more than you actually can |
| 21627 | We have inexact knowledge when we include margins of error [Williamson] |
| Full Idea: Inexact knowledge is a widespread and easily recognised cognitive phenomenon, whose underlying nature turns out to be characterised by the holding of margin of error principles. | |
| From: Timothy Williamson (Vagueness [1994], 8.3) | |
| A reaction: Williamson is invoking this as a tool in developing his epistemic view of vagueness. It obviously invites the question of how it can be knowledge if error is a possibility. A very large margin of error would obviously invalidate it. |
| 21626 | Knowing you know (KK) is usually denied if the knowledge concept is missing, or not considered [Williamson] |
| Full Idea: The failure of the KK principle is not news. The standard counterexamples involve knowing subjects who lack the concept of knowledge, or have not reflected on their knowledge, and therefore do not know that they know. | |
| From: Timothy Williamson (Vagueness [1994], 8.2) | |
| A reaction: There is also the timid but knowledgeable pupil, who can't believe they know so much. The simplest case would be if we accept that animals know lots of things, but are largely devoid of any metathinking. |
| 21631 | To know, believe, hope or fear, one must grasp the thought, but not when you fail to do them [Williamson] |
| Full Idea: To know, believe, hope, or fear that A, one must grasp the thought that A. In contrast, to fail to know, believe, hope or fear that A, one need not grasp the thought that A. | |
| From: Timothy Williamson (Vagueness [1994], 9.3 c) | |
| A reaction: A simple point, which at least shows that propositional attitudes are a two-stage operation. |
| 21600 | 'Blue' is not a family resemblance, because all the blues resemble in some respect [Williamson] |
| Full Idea: 'Blue' is vague by some standards, for it has borderline cases, but that does not make it a family resemblance term, for all the shades of blue resemble each other in some respect. | |
| From: Timothy Williamson (Vagueness [1994], 3.3) | |
| A reaction: Presumably the point of family resemblance is that fringe members as still linked to the family, despite having lost the main features. A bit of essentialism seems needed here. |
| 23646 | Reference is by name, or a term-plus-circumstance, or ostensively, or by description [Reid] |
| Full Idea: An individual is expressed by a proper name, or by a general word joined to distinguishing circumstances; if unknown, it may be pointed out to the senses; when beyond the reach of the senses it may be picked out by an imperfect but true description. | |
| From: Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], 1) | |
| A reaction: [compressed] If Putnam, Kripke and Donnellan had read this paragraph they could have save themselves a lot of work! I take reference to be the activity of speakers and writers, and these are the main tools of the trade. |
| 21615 | References to the 'greatest prime number' have no reference, but are meaningful [Williamson] |
| Full Idea: The predicate 'is a prime number greater than all other prime numbers' is necessarily not true of anything, but it is not semantically defective, for it occurs in sentences that constitute a sound proof that there is no such number. | |
| From: Timothy Williamson (Vagueness [1994], 6.2) | |
| A reaction: One might reply that the description can be legitimately mentioned, but not legitimately used. |
| 23645 | A word's meaning is the thing conceived, as fixed by linguistic experts [Reid] |
| Full Idea: The meaning of a word (such as 'felony') is the thing conceived; and that meaning is the conception affixed to it by those who best understand the language. | |
| From: Thomas Reid (Essays on Intellectual Powers 4: Conception [1785], 1) | |
| A reaction: He means legal experts. This is precisely that same as Putnam's account of the meaning of 'elm tree'. His discussion here of reference is the earliest I have encountered, and it is good common sense (for which Reid is famous). |
| 18038 | The 't' and 'f' of formal semantics has no philosophical interest, and may not refer to true and false [Williamson] |
| Full Idea: In a formal semantics we can label two properties 't' and 'f' and suppose that some sentences have neither (or both). Such a manoeuvre shows nothing of philosophical interest. No connection has been made between 't' and 'f' and truth and falsity. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This is right, and means there is a huge gulf between 'formal' semantics (which could be implemented on a computer), and seriously interesting semantics about how language refers to and describes the world. |
| 21624 | It is known that there is a cognitive loss in identifying propositions with possible worlds [Williamson] |
| Full Idea: It is well known that when a proposition is identified with the set of possible worlds at which it is true, a region in the space of possible worlds, cognitively significant distinctions are lost. | |
| From: Timothy Williamson (Vagueness [1994], 7.6) | |
| A reaction: Alas, he doesn't specify which distinctions get lost, so this is just a pointer. It would seem likely that two propositions could have identical sets of possible worlds, while not actually saying the same thing. Equilateral/equiangular. |