87 ideas
| 6267 | A culture needs to admit that knowledge is more extensive than just 'science' [Putnam] |
| Full Idea: The acknowledgement that the sphere of knowledge is wider than the sphere of 'science' seems to me to be a cultural necessity if we are to arrive at a sane and human view of ourselves or of science. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
| A reaction: A very nice remark, with which I intuitively agree, but then you are left with the problem of explaining how something can qualify as knowledge when it can't pass the stringent tests of science. How wide to we spread, and why? |
| 6272 | 'True' and 'refers' cannot be made scientically precise, but are fundamental to science [Putnam] |
| Full Idea: Some non-scientific knowledge is presupposed by science; for example, I have been arguing that 'refers' and 'true' cannot be made scientifically precise; yet truth is a fundamental term in logic - a precise science. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec VI) | |
| A reaction: We might ask whether we 'know' what 'true' and 'refers' mean, as opposed to being able to use them. If their usage doesn't count as knowledge, then we could still end up with all actual knowledge being somehow 'scientific'. |
| 6276 | 'The rug is green' might be warrantedly assertible even though the rug is not green [Putnam] |
| Full Idea: 'The rug is green' might be warrantedly assertible even though the rug is not green. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
| A reaction: The word 'warranted' seems to be ambiguous in modern philosophy. See Idea 6150. There seem to be internalist and externalist versions. It seems clear to say that a belief could be well-justified and yet false. |
| 6266 | We need the correspondence theory of truth to understand language and science [Putnam] |
| Full Idea: A correspondence theory of truth is needed to understand how language works, and how science works. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
| A reaction: Putnam retreated from this position to a more pragmatic one later on, but all my sympathies are with the present view, despite being repeatedly told by modern philosophers that I am wrong. See McGinn (Idea 6085) and Searle (Idea 3508). |
| 6277 | Correspondence between concepts and unconceptualised reality is impossible [Putnam] |
| Full Idea: The great nineteenth century argument against the correspondence theory of truth was that one cannot think of truth as correspondence to facts (or 'reality') because one would need to compare concepts directly with unconceptualised reality. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
| A reaction: Presumably the criticism was offered by idealists, who preferred a coherence theory. The defence is to say that there is a confusion here between a concept and the contents of a concept. The contents of a concept are designed to be facts. |
| 6264 | In Tarski's definition, you understand 'true' if you accept the notions of the object language [Putnam] |
| Full Idea: Anyone who accepts the notions of whatever object language is in question - and this can be chosen arbitrarily - can also understand 'true' as defined by Tarski for that object language. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
| A reaction: Thus if we say "'snow is white' is true iff snow is white", then if you 'accept the notion' that snow is white in English, you understand what 'true' means. This seems to leave you with the meaning of 'snow is white' being its truth conditions. |
| 6265 | Tarski has given a correct account of the formal logic of 'true', but there is more to the concept [Putnam] |
| Full Idea: What Tarski has done is to give us a perfectly correct account of the formal logic of the concept 'true', but the formal logic of the concept is not all there is to the notion of truth. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Intro) | |
| A reaction: I find this refreshing. A lot of modern philosophers seem to think that truth is no longer an interesting philosophical topic, because deflationary accounts have sidelined it, but I take the concept to be at the heart of metaphysics. |
| 6269 | Only Tarski has found a way to define 'true' [Putnam] |
| Full Idea: There is only one way anyone knows how to define 'true' and that is Tarski's way. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec II.5) | |
| A reaction: However, Davidson wrote a paper called 'On the Folly of Trying to Define Truth', which seems to reject even Tarski. Also bear in mind Putnam's earlier remark (Idea 6265) that there is more to truth than Tarski's definition. Just take 'true' as primitive. |
| 8166 | Truth is part of semantics, since valid inference preserves truth [Dummett] |
| Full Idea: The concept of truth belongs to semantics, since after all truth is what must be preserved by a valid deductive inference. | |
| From: Michael Dummett (Thought and Reality [1997], 2) | |
| A reaction: Does this conclusion follow? Compare 'nice taste belongs to cooking, since that is what cooking must preserve'. I don't like this. I take 'truth' to be a relevant concept to a discussion of a dog's belief that it is going to be taken for a walk. |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| Full Idea: By Gödel's First Incompleteness Theorem, there cannot be a negation-complete set theory. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.3) | |
| A reaction: This means that we can never prove all the truths of a system of set theory. |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| Full Idea: Going second-order in arithmetic enables us to prove new first-order arithmetical sentences that we couldn't prove before. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.4) | |
| A reaction: The wages of Satan, perhaps. We can prove things about objects by proving things about their properties and sets and functions. Smith says this fact goes all the way up the hierarchy. |
| 8173 | Language can violate bivalence because of non-referring terms or ill-defined predicates [Dummett] |
| Full Idea: Two features of natural languages cause them to violate bivalence: singular terms (or proper names) which have a sense but fail to denote an object ('the centre of the universe'); and predicates which are not well defined for every object. | |
| From: Michael Dummett (Thought and Reality [1997], 4) | |
| A reaction: If we switch from sentences to propositions these problems might be avoided. If there is no reference, or a vague predicate, then there is (maybe) just no proposition being expressed which could be evaluated for truth. |
| 8179 | The law of excluded middle is the logical reflection of the principle of bivalence [Dummett] |
| Full Idea: The law of excluded middle is the reflection, within logic, of the principle of bivalence. It states that 'For any statement A, the statement 'A or not-A' is true'. | |
| From: Michael Dummett (Thought and Reality [1997], 5) | |
| A reaction: True-or-not-true is an easier condition to fulfil than true-or-false. The second says that 'false' is the only alternative, but the first allows other alternatives to 'true' (such as 'undecidable'). It is hard to challenge excluded middle. Somewhat true? |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| Full Idea: The 'range' of a function is the set of elements in the output set that are values of the function for elements in the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: In other words, the range is the set of values that were created by the function. |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| Full Idea: We count two functions as being the same if they have the same extension, i.e. if they pair up arguments with values in the same way. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 11.3) | |
| A reaction: So there's only one way to skin a cat in mathematical logic. |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| Full Idea: A 'partial function' is one which maps only some elements of a domain to elements in another set. For example, the reciprocal function 1/x is not defined for x=0. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1 n1) |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| Full Idea: A 'total function' is one which maps every element of a domain to exactly one corresponding value in another set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| Full Idea: If a function f maps the argument a back to a itself, so that f(a) = a, then a is said to be a 'fixed point' for f. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 20.5) |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| Full Idea: The so-called Comprehension Schema ∃X∀x(Xx ↔ φ(x)) says that there is a property which is had by just those things which satisfy the condition φ. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 22.3) |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| Full Idea: 'Theorem': given a derivation of the sentence φ from the axioms of the theory T using the background logical proof system, we will say that φ is a 'theorem' of the theory. Standard abbreviation is T |- φ. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| Full Idea: A 'natural deduction system' will have no logical axioms but may rules of inference. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 09.1) | |
| A reaction: He contrasts this with 'Hilbert-style systems', which have many axioms but few rules. Natural deduction uses many assumptions which are then discharged, and so tree-systems are good for representing it. |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| Full Idea: No nice theory can define truth for its own language. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 21.5) | |
| A reaction: This leads on to Tarski's account of truth. |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| Full Idea: An 'injective' function is 'one-to-one' - each element of the output set results from a different element of the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: That is, two different original elements cannot lead to the same output element. |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| Full Idea: A 'surjective' function is 'onto' - the whole of the output set results from the function being applied to elements of the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| Full Idea: A 'bijective' function has 'one-to-one correspondence' - it is both surjective and injective, so that every element in each of the original and the output sets has a matching element in the other. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: Note that 'injective' is also one-to-one, but only in the one direction. |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| Full Idea: If everything that a theory proves must be true, then it is a 'sound' theory. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| Full Idea: Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
| A reaction: The only exception I can think of is if a theory consisted of nothing but the axioms. |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| Full Idea: A theory is 'sound' iff every theorem of it is true (i.e. true on the interpretation built into its language). Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| Full Idea: A theory is 'negation complete' if it decides every sentence of its language (either the sentence, or its negation). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| Full Idea: There is an annoying double-use of 'complete': a logic may be semantically complete, but there may be an incomplete theory expressed in it. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| Full Idea: Logicians say that a theory T is '(negation) complete' if, for every sentence φ in the language of the theory, either φ or ¬φ is deducible in T's proof system. If this were the case, then truth could be equated with provability. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
| A reaction: The word 'negation' seems to be a recent addition to the concept. Presumable it might be the case that φ can always be proved, but not ¬φ. |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| Full Idea: There are two routes to Incompleteness results. One goes via the semantic assumption that we are dealing with sound theories, using a result about what they can express. The other uses the syntactic notion of consistency, with stronger notions of proof. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 18.1) |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| Full Idea: An 'effectively decidable' (or 'computable') algorithm will be step-by-small-step, with no need for intuition, or for independent sources, with no random methods, possible for a dumb computer, and terminates in finite steps. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.2) | |
| A reaction: [a compressed paragraph] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| Full Idea: A theory is 'decidable' iff there is a mechanical procedure for determining whether any sentence of its language can be proved. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
| A reaction: Note that it doesn't actually have to be proved. The theorems of the theory are all effectively decidable. |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| Full Idea: Any consistent, axiomatized, negation-complete formal theory is decidable. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.6) |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| Full Idea: A set is 'enumerable' iff either the set is empty, or there is a surjective function to the set from the set of natural numbers, so that the set is in the range of that function. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.3) |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| Full Idea: A set is 'effectively enumerable' if an (idealised) computer could be programmed to generate a list of its members such that any member will eventually be mentioned (even if the list is empty, or without end, or contains repetitions). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| Full Idea: A finite set of finitely specifiable objects is always effectively enumerable (for example, the prime numbers). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| Full Idea: The set of ordered pairs of natural numbers (i,j) is effectively enumerable, as proven by listing them in an array (across: <0,0>, <0,1>, <0,2> ..., and down: <0,0>, <1,0>, <2,0>...), and then zig-zagging. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.5) |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| Full Idea: The theorems of any properly axiomatized theory can be effectively enumerated. However, the truths of any sufficiently expressive arithmetic can't be effectively enumerated. Hence the theorems and truths of arithmetic cannot be the same. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 05 Intro) |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| Full Idea: Whether a property is 'expressible' in a given theory depends on the richness of the theory's language. Whether the property can be 'captured' (or 'represented') by the theory depends on the richness of the axioms and proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 04.7) |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| Full Idea: For prime numbers we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))). That is, the only way to multiply two numbers and a get a prime is if one of them is 1. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 04.5) |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| Full Idea: It has been proved (by Tarski) that the real numbers R is a complete theory. But this means that while the real numbers contain the natural numbers, the pure theory of real numbers doesn't contain the theory of natural numbers. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 18.2) |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| Full Idea: The truths of arithmetic are just the true equations involving particular numbers, and universally quantified versions of such equations. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 27.7) | |
| A reaction: Must each equation be universally quantified? Why can't we just universally quantify over the whole system? |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| Full Idea: The number of Fs is the 'successor' of the number of Gs if there is an object which is an F, and the remaining things that are F but not identical to the object are equinumerous with the Gs. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 14.1) |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| Full Idea: All numbers are related to zero by the ancestral of the successor relation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
| A reaction: The successor relation only ties a number to the previous one, not to the whole series. Ancestrals are a higher level of abstraction. |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| Full Idea: Baby Arithmetic 'knows' the addition of particular numbers and multiplication, but can't express general facts about numbers, because it lacks quantification. It has a constant '0', a function 'S', and functions '+' and 'x', and identity and negation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.1) |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| Full Idea: Baby Arithmetic is negation complete, so it can prove every claim (or its negation) that it can express, but it is expressively extremely impoverished. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| Full Idea: Robinson Arithmetic (Q) is not negation complete | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.4) |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| Full Idea: We can beef up Baby Arithmetic into Robinson Arithmetic (referred to as 'Q'), by restoring quantifiers and variables. It has seven generalised axioms, plus standard first-order logic. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| Full Idea: The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
| A reaction: These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems. |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| Full Idea: If the logic of arithmetic doesn't have second-order quantifiers to range over properties of numbers, how can it handle induction? | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.1) |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| Full Idea: Multiplication in itself isn't is intractable. In 1929 Skolem showed a complete theory for a first-order language with multiplication but lacking addition (or successor). Multiplication together with addition and successor produces incompleteness. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7 n8) |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| Full Idea: Putting multiplication together with addition and successor in the language of arithmetic produces incompleteness. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7) | |
| A reaction: His 'Baby Arithmetic' has all three and is complete, but lacks quantification (p.51) |
| 6280 | Realism is a theory, which explains the convergence of science and the success of language [Putnam] |
| Full Idea: Realism is an empirical theory; it explains the convergence of scientific theories, where earlier theories are often limiting cases of later theories (which is why theoretical terms preserve their reference); and it explains the success of language. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
| A reaction: I agree. Personally, I think of Plato's Theory of Forms and all religions as empirical theories. The response from anti-realists is generally to undermine confidence in the evidence which these 'empirical theories' are said to explain. |
| 8184 | Philosophers should not presume reality, but only invoke it when language requires it [Dummett] |
| Full Idea: The philosopher's task is not to make a prior commitment for or against realism, but to discover how far realist considerations must be invoked in order to describe our understanding of our language: they may be invoked only if they must be invoked. | |
| From: Michael Dummett (Thought and Reality [1997], 6) | |
| A reaction: I don't see why the default position should be solipsism, or a commitment to Ockham's Razor. This is the Cartesian 'Enlightenment Project' approach to philosophy - that everything has to be proved. There is more to ontology than language. |
| 8185 | We can't make sense of a world not apprehended by a mind [Dummett] |
| Full Idea: We can make no clear sense of there being a world that is not apprehended by any mind. | |
| From: Michael Dummett (Thought and Reality [1997], 8) | |
| A reaction: I find Dummett's view quite baffling. It is no coincidence that Dummett is a theist, along (it seems) Berkeleian lines. I see no more problem with imagining such worlds than with imagining ships sunken long ago which will never be found. |
| 8163 | Since 'no bird here' and 'no squirrel here' seem the same, we must talk of 'atomic' facts [Dummett] |
| Full Idea: What complex of objects constitutes the fact that there is no bird on the bough, and how is that distinct from no squirrel on the bough? This drives us to see the world as composed of 'atomic' facts, making complexes into compounds, not reality itself. | |
| From: Michael Dummett (Thought and Reality [1997], 1) | |
| A reaction: [He cites early Wittgenstein as an example] But 'no patch of red here' (or sense-datum) seems identical to 'no patch of green here'. I suppose you could catalogue all the atomic facts, and note that red wasn't among them. But you could do that for birds. |
| 8161 | We know we can state facts, with true statements [Dummett] |
| Full Idea: One thing we know about facts, namely that we can state them. Whenever we make some true statement, we state some fact. | |
| From: Michael Dummett (Thought and Reality [1997], 1) | |
| A reaction: Then facts become boring, and are subsumed within the problem of what 'true' means. Personally I have a concept of facts which includes unstatable facts. The physical basis of melancholy I take to be a complex fact which is beyond our powers. |
| 8180 | 'That is red or orange' might be considered true, even though 'that is red' and 'that is orange' were not [Dummett] |
| Full Idea: A statement of the form 'that is red or orange', said of something on the borderline between the two colours, might rank as true, although neither 'that is red' nor 'that is orange' was true. | |
| From: Michael Dummett (Thought and Reality [1997], 5) | |
| A reaction: It seems to me that the problem here would be epistemological rather than ontological. One of the two is clearly true, but sometimes we can't decide which. How can anyone say 'It isn't red and it isn't orange, but it is either red or orange'? |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| Full Idea: The 'ancestral' of a relation is that relation which holds when there is an indefinitely long chain of things having the initial relation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
| A reaction: The standard example is spotting the relation 'ancestor' from the receding relation 'parent'. This is a sort of abstraction derived from a relation which is not equivalent (parenthood being transitive but not reflexive). The idea originated with Frege. |
| 6284 | If a tautology is immune from revision, why would that make it true? [Putnam] |
| Full Idea: If we held, say, 'All unmarried men are unmarried' as absolutely immune from revision, why would this make it true? | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
| A reaction: A very nice question. Like most American philosophers, Putnam accepts Quine's attack on the unrevisability of analytic truths. His point here is that defenders of analytic truths are probably desperate to preserve basic truths, but it won't work. |
| 8178 | Empirical and a priori knowledge are not distinct, but are extremes of a sliding scale [Dummett] |
| Full Idea: Our sentences cannot be divided into two classes, empirical and a priori, the truth of one to be decided by observation, the other by ratiocination. They lie on a scale, with observational sentences at one end, and mathematical ones at the other. | |
| From: Michael Dummett (Thought and Reality [1997], 5) | |
| A reaction: The modern post-Kantian dissolution of the rationalist-empiricist debate. I would say that mathematical sentences require no empirical evidence (for their operation, rather than foundation), but a bit of reasoning is involved in observation. |
| 6273 | Knowledge depends on believing others, which must be innate, as inferences are not strong enough [Putnam] |
| Full Idea: Our ability to picture how people are likely to respond may well be innate; indeed, our disposition to believe what other people tell us (which is fundamental to knowledge) could hardly be an inference, as that isn’t good enough for knowledge. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec VI) | |
| A reaction: An interesting claim. There could be an intermediate situation, which is a hard-wired non-conscious inference. When dismantled, the 'innate' brain circuits for assessing testimony could turn out to work on logic and evidence. |
| 6274 | Empathy may not give knowledge, but it can give plausibility or right opinion [Putnam] |
| Full Idea: Empathy with others may give less than 'Knowledge', but it gives more than mere logical or physical possibility; it gives plausibility, or (to revive Platonic terminology) it provides 'right opinion'. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec VI) | |
| A reaction: See Ideas 174 and 2140 for Plato. Putnam is exploring areas of knowledge outside the limits of strict science. Behind this claim seems to lie the Principle of Charity (3971), but a gang of systematic liars (e.g. evil students) would be a problem case. |
| 17084 | You can't decide which explanations are good if you don't attend to the interest-relative aspects [Putnam] |
| Full Idea: Explanation is an interest-relative notion …explanation has to be partly a pragmatic concept. To regard the 'pragmatics' of explanation as no part of the concept is to abdicate the job of figuring out what makes an explanation good. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], p. 41-2), quoted by David-Hillel Ruben - Explaining Explanation Ch 1 | |
| A reaction: I suppose this is just obvious, depending on how far you want to take the 'interest-relative' bit. If a fool is fobbed off with a trivial explanation, there must be some non-relative criterion for assessing that. |
| 8175 | A theory of thought will include propositional attitudes as well as propositions [Dummett] |
| Full Idea: A comprehensive theory of thought will include such things as judgement and belief, as well as the mere grasp of propositions. | |
| From: Michael Dummett (Thought and Reality [1997], 4) | |
| A reaction: This seems to make any theory of thought a neat two-stage operation. Beware of neatness. While propositions might be explained using concepts, syntax and truth, the second stage looks faintly daunting. See Idea 2209, for example. |
| 8174 | The theories of meaning and understanding are the only routes to an account of thought [Dummett] |
| Full Idea: For the linguistic philosopher, the theory of meaning, and the theory of understanding that is built upon it, form the only route to a philosophical account of thought. | |
| From: Michael Dummett (Thought and Reality [1997], 4) | |
| A reaction: I am of the party that thinks thought is prior to language (esp. because of animals), but Dummett's idea does not deny this. He may well be right that this is the 'only route'. We can only hope to give an account of human thought. |
| 8165 | To 'abstract from' is a logical process, as opposed to the old mental view [Dummett] |
| Full Idea: The phrase 'abstracted from' does not refer to the mental process of abstraction by disregarding features of concrete objects, in which many nineteenth century thinkers believed; it is a logical (not mental) process of concept-formation. | |
| From: Michael Dummett (Thought and Reality [1997], 1) | |
| A reaction: I take Frege's attack on 'psychologism' to be what dismissed the old view (Idea 5816). Could one not achieve the same story by negating properties in quantified logical expressions, instead of in the mind? |
| 6282 | Theory of meaning presupposes theory of understanding and reference [Putnam] |
| Full Idea: Theory of meaning presupposes theory of understanding and reference. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
| A reaction: How can you have a theory of understanding without a meaning that requires to be understood? Personally I think about the minds of small animals when pondering this, and that seems to put reference and truth at the front of the queue. |
| 6281 | Truth conditions can't explain understanding a sentence, because that in turn needs explanation [Putnam] |
| Full Idea: You can't treat understanding a sentence as knowing its truth conditions, because it then becomes unintelligible what that knowledge in turn consists in. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
| A reaction: The implication, I take it, is circularity; how can you specify truth conditions if you don't understand sentences? Putnam here agrees with Dummett that verification must be involved. Something has to be taken as axiomatic in all this. |
| 6278 | We should reject the view that truth is prior to meaning [Putnam] |
| Full Idea: I am suggesting that we reject the view that truth (based on the semantic theory) is prior to meaning. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
| A reaction: It is a nice question which of truth or meaning has logical priority. One might start by speculating about how and why animals think. A moth attracted to flame is probably working on truth without much that could be called 'meaning'. |
| 8168 | To know the truth-conditions of a sentence, you must already know the meaning [Dummett] |
| Full Idea: You can know the condition for a sentence to be true only when you know what the sentence means. | |
| From: Michael Dummett (Thought and Reality [1997], 3) | |
| A reaction: This makes the truth-conditions theory of meaning circular, and is Dummett's big objection to Davidson's view. The composition of a sentence creates a model of a world. Truth-conditions may only presuppose knowledge of concepts. |
| 8181 | A justificationist theory of meaning leads to the rejection of classical logic [Dummett] |
| Full Idea: If we adopt a justificationist theory of meaning, we must reject the universal law of excluded middle, and with it classical logic (which rests on the two-valued semantics of bivalence). We admit only intuitionist logic, which preserves justifiability. | |
| From: Michael Dummett (Thought and Reality [1997], 5) | |
| A reaction: This is Dummett's philosophy in a very neat nutshell. He seems to have started by accepting Brouwer's intuitionism, and then working back to language. It all implies anti-realism. I don't buy it. |
| 8182 | Verificationism could be realist, if we imagined the verification by a superhuman power [Dummett] |
| Full Idea: There is a possible route to realism, which has been called 'ideal verificationism', if we base our grasping the understanding and truth of a range of sentences on the procedure that would be available to an imagined being with superhuman powers. | |
| From: Michael Dummett (Thought and Reality [1997], 5) | |
| A reaction: This is actually a slippery slope for verificationists, as soon as they allow that verification could be done by other people. A verifier might turn up who had telepathy, or x-ray vision, or could see quarks... |
| 8183 | If truths about the past depend on memories and current evidence, the past will change [Dummett] |
| Full Idea: If justificationists succumb to the temptation for statements in the past, we shall view their senses as given by present memories and present traces of past events; but this will force us into a view of the past as itself changing. | |
| From: Michael Dummett (Thought and Reality [1997], 6) | |
| A reaction: Obviously Dummett attempts to sidestep this problem, but it strikes me as powerful support for the realist view about the past. How can we not be committed to the view that there are facts about the past quite unconnected to our verifying abilities? |
| 8176 | We could only guess the meanings of 'true' and 'false' when sentences were used [Dummett] |
| Full Idea: Even if we guessed that the two words denoted the two truth-values, we should not know which stood for the value 'true' and which for the value 'false' until we knew how the sentences were in practice used. | |
| From: Michael Dummett (Thought and Reality [1997], 4) | |
| A reaction: These types of problem are always based on the idea that some one item must have logical priority in the process, but there is a lot of room for benign circularity in the development of mental and linguistic functions. |
| 8170 | Sentences are the primary semantic units, because they can say something [Dummett] |
| Full Idea: While words are semantic atoms, sentences remain the primary semantic units, in the sense of the smallest bits of language by means of which it is possible to say anything. | |
| From: Michael Dummett (Thought and Reality [1997], 3) | |
| A reaction: Syncategorematic terms (look it up!) may need sentences, but most nouns and verbs can communicate quite a lot on their own. Whether words or sentences come first may not be a true/false issue. |
| 6271 | How reference is specified is not what reference is [Putnam] |
| Full Idea: A theory of how reference is specified isn't a theory of what reference is. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec V) | |
| A reaction: A simple and important point. We may achieve reference by naming, describing, grunting or pointing, but the question is, what have we achieved when we get there? |
| 6268 | The claim that scientific terms are incommensurable can be blocked if scientific terms are not descriptions [Putnam] |
| Full Idea: The line of reasoning of Kuhn and Feyerabend can be blocked by arguing (as I have in various places, and as Saul Kripke has) that scientific terms are not synonymous with descriptions. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec II.2) | |
| A reaction: A nice clear statement of the motivation for creating the causal theory of reference. See Idea 6162. We could still go back and ask whether we could block scientific relativism by rethinking how descriptions work, instead of abandoning them. |
| 8169 | We can't distinguish a proposition from its content [Dummett] |
| Full Idea: No distinction can be drawn between a proposition and its content; no two distinct propositions can have the same content. | |
| From: Michael Dummett (Thought and Reality [1997], 3) | |
| A reaction: And one proposition cannot have two possible contents (ambiguity). Are we to say that a proposition supervenes on its content, or that proposition and content are identical? Ockham favours the latter. |
| 6279 | A private language could work with reference and beliefs, and wouldn't need meaning [Putnam] |
| Full Idea: A language made up and used by a being who belonged to no community would have no need for such a concept as the 'meaning' of a term. To state the reference of each term and what the language speaker believes is to tell the whole story. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
| A reaction: A subtle response to Wittgenstein's claim (e.g. Ideas 4152,4158), but I am not sure what Putnam means. I would have thought that beliefs had to be embodied in propositions. They may not need 'meaning' quite as urgently as sentences, but still… |
| 6270 | The correct translation is the one that explains the speaker's behaviour [Putnam] |
| Full Idea: What it is to be a correct translation is to be the translation that best explains the behaviour of the speaker. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Lec III) | |
| A reaction: This seems fairly close to Quine, but rather puzzlingly uses the word 'correct'. If our criteria of translation are purely behavioural, there is no way we can be correct after one word ('gavagai'), so at what point does it become 'correct'? |
| 6283 | Language maps the world in many ways (because it maps onto other languages in many ways) [Putnam] |
| Full Idea: We could say that the language has more than one correct way of being mapped onto the world (it must, since it has more than one way of being correctly mapped onto a language which is itself correctly mapped onto the world). | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Four) | |
| A reaction: This spells out nicely the significance of Quine's 'indeterminacy of translation'. Others have pointed out that the fact that language maps onto world in many ways need not be anti-realist; the world is endless, and language is limited. |
| 6275 | You can't say 'most speaker's beliefs are true'; in some areas this is not so, and you can't count beliefs [Putnam] |
| Full Idea: The maxim that 'most of a speaker's beliefs are true' as an a priori principle governing radical translation seems to me to go too far; first, I don't know how to count beliefs; second, most people's beliefs on some topics (philosophy) are probably false. | |
| From: Hilary Putnam (Meaning and the Moral Sciences [1978], Pt Three) | |
| A reaction: Putnam prefers a pragmatic view, where charity is applicable if behaviour is involved. Philosophy is too purely theoretical. The extent to which Charity should apply in philosophy seminars is a nice question, which all students should test in practice. |
| 8186 | Time is the measure of change, so we can't speak of time before all change [Dummett] |
| Full Idea: Time is the measure of change, and it makes no sense to speak of how things were before there was anything that changed. | |
| From: Michael Dummett (Thought and Reality [1997], 8) | |
| A reaction: Something creating its own measure sounds like me marking my own exam papers. If an object appears, then inverts five seconds later, how can the inversion create the five seconds? How does that differ from inverting ten seconds later? |
| 8167 | If Presentism is correct, we cannot even say that the present changes [Dummett] |
| Full Idea: If Presentism is correct - the doctrine that there is nothing at all, save what holds good at the present moment - then we cannot even say that the present changes, because that requires that things are not now as they were some time ago. | |
| From: Michael Dummett (Thought and Reality [1997], 2) | |
| A reaction: Presumably we can compare our present memory with our present experience. See Idea 6668. The logic (very ancient!) is that the present has not duration at all, and so no experiences can occur during it. |