43 ideas
| 16877 | A 'constructive' (as opposed to 'analytic') definition creates a new sign [Frege] |
| Full Idea: We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a 'constructive definition', but we prefer to call it a 'definition' tout court. It contrasts with an 'analytic' definition. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.210) | |
| A reaction: An analytic definition is evidently a deconstruction of a past constructive definition. Fregean definition is a creative activity. |
| 11219 | Frege suggested that mathematics should only accept stipulative definitions [Frege, by Gupta] |
| Full Idea: Frege has defended the austere view that, in mathematics at least, only stipulative definitions should be countenanced. | |
| From: report of Gottlob Frege (Logic in Mathematics [1914]) by Anil Gupta - Definitions 1.3 | |
| A reaction: This sounds intriguingly at odds with Frege's well-known platonism about numbers (as sets of equinumerous sets). It makes sense for other mathematical concepts. |
| 16878 | We must be clear about every premise and every law used in a proof [Frege] |
| Full Idea: It is so important, if we are to have a clear insight into what is going on, for us to be able to recognise the premises of every inference which occurs in a proof and the law of inference in accordance with which it takes place. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.212) | |
| A reaction: Teachers of logic like natural deduction, because it reduces everything to a few clear laws, which can be stated at each step. |
| 10455 | Free logic at least allows empty names, but struggles to express non-existence [Bach] |
| Full Idea: Unlike standard first-order logic, free logic can allow empty names, but still has to deny existence by either representing it as a predicate, or invoke some dubious distinction such as between existence and being. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1) |
| 16867 | Logic not only proves things, but also reveals logical relations between them [Frege] |
| Full Idea: A proof does not only serve to convince us of the truth of what is proved: it also serves to reveal logical relations between truths. Hence we find in Euclid proofs of truths that appear to stand in no need of proof because they are obvious without one. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204) | |
| A reaction: This is a key idea in Frege's philosophy, and a reason why he is the founder of modern analytic philosophy, with logic placed at the centre of the subject. I take the value of proofs to be raising questions, more than giving answers. |
| 16863 | Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege] |
| Full Idea: Are there perhaps modes of inference peculiar to mathematics which …do not belong to logic? Here one may point to inference by mathematical induction from n to n+1. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
| A reaction: He replies that it looks as if induction can be reduced to general laws, and those can be reduced to logic. |
| 16862 | The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege] |
| Full Idea: Mathematics has closer ties with logic than does almost any other discipline; for almost the entire activity of the mathematician consists in drawing inferences. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
| A reaction: The interesting question is who is in charge - the mathematician or the logician? |
| 10454 | In first-order we can't just assert existence, and it is very hard to deny something's existence [Bach] |
| Full Idea: In standard logic we can't straightforwardly say that n exists. We have to resort to using a formula like '∃x(x=n)', but we can't deny n's existence by negating that formula, because standard first-order logic disallows empty names. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1) |
| 10453 | In logic constants play the role of proper names [Bach] |
| Full Idea: In standard first-order logic the role of proper names is played by individual constants. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1) |
| 16865 | 'Theorems' are both proved, and used in proofs [Frege] |
| Full Idea: Usually a truth is only called a 'theorem' when it has not merely been obtained by inference, but is used in turn as a premise for a number of inferences in the science. ….Proofs use non-theorems, which only occur in that proof. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204) |
| 10452 | Proper names can be non-referential - even predicate as well as attributive uses [Bach] |
| Full Idea: Like it or not, proper names have non-referential uses, including not only attributive but even predicate uses. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1) | |
| A reaction: 'He's a right little Hitler'. 'You're doing a George Bush again'. 'Try to live up to the name of Churchill'. |
| 10456 | Millian names struggle with existence, empty names, identities and attitude ascription [Bach] |
| Full Idea: The familiar problems with the Millian view of names are the problem of positive and negative existential statements, empty names, identity sentences, and propositional attitude ascription. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1) | |
| A reaction: I take this combination of problems to make an overwhelming case against the daft idea that the semantics of a name amounts to the actual object it picks out. It is a category mistake to attempt to insert a person into a sentence. |
| 10440 | An object can be described without being referred to [Bach] |
| Full Idea: An object can be described without being referred to. | |
| From: Kent Bach (What Does It Take to Refer? [2006], Intro) | |
| A reaction: I'm not clear how this is possible for a well-known object, though it is clearly possible for a speculative object, such as a gadget I would like to buy. In the former case reference seems to occur even if the speaker is trying to avoid it. |
| 10444 | Definite descriptions can be used to refer, but are not semantically referential [Bach] |
| Full Idea: If Russell is, as I believe, basically right, then definite descriptions are the paradigm of singular terms that can be used to refer but are not linguistically (semantically) referential. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s5) | |
| A reaction: I'm not sure that we can decide what is 'semantically referential'. Most of the things we refer to don't have names. We don't then 'use' definite descriptions (I'm thinking) - they actually DO the job. If we use them, we can 'use' names too? |
| 16866 | Tracing inference backwards closes in on a small set of axioms and postulates [Frege] |
| Full Idea: We can trace the chains of inference backwards, …and the circle of theorems closes in more and more. ..We must eventually come to an end by arriving at truths can cannot be inferred, …which are the axioms and postulates. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204) | |
| A reaction: The rival (more modern) view is that that all theorems are equal in status, and axioms are selected for convenience. |
| 16868 | The essence of mathematics is the kernel of primitive truths on which it rests [Frege] |
| Full Idea: Science must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in this kernel. The essence of mathematics has to be defined by this kernel of truths. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.204-5) | |
| A reaction: [compressed] I will make use of this thought, by arguing that mathematics may be 'explained' by this kernel. |
| 16871 | A truth can be an axiom in one system and not in another [Frege] |
| Full Idea: It is possible for a truth to be an axiom in one system and not in another. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.205) | |
| A reaction: Frege aspired to one huge single system, so this is a begrudging concession, one which modern thinkers would probably take for granted. |
| 16870 | Axioms are truths which cannot be doubted, and for which no proof is needed [Frege] |
| Full Idea: The axioms are theorems, but truths for which no proof can be given in our system, and no proof is needed. It follows from this that there are no false axioms, and we cannot accept a thought as an axiom if we are in doubt about its truth. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.205) | |
| A reaction: He struggles to be as objective as possible, but has to concede that whether we can 'doubt' the axiom is one of the criteria. |
| 16869 | To create order in mathematics we need a full system, guided by patterns of inference [Frege] |
| Full Idea: We cannot long remain content with the present fragmentation [of mathematics]. Order can be created only by a system. But to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.205) |
| 16864 | If principles are provable, they are theorems; if not, they are axioms [Frege] |
| Full Idea: If the law [of induction] can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.203) | |
| A reaction: This links Frege with the traditional Euclidean view of axioms. The question, then, is how do we know them, given that we can't prove them. |
| 9388 | Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege] |
| Full Idea: Of any concept, we must require that it have a sharp boundary. Of any object it must hold either that it falls under the concept or it does not. We may not allow a third case in which it is somehow indeterminate whether an object falls under a concept. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.229), quoted by Ian Rumfitt - The Logic of Boundaryless Concepts p.1 n1 | |
| A reaction: This is the voice of the classical logician, which has echoed by Russell. I'm with them, I think, in the sense that logic can only work with precise concepts. The jury is still out. Maybe we can 'precisify', without achieving total precision. |
| 16876 | We need definitions to cram retrievable sense into a signed receptacle [Frege] |
| Full Idea: If we need such signs, we also need definitions so that we can cram this sense into the receptacle and also take it out again. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.209) | |
| A reaction: Has anyone noticed that Frege is the originator of the idea of the mental file? Has anyone noticed the role that definition plays in his account? |
| 16875 | We use signs to mark receptacles for complex senses [Frege] |
| Full Idea: We often need to use a sign with which we associate a very complex sense. Such a sign seems a receptacle for the sense, so that we can carry it with us, while being always aware that we can open this receptacle should we need what it contains. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.209) | |
| A reaction: This exactly the concept of a mental file, which I enthusiastically endorse. Frege even talks of 'opening the receptacle'. For Frege a definition (which he has been discussing) is the assigment of a label (the 'definiendum') to the file (the 'definiens'). |
| 16879 | A sign won't gain sense just from being used in sentences with familiar components [Frege] |
| Full Idea: No sense accrues to a sign by the mere fact that it is used in one or more sentences, the other constituents of which are known. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.213) | |
| A reaction: Music to my ears. I've never grasped how meaning could be grasped entirely through use. |
| 10446 | Fictional reference is different inside and outside the fiction [Bach] |
| Full Idea: We must distinguish 'reference' in a fiction from reference outside the fiction to fictional entities. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.1) | |
| A reaction: This may be more semantically than ontologically significant. It is perhaps best explicated by Coleridge's distinction over whether or not I am 'suspending my disbelief' when I am discussing a character. |
| 10447 | We can refer to fictional entities if they are abstract objects [Bach] |
| Full Idea: If fictional entities, such as characters in a play, are real, albeit abstract entities, then we can genuinely refer to them. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.1) | |
| A reaction: [He cites Nathan Salmon 1998] Personally I would prefer to say that abstract entities are fictions. Fictional characters have uncertain identity conditions. Do they all have a pancreas, if this is never mentioned? |
| 10443 | You 'allude to', not 'refer to', an individual if you keep their identity vague [Bach] |
| Full Idea: If you say 'a special person is coming to visit', you are not referring to but merely 'alluding to' that individual. This does not count as referring because you are not expressing a singular proposition about it. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s2) | |
| A reaction: If you add 'I hope he doesn't wear his red suit, but I hope he plays his tuba', you seem to be expressing singular propositions about the person. Bach seems to want a very strict notion of reference, as really attaching listeners to individuals. |
| 10439 | What refers: indefinite or definite or demonstrative descriptions, names, indexicals, demonstratives? [Bach] |
| Full Idea: Philosophers agree that some expressions refer, but disagree over which ones. Few include indefinite descriptions, but some include definite descriptions, or only demonstrative descriptions. Some like proper names, some only indexicals and demonstratives. | |
| From: Kent Bach (What Does It Take to Refer? [2006], Intro) | |
| A reaction: My initial prejudice is rather Strawsonian - that people refer, not language, and it can be done in all sorts of ways. But Bach argues well that only language intrinsically does it. Even pointing fails without linguistic support. |
| 10441 | If we can refer to things which change, we can't be obliged to single out their properties [Bach] |
| Full Idea: We can refer to things which change over time, which suggests that in thinking of and in referring to an individual we are not constrained to represent it as that which has certain properties. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.1) | |
| A reaction: This seems a good argument against the descriptive theory of reference which is not (I think) in Kripke. Problems like vagueness and the Ship of Theseus rear their heads. |
| 10442 | We can think of an individual without have a uniquely characterizing description [Bach] |
| Full Idea: Being able to think of an individual does not require being able to identify that individual by means of a uniquely characterizing description. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s1) | |
| A reaction: There is a bit of an equivocation over 'recognise' here. His example is 'the first child born in the 4th century'. We can't visually recognise such people, but the description does fix them, and a records office might give us 'recognition'. |
| 10445 | It can't be real reference if it could refer to some other thing that satisfies the description [Bach] |
| Full Idea: If one is referring to whatever happens to satisfy a description, and one would be referring to something else were it to have satisfied the description instead, this is known as 'weak' reference,...but surely this is not reference at all. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s7) | |
| A reaction: Bach wants a precise notion of reference, as success in getting the audience to focus on the correct object. He talks of this case as 'singling out' some unfixed thing, and he also has 'alluding to' an unstated thing. Plausible view. |
| 10457 | Since most expressions can be used non-referentially, none of them are inherently referential [Bach] |
| Full Idea: An embarrassingly simple argument is that most expressions can be used literally but not referentially, no variation in meaning explains this fact, so its meaning is compatible with being non-referential, so no expression is inherently referential. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L2) | |
| A reaction: I think I have decided that no expression is 'inherently referential', and that it is all pragmatics. |
| 10463 | Just alluding to or describing an object is not the same as referring to it [Bach] |
| Full Idea: Much of what speakers do that passes for referring is merely alluding or describing. ...It is one thing for a speaker to express a thought about a certain object using an expression, and quite another for the expression to stand for that object. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.3) | |
| A reaction: Bach builds up a persuasive case for this view. If the question, though, is 'what are you talking about?', then saying what is being alluded to or singled out or described seems fine. Bach is being rather stipulative. |
| 10459 | Context does not create reference; it is just something speakers can exploit [Bach] |
| Full Idea: Context does not determine or constitute reference; rather, it is something for the speaker to exploit to enable the listener to determine the intended reference. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L3) | |
| A reaction: Bach thinks linguistic reference is a matter of speaker's intentions, and I think he is right. And this idea is right too. The domain of quantification constantly shifts in a conversation, and good speakers and listeners are sensitive to this. |
| 10460 | 'That duck' may not refer to the most obvious one in the group [Bach] |
| Full Idea: If one ducks starts quacking furiously, and you say 'that duck is excited', it isn't context that makes me take it that you are referring to the quacking duck. You could be referring to a quiet duck you recognise by its distinctive colour. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L3) | |
| A reaction: A persuasive example to make his point against the significance of context in conversational reference. Speaker's intended reference must always trump any apparent reference suggested by context. |
| 10461 | What a pronoun like 'he' refers back to is usually a matter of speaker's intentions [Bach] |
| Full Idea: To illustrate speakers' intentions, consider the anaphoric reference using pronouns in these: "A cop arrested a robber; he was wearing a badge", and "A cop arrested a robber; he was wearing a mask". The natural supposition is not the inevitable one. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L4) | |
| A reaction: I am a convert to speakers' intentions as the source of all reference, and this example seems to illustrate it very well. 'He said..' 'Who said?' |
| 10462 | Information comes from knowing who is speaking, not just from interpretation of the utterance [Bach] |
| Full Idea: It is a fallacy that all the information in an utterance must come from its interpretation, which ignores the essentially pragmatic fact that the speaker is making the utterance. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L4) | |
| A reaction: [He cites Barwise and Perry 1983:34] This is blatantly obvious in indexical remarks like 'I am tired', where the words don't tell you who is tired. But also 'the car has broken down, dear'. |
| 16873 | Thoughts are not subjective or psychological, because some thoughts are the same for us all [Frege] |
| Full Idea: A thought is not something subjective, is not the product of any form of mental activity; for the thought that we have in Pythagoras's theorem is the same for everybody. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.206) | |
| A reaction: When such thoughts are treated as if the have objective (platonic) existence, I become bewildered. I take a thought (or proposition) to be entirely psychological, but that doesn't stop two people from having the same thought. |
| 16872 | A thought is the sense expressed by a sentence, and is what we prove [Frege] |
| Full Idea: The sentence is of value to us because of the sense that we grasp in it, which is recognisably the same in a translation. I call this sense the thought. What we prove is not a sentence, but a thought. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.206) | |
| A reaction: The 'sense' is presumably the German 'sinn', and a 'thought' in Frege is what we normally call a 'proposition'. So the sense of a sentence is a proposition, and logic proves propositions. I'm happy with that. |
| 16874 | The parts of a thought map onto the parts of a sentence [Frege] |
| Full Idea: A sentence is generally a complex sign, so the thought expressed by it is complex too: in fact it is put together in such a way that parts of a thought correspond to parts of the sentence. | |
| From: Gottlob Frege (Logic in Mathematics [1914], p.207) | |
| A reaction: This is the compositional view of propositions, as opposed to the holistic view. |
| 10458 | People slide from contextual variability all the way to contextual determination [Bach] |
| Full Idea: People slide from contextual variability to context relativity to context sensitivity to context dependence to contextual determination. | |
| From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L3) | |
| A reaction: This is reminiscent of the epistemological slide from cultural or individual relativity of some observed things, to a huge metaphysical denial of truth. Bach's warning applies to me, as I have been drifting down his slope lately. Nice. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |