46 ideas
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
| A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
| A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
| A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro) | |
| A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything. |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3) | |
| A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going. |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2) | |
| A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh????? |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set. |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third.... | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different. |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal). | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1) | |
| A reaction: There has to be an well-founded ordering for inductive proofs to be possible. |
| 6504 | For physicalists, the only relations are spatial, temporal and causal [Robinson,H] |
| Full Idea: Spatial, temporal and causal relations are the only respectable candidates for relations for a physicalist. | |
| From: Howard Robinson (Perception [1994], V.4) | |
| A reaction: This seems to be true, and is an absolutely crucial principle upon which any respectable physicalist account of the world must be built. It means that physicalists must attempt to explain all mental events in causal terms. |
| 6520 | If reality just has relational properties, what are its substantial ontological features? [Robinson,H] |
| Full Idea: Some thinkers claim the physical world consists just of relational properties - generally of active powers or fields; ..but an ontology of mutual influences is not an ontology at all unless the possessors of the influence have more substantial features. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: I think this idea is one of the keys to wisdom. It is the same problem with functional explanations - you are left asking WHY this thing can have this particular function. Without the buck stopping at essences you are chasing your explanatory tail. |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4) |
| 6485 | When a red object is viewed, the air in between does not become red [Robinson,H] |
| Full Idea: When the form of red passes from an object to the eye, the air in between does not become red. | |
| From: Howard Robinson (Perception [1994], 1.2) | |
| A reaction: This strikes me as a crucial and basic fact which must be faced by any philosopher offering a theory of perception. I would have thought it instantly eliminated any sort of direct or naïve realism. The quale of red is created by my brain. |
| 6521 | Representative realists believe that laws of phenomena will apply to the physical world [Robinson,H] |
| Full Idea: One thing which is meant by saying that the phenomenal world represents or resembles the transcendental physical world is that the scientific laws devised to apply to the former, if correct, also apply (at least approximately) to the latter. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: This is not, of course, an argument, or a claim which can be easily substantiated, but it does seem to be a nice statement of a central article of faith for representative realists. The laws of the phenomenal world are the only ones we are going to get. |
| 6509 | Representative realists believe some properties of sense-data are shared by the objects themselves [Robinson,H] |
| Full Idea: A representative realist believes that at least some of the properties that are ostensively demonstrable in virtue of being exemplified in sense-data are of the same kind as some of those exemplified in physical objects. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: It is hard to pin down exactly what is being claimed here. Locke's primary qualities will obviously qualify, but could properties be 'exemplified' in sense-data without them actually being the same as those of the objects? |
| 6522 | Phenomenalism can be theistic (Berkeley), or sceptical (Hume), or analytic (20th century) [Robinson,H] |
| Full Idea: It is useful to identify three kinds of phenomenalism: theistic, sceptical and analytic; the first is represented by Berkeley, the second by Hume, and the third by most twentieth-century phenomenalists. | |
| From: Howard Robinson (Perception [1994], IX.4) | |
| A reaction: In Britain the third group is usually represented by A.J.Ayer. My simple objection to all phenomenalists is that they are intellectual cowards because they won't venture to give an explanation of the phenomena which confront them. |
| 6502 | Can we reduce perception to acquisition of information, which is reduced to causation or disposition? [Robinson,H] |
| Full Idea: Many modern physicalists first analyse perception as no more than the acquisition of beliefs or information through the senses, and then analyse belief and the possession of information in causal or dispositional terms. | |
| From: Howard Robinson (Perception [1994], V.1) | |
| A reaction: (He mentions Armstrong, Dretske and Pitcher). A reduction to dispositions implies behaviourism. This all sounds more like an eliminativist strategy than a reductive one. I would start by saying that perception is only information after interpretation. |
| 6513 | Would someone who recovered their sight recognise felt shapes just by looking? [Robinson,H] |
| Full Idea: Molyneux's Problem is whether someone who was born blind and acquired sight would be able to recognise, on sight, which shapes were which; that is, would they see which shape was the one that felt so-and-so? | |
| From: Howard Robinson (Perception [1994], VIII.7) | |
| A reaction: (Molyneux wrote a letter to John Locke about this). It is a good question, and much discussed in modern times. My estimation is that the person would recognise the shapes. We are partly synaesthetic, and see sharpness as well as feeling it. |
| 6512 | Secondary qualities have one sensory mode, but primary qualities can have more [Robinson,H] |
| Full Idea: Primary qualities and secondary qualities are often distinguished on the grounds that secondaries are restricted to one sensory modality, but primaries can appear in more. | |
| From: Howard Robinson (Perception [1994], VIII.7) | |
| A reaction: This distinction seems to me to be accurate and important. It is not just that the two types are phenomenally different - it is that the best explanation is that the secondaries depend on their one sense, but the primaries are independent. |
| 6497 | We say objects possess no intrinsic secondary qualities because physicists don't need them [Robinson,H] |
| Full Idea: The idea that objects do not possess secondary qualities intrinsically rests on the thought that they do not figure in the physicist's account of the world; ..as they are causally idle, no purpose is served by attributing them to objects. | |
| From: Howard Robinson (Perception [1994], III.1) | |
| A reaction: On the whole I agree with this, but colours (for example) are not causally idle, as they seem to affect the behaviour of insects. They are properties which can only have a causal effect if there is a brain in their vicinity. Physicists ignore brains. |
| 6494 | If objects are not coloured, and neither are sense-contents, we are left saying that nothing is coloured [Robinson,H] |
| Full Idea: If there are good reasons for thinking that physical objects are not literally coloured, and one also refuses to attribute them to sense-contents, then one will have the bizarre theory (which has been recently adopted) that nothing is actually coloured. | |
| From: Howard Robinson (Perception [1994], 1.7) | |
| A reaction: It seems to me that objects are not literally coloured, that the air in between does not become coloured, and that my brain doesn't turn a funny colour, so that only leaves colour as an 'interior' feature of certain brain states. That's how it is. |
| 6499 | Shape can be experienced in different ways, but colour and sound only one way [Robinson,H] |
| Full Idea: Shape can be directly experienced by either touch or sight, which are subjectively different; but colour and sound can be directly experienced only through experiences which are subjectively like sight and hearing. | |
| From: Howard Robinson (Perception [1994], III.1) | |
| A reaction: This seems to be a key argument in support of the distinction between primary and secondary qualities. It seems to me that the distinction may be challenged and questioned, but to deny it completely (as Berkeley and Hume do) is absurd. |
| 6500 | If secondary qualities match senses, would new senses create new qualities? [Robinson,H] |
| Full Idea: As secondary qualities are tailored to match senses, a proliferation of senses would lead to a proliferation of secondary qualities. | |
| From: Howard Robinson (Perception [1994], III.1) | |
| A reaction: One might reply that if we experienced, say, magnetism, we would just be discerning a new fine grained primary quality, not adding something new to the ontological stock of properties in the world. It is a matter of HOW we experience the magnetism. |
| 6484 | Most moderate empiricists adopt Locke's representative theory of perception [Robinson,H] |
| Full Idea: The representative theory of perception is found in Locke, and is adopted by most moderate empiricists. | |
| From: Howard Robinson (Perception [1994], 1.2) | |
| A reaction: This is, I think, my own position. Anything less than fairly robust realism strikes me as being a bit mad (despite Berkeley's endless assertions that he is preaching common sense), and direct realism seems obviously false. |
| 6508 | Sense-data leads to either representative realism or phenomenalism or idealism [Robinson,H] |
| Full Idea: The sense-datum theorist is either a representative realist or a phenomenalist (with which we can classify idealism for present purposes). | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: The only alternative to these two positions seems to be some sort of direct realism. I class myself as a representative realist, as this just seems (after a very little thought about colour blindness) to be common sense. I'm open to persuasion. |
| 6480 | Sense-data do not have any intrinsic intentionality [Robinson,H] |
| Full Idea: I understand sense-data as having no intrinsic intentionality; that is, though it may suggest, by habit, things beyond it, in itself it possesses only sensible qualities which do not refer beyond themselves. | |
| From: Howard Robinson (Perception [1994], 1.1) | |
| A reaction: This seems right, as the whole point of proposing sense-data was as something neutral between realism and anti-realism |
| 6482 | For idealists and phenomenalists sense-data are in objects; representative realists say they resemble objects [Robinson,H] |
| Full Idea: For idealists and phenomenalists sense-data are part of physical objects, for objects consist only of actual or actual and possible sense-data; representative realists say they just have an abstract and structural resemblance to objects. | |
| From: Howard Robinson (Perception [1994], 1.1) | |
| A reaction: He puts Berkeley, Hume and Mill in the first group, and Locke in the second. Russell belongs in the second. The very fact that there can be two such different theories about the location of sense-data rather discredits the whole idea. |
| 6505 | Sense-data are rejected because they are a veil between us and reality, leading to scepticism [Robinson,H] |
| Full Idea: Resistance to the sense-datum theory is inspired mainly by the fear that such data constitute a veil of perception which stands between the observer and the external world, threatening scepticism, or even solipsism. | |
| From: Howard Robinson (Perception [1994], VII.1) | |
| A reaction: It is very intellectually dishonest to reject any theory because it leads to scepticism or relativism. This is a common failing among quite good professional philosophers. See Idea 241. |
| 6506 | 'Sense redly' sounds peculiar, but 'senses redly-squarely tablely' sounds far worse [Robinson,H] |
| Full Idea: 'Sense redly' sounds peculiar, but 'senses redly-squarely' or 'red-squarely' or 'senses redly-squarely-tablely' and other variants sound far worse. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: This is a comment on the adverbial theory, which is meant to replace representative theories based on sense-data. The problem is not that it sounds weird; it is that while plain red can be a mode of perception, being a table obviously can't. |
| 6507 | Adverbialism sees the contents of sense-experience as modes, not objects [Robinson,H] |
| Full Idea: The defining claim of adverbialism is that the contents of sense-experience are modes, not objects, of sensory activity. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: This seems quite a good account of simple 'modes' like colour, but not so good when you instantly perceive a house. It never seems wholly satisfactory to sidestep the question of 'what are you perceiving when you perceive red or square?' |
| 6511 | If there are only 'modes' of sensing, then an object can no more be red or square than it can be proud or lazy. [Robinson,H] |
| Full Idea: If only modes of sensing are ostensively available, ..then it is a category mistake to see any resemblance between what is available and properties of bodies; one could as sensibly say that a physical body is proud or lazy as that it is red or square. | |
| From: Howard Robinson (Perception [1994], VII.5) | |
| A reaction: This is an objection to the 'adverbial' theory of perception. It looks to me like a devastating objection, if the theory is meant to cover primary qualities as well as secondary. Red could be a mode of perception, but not square, surely? |
| 6515 | An explanation presupposes something that is improbable unless it is explained [Robinson,H] |
| Full Idea: Any search for an explanation presupposes that there is something in need of an explanation - that is, something which is improbable unless explained. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: Elementary enough, but it underlines the human perspective of all explanations. I may need an explanation of baseball, where you don't. |
| 6517 | If all possibilities are equal, order seems (a priori) to need an explanation - or does it? [Robinson,H] |
| Full Idea: The fact that order requires an explanation seems to be an a priori principle; ..we assume all possibilities are equally likely, and so no striking regularities should emerge; the sceptic replies that a highly ordered sequence is as likely as any other. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: An independent notion of 'order' is required. If I write down '14356', and then throw 1 4 3 5 6 on a die, the match is the order; instrinsically 14356 is nothing special. If you threw the die a million times, a run of six sixes seems quite likely. |
| 6481 | If intentional states are intrinsically about other things, what are their own properties? [Robinson,H] |
| Full Idea: Intentional states are mysterious things; if they are intrinsically about other things, what properties, if any, do they possess intrinsically? | |
| From: Howard Robinson (Perception [1994], 1.1) | |
| A reaction: A very nice question, which I suspect to be right at the heart of the tendency towards externalist accounts of the mind. Since you can only talk about the contents of the thoughts, you can't put forward a decent internalist account of what is going on. |
| 6503 | Physicalism cannot allow internal intentional objects, as brain states can't be 'about' anything [Robinson,H] |
| Full Idea: It is generally conceded by reductive physicalists that a state of the brain cannot be intrinsically about anything, for intentionality is not an intrinsic property of anything, so there can be no internal objects for a physicalist. | |
| From: Howard Robinson (Perception [1994], V.4) | |
| A reaction: Perhaps it is best to say that 'aboutness' is not a property of physics. We may say that a brain state 'represents' something, because the something caused the brain state, but representations have to be recognised |
| 18265 | We don't judge by combining subject and concept; we get a concept by splitting up a judgement [Frege] |
| Full Idea: Instead of putting a judgement together out of an individual as subject and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of possible judgement. | |
| From: Gottlob Frege (Boole calculus and the Concept script [1881], p.17) | |
| A reaction: This is behind holistic views of sentences, and hence of whole languages, and behind Quine's rejection of 'properties' inferred from the predicates in judgements. |
| 6519 | Locke's solidity is not matter, because that is impenetrability and hardness combined [Robinson,H] |
| Full Idea: Notoriously, Locke's filler for Descartes's geometrical matter, solidity, will not do, for that quality collapses on examination into a composite of the dispositional-cum-relational propery of impenetrability, and the secondary quality of hardness. | |
| From: Howard Robinson (Perception [1994], IX.3) | |
| A reaction: I would have thought the problem was that 'matter is solidity' turns out on analysis to be a tautology. We have a handful of nearly synonymous words for matter and our experiences of it, but they boil down to some 'given' thing for which we lack words. |