45 ideas
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 6548 | Physicalism requires the naturalisation or rejection of set theory [Lycan] |
| Full Idea: Eventually set theory will have to be either naturalised or rejected, if a thoroughgoing physicalism is to be maintained. | |
| From: William Lycan (Consciousness [1987], 8.4) | |
| A reaction: Personally I regard Platonism as a form of naturalism (though a rather bold and dramatic one). The central issue seems to be the ability of the human main/brain to form 'abstract' notions about the physical world in which it lives. |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 16014 | It is controversial whether only 'numerical identity' allows two things to be counted as one [Noonan] |
| Full Idea: 'Numerical identity' implies the controversial view that it is the only identity relation in accordance with which we can properly count (or number) things: x and y are to be properly counted as one just in case they are numerically identical. | |
| From: Harold Noonan (Identity [2009], §1) | |
| A reaction: Noonan cites Geach, presumably to remind us of relative identity, where two things may be one or two, depending on what they are relative to. The one 'guard on the gate' may actually be two men. |
| 6531 | Institutions are not reducible as types, but they are as tokens [Lycan] |
| Full Idea: Institutional types are irreducible, though I assume that institutional tokens are reducible in the sense of strict identity, all the way down to the subatomic level. | |
| From: William Lycan (Consciousness [1987], 4.3) | |
| A reaction: This seems a promising distinction, as the boundaries of 'institutions' disappear when you begin to reduce them to lower levels (cf. Idea 4601), and yet plenty of institutions are self-evidently no more than physics. Plants are invisible as physics. |
| 6532 | Types cannot be reduced, but levels of reduction are varied groupings of the same tokens [Lycan] |
| Full Idea: If types cannot be reduced to more physical levels, this is not an embarrassment, as long as our institutional categories, our physiological categories, and our physical categories are just alternative groupings of the same tokens. | |
| From: William Lycan (Consciousness [1987], 4.3) | |
| A reaction: This is a self-evident truth about a car engine, so I don't see why it wouldn't apply equally to a brain. Lycan's identification of the type as the thing which cannot be reduced seems a promising explanation of much confusion among philosophers. |
| 6534 | One location may contain molecules, a metal strip, a key, an opener of doors, and a human tragedy [Lycan] |
| Full Idea: One space-time slice may be occupied by a collection of molecules, a metal strip, a key, an allower of entry to hotel rooms, a facilitator of adultery, and a destroyer souls. | |
| From: William Lycan (Consciousness [1987], 4.3) | |
| A reaction: Desdemona's handkerchief is a nice example. This sort of remark seems to be felt by some philosophers to be heartless wickedness, and yet it so screamingly self-evident that it is impossible to deny. |
| 6529 | I see the 'role'/'occupant' distinction as fundamental to metaphysics [Lycan] |
| Full Idea: I see the 'role'/'occupant' distinction as fundamental to metaphysics. | |
| From: William Lycan (Consciousness [1987], 4.0) | |
| A reaction: A passing remark in a discussion of functionalism about the mind, but I find it appealing. Causation is basic to materialistic metaphysics, and it creates networks of regular causes. It leaves open the essentialist question of WHY it has that role. |
| 16024 | I could have died at five, but the summation of my adult stages could not [Noonan] |
| Full Idea: Persons have different modal properties from the summations of person-stages. …I might have died when I was five. But the maximal summation of person-stages which perdurantists say is me could not have had a temporal extent of a mere five years. | |
| From: Harold Noonan (Identity [2009], §5) | |
| A reaction: Thus the summation of stages seems to fail Leibniz's Law, since truths about a part are not true of the whole. But my foot might be amputated without me being amputated. The objection is the fallacy of composition? |
| 16023 | Stage theorists accept four-dimensionalism, but call each stage a whole object [Noonan] |
| Full Idea: Stage theorists, accepting the ontology of perdurance, modify the semantics to secure the result that fatness is a property of a cat. Every temporal part of a cat (such as Tabby-on-Monday) is a cat. …(but they pay a price over the counting of cats). | |
| From: Harold Noonan (Identity [2009], §5) | |
| A reaction: [Noonan cites Hawley and Sider for this view. The final parenthesis compresses Noonan] I would take the difficulty over counting cats to be fatal to the view. It produces too many cats, or too few, or denies counting altogether. |
| 16015 | Problems about identity can't even be formulated without the concept of identity [Noonan] |
| Full Idea: If identity is problematic, it is difficult to see how the problem could be resolved, since it is difficult to see how a thinker could have the conceptual resources with which to explain the concept of identity whilst lacking that concept itself. | |
| From: Harold Noonan (Identity [2009], §1) | |
| A reaction: I don't think I accept this. We can comprehend the idea of a mind that didn't think in terms of identities (at least for objects). I suppose any relation of a mind to the world has to distinguish things in some way. Does the Parmenidean One have identity? |
| 16017 | Identity is usually defined as the equivalence relation satisfying Leibniz's Law [Noonan] |
| Full Idea: Numerical identity is usually defined as the equivalence relation (or: the reflexive relation) satisfying Leibniz's Law, the indiscernibility of identicals, where everything true of x is true of y. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: Noonan says this must include 'is identical to x' among the truths, and so is circular |
| 16016 | Identity definitions (such as self-identity, or the smallest equivalence relation) are usually circular [Noonan] |
| Full Idea: Identity can be circularly defined, as 'the relation everything has to itself and to nothing else', …or as 'the smallest equivalence relation'. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: The first one is circular because 'nothing else' implies identity. The second is circular because it has to quantify over all equivalence relations. (So says Noonan). |
| 16020 | Identity can only be characterised in a second-order language [Noonan] |
| Full Idea: There is no condition in a first-order language for a predicate to express identity, rather than indiscernibility within the resources of the language. Leibniz's Law is statable in a second-order language, so identity can be uniquely characterised. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: The point is that first-order languages only refer to all objects, but you need to refer to all properties to include Leibniz's Law. Quine's 'Identity, Ostension and Hypostasis' is the source of this idea. |
| 16018 | Indiscernibility is basic to our understanding of identity and distinctness [Noonan] |
| Full Idea: Leibniz's Law (the indiscernibility of identicals) appears to be crucial to our understanding of identity, and, more particularly, to our understanding of distinctness. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: True, but indiscernibility concerns the epistemology, and identity concerns the ontology. |
| 16019 | Leibniz's Law must be kept separate from the substitutivity principle [Noonan] |
| Full Idea: Leibniz's Law must be clearly distinguished from the substitutivity principle, that if 'a' and 'b' are codesignators they are substitutable salva veritate. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: He gives a bunch of well-known problem cases for substitutivity. The Morning Star, Giorgione, and the number of planets won't work. Belief contexts, or facts about spelling, may not be substitutable. |
| 6549 | I think greenness is a complex microphysical property of green objects [Lycan] |
| Full Idea: Personally I favour direct realism regarding secondary qualities, and identify greenness with some complex microphysical property exemplified by green physical objects. | |
| From: William Lycan (Consciousness [1987], 8.4) | |
| A reaction: He cites D.M.Armstrong (1981) as his source. Personally I find this a bewildering proposal. Does he think there is greenness in grass AS WELL AS the emission of that wavelength of electro-magnetic radiation? Is greenness zooming through the air? |
| 6543 | Intentionality comes in degrees [Lycan] |
| Full Idea: Intentionality comes in degrees. | |
| From: William Lycan (Consciousness [1987], 5.4) | |
| A reaction: I agree. A footprint is 'about' a foot, in the sense of containing concentrated information about it. Can we, though, envisage a higher degree than human thought? Is there a maximum degree? Everything is 'about' everything, in some respect. |
| 6537 | Teleological views allow for false intentional content, unlike causal and nomological theories [Lycan] |
| Full Idea: The teleological view begins to explain intentionality, and in particular allows brain states and events to have false intentional content; causal and nomological theories of intentionality tend to falter on this last task. | |
| From: William Lycan (Consciousness [1987], 4.4) | |
| A reaction: Certainly if you say thought is 'caused' by the world, false thought become puzzling. I'm not sure I understand the rest of this, but it is an intriguing remark about a significant issue… |
| 6546 | Pain is composed of urges, desires, impulses etc, at different levels of abstraction [Lycan] |
| Full Idea: Our phenomenal experience of pain has components - it is a complex, consisting (perhaps) of urges, desires, impulses, and beliefs, probably occurring at quite different levels of institutional abstraction. | |
| From: William Lycan (Consciousness [1987], 5.5) | |
| A reaction: This seems to be true, and offers the reductionist a strategy for making inroads into the supposed irreducable and fundamental nature of qualia. What's it like to be a complex hierarchically structured multi-functional organism? |
| 6547 | The right 'level' for qualia is uncertain, though top (behaviourism) and bottom (particles) are false [Lycan] |
| Full Idea: It is just arbitrary to choose a level of nature a priori as the locus of qualia, even though we can agree that high levels (such as behaviourism) and low-levels (such as the subatomic) can be ruled out as totally improbable. | |
| From: William Lycan (Consciousness [1987], 5.6) | |
| A reaction: Very good. People scream 'qualia!' whenever the behaviour level or the atomic level are proposed as the locations of the mind, but the suggestion that they are complex, and are spread across many functional levels in the middle sounds good. |
| 6527 | If energy in the brain disappears into thin air, this breaches physical conservation laws [Lycan] |
| Full Idea: By interacting causally, Cartesian dualism seems to violate the conservation laws of physics (concerning matter and energy). This seems testable, and afferent and efferent pathways disappearing into thin air would suggest energy is not conserved. | |
| From: William Lycan (Consciousness [1987], 1.1) | |
| A reaction: It would seem to be no problem as long as outputs were identical in energy to inputs. If the experiment could actually be done, the result might astonish us. |
| 6528 | In lower animals, psychology is continuous with chemistry, and humans are continuous with animals [Lycan] |
| Full Idea: Evolution has proceeded in all other known species by increasingly complex configurations of molecules and organs, which support primitive psychologies; our human psychologies are more advanced, but undeniably continuous with lower animals. | |
| From: William Lycan (Consciousness [1987], 1.1) | |
| A reaction: Personally I find the evolution objection to dualism highly persuasive. I don't see how anyone can take evolution seriously and be a dualist. If there is a dramatic ontological break at some point, a plausible reason would be needed for that. |
| 6554 | Two behaviourists meet. The first says,"You're fine; how am I?" [Lycan] |
| Full Idea: Old joke: two Behaviourists meet in the street, and the first says,"You're fine; how am I?" | |
| From: William Lycan (Consciousness [1987], n1.6) | |
| A reaction: This invites the response that introspection is uniquely authoritative about 'how we are', but this has been challenged quite a lot recently, which pushes us to consider whether these stupid behaviourists might actually have a good point. |
| 6545 | If functionalism focuses on folk psychology, it ignores lower levels of function [Lycan] |
| Full Idea: 'Analytical functionalists', who hold that meanings of mental terms are determined by the causal roles associated with them by 'folk psychology', deny themselves appeals to lower levels of functional organisation. | |
| From: William Lycan (Consciousness [1987], 5.4) | |
| A reaction: Presumably folk psychology can fit into the kind of empirical methodology favoured by behaviourists, whereas 'lower levels' are going to become rather speculative and unscientific. |
| 6541 | Functionalism must not be too abstract to allow inverted spectrum, or so structural that it becomes chauvinistic [Lycan] |
| Full Idea: The functionalist must find a level of characterisation of mental states that is not so abstract or behaviouristic as to rule out the possibility of inverted spectrum etc., nor so specific and structural as to fall into chauvinism. | |
| From: William Lycan (Consciousness [1987], 5.4) | |
| A reaction: If too specific then animals and aliens won't be able to implement the necessary functions; if the theory becomes very behaviouristic, then it loses interest in the possibility of an inverted spectrum. He is certainly right to hunt for a middle ground. |
| 6539 | The distinction between software and hardware is not clear in computing [Lycan] |
| Full Idea: Even the software/hardware distinction as it is literally applied within computer science is philosophically unclear. | |
| From: William Lycan (Consciousness [1987], 4.4) | |
| A reaction: This is true, and very important for functionalist theories of the mind. Even very volatile software is realised in 'hard' physics, and rewritable discs etc blur the distinction between 'programmable' and 'hardwired'. |
| 6533 | Mental types are a subclass of teleological types at a high level of functional abstraction [Lycan] |
| Full Idea: I am taking mental types to form a small subclass of teleological types occurring for the most part at a high level of functional abstraction. | |
| From: William Lycan (Consciousness [1987], 4.3) | |
| A reaction: He goes on to say that he understand teleology in evolutionary terms. There is always a gap between how you characterise or individuate something, and what it actually is. To say spanners are 'a small subclass of tools' is not enough. |
| 6535 | Teleological characterisations shade off smoothly into brutely physical ones [Lycan] |
| Full Idea: Highly teleological characterisations, unlike naïve and explicated mental characterisations, have the virtue of shading off fairly smoothly into (more) brutely physical ones. | |
| From: William Lycan (Consciousness [1987], 4.3) | |
| A reaction: Thus the purpose of a car engine, and a spark plug, and the spark, and the temperature, and the vibration of molecules show a fading away of the overt purpose, disappearing into the pointless activity of electrons and quantum levels. |
| 6544 | Identity theory is functionalism, but located at the lowest level of abstraction [Lycan] |
| Full Idea: 'Neuron' may be understood as a physiological term or a functional term, so even the Identity Theorist is a Functionalist - one who locates mental entities at a very low level of abstraction. | |
| From: William Lycan (Consciousness [1987], 5.4) | |
| A reaction: This is a striking observation, and somewhat inclines me to switch from identity theory to functionalism. If you ask what is the correct level of abstraction, Lycan's teleological-homuncular version refers you to all the levels. |
| 6530 | We reduce the mind through homuncular groups, described abstractly by purpose [Lycan] |
| Full Idea: I am explicating the mental in a reductive way, by reducing mental characterizations to homuncular institutional ones, which are teleological characterizations at various levels of functional abstraction. | |
| From: William Lycan (Consciousness [1987], 4.3) | |
| A reaction: I think this is the germ of a very good physicalist account of the mind. More is needed than a mere assertion about what the mind reduces to at the very lowest level; this offers a decent account of the descending stages of reduction. |
| 6536 | Teleological functionalism helps us to understand psycho-biological laws [Lycan] |
| Full Idea: Teleological functionalism helps us to understand the nature of biological and psychological laws, particularly in the face of Davidsonian scepticism about the latter. | |
| From: William Lycan (Consciousness [1987], 4.4) | |
| A reaction: Personally I doubt the existence of psycho-physical laws, but only because of the vast complexity. They would be like the laws of weather. 'Psycho-physical' laws seem to presuppose some sort of dualism. |
| 6542 | A Martian may exhibit human-like behaviour while having very different sensations [Lycan] |
| Full Idea: Quite possibly a Martian's humanoid behaviour is prompted by his having sensations somewhat unlike ours, despite his superficial behavioural similarities to us. | |
| From: William Lycan (Consciousness [1987], 5.4) | |
| A reaction: I think this firmly refutes the multiple realisability objection to type-type physicalism. Mental events are individuated by their phenomenal features (known only to the user), and by their causal role (publicly available). These are separate. |
| 6538 | We need a notion of teleology that comes in degrees [Lycan] |
| Full Idea: We need a notion of teleology that comes in degrees. | |
| From: William Lycan (Consciousness [1987], 4.4) | |
| A reaction: Anyone who says that key concepts, such as those concerning the mind, should come 'in degrees' wins my instant support. A whole car engine requires a very teleological explanation, the spark in the sparkplug far less so. |
| 6551 | 'Physical' means either figuring in physics descriptions, or just located in space-time [Lycan] |
| Full Idea: An object is specifically physical if it figures in explanations and descriptions of features of ordinary non-living matter, as in current physics; it is more generally physical if it is simply located in space-time. | |
| From: William Lycan (Consciousness [1987], 8.5) | |
| A reaction: This gives a useful distinction when trying to formulate a 'physicalist' account of the mind, where type-type physicalism says only the 'postulates of physics' can be used, whereas 'naturalism' about the mind uses the more general concept. |