33 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) |
| 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) |
| 6375 | The taste of chocolate is a 'finer-grained' sensation than the taste of sweetness [Polger] |
| Full Idea: The taste of chocolate is presumably a 'finer-grained' sensation than the taste of sweetness. | |
| From: Thomas W. Polger (Natural Minds [2004], Ch.1.4) | |
| A reaction: An interesting distinction when it comes to what they are like, and whether two very different brains can realise them. Sweetness might be the same for most creatures, but the tast of chocolate subtly different. |
| 6381 | The mind and the self are one, and the mind-self is a biological phenomenon [Polger] |
| Full Idea: We should return to the old idea that the mind and the self are one and combine it with the new idea that the mind-self is a biological phenomenon. | |
| From: Thomas W. Polger (Natural Minds [2004], §8.3) | |
| A reaction: This doesn't make allowance for the fact that some parts of my mind seem like irritating visitors, and other parts seem like the home-owner. Personally I take the self to be the brain's central controller, or the centre (forum) of brain integration. |
| 6378 | Teleological functions explain why a trait exists; causal-role functions say what it does [Polger] |
| Full Idea: Teleological functions help explain why a trait has come to exist; causal-role functions tell what a trait does or is apt to do. | |
| From: Thomas W. Polger (Natural Minds [2004], §5.4) | |
| A reaction: The teleological view has the merit of nesting nicely with the theory of evolution, and with Aristotelian virtue ethics (which I like). Causal-role functionalism focuses better on what is actually happening inside the head. |
| 6380 | Identity theory says consciousness is an abstraction: a state, event, process or property [Polger] |
| Full Idea: Identity theories locate consciousness at a certain order of abstraction, typically among neurophysiological states, events, processes, or properties. | |
| From: Thomas W. Polger (Natural Minds [2004], Ch.7.6) | |
| A reaction: I increasingly think that processes are the answer. My new analogy for the mind is a waterfall: its physical ontology is simple, it only exists because there is a sustained process, and it is far too complex to predict individual droplet outcomes. |
| 20014 | Actions include: the involuntary, the purposeful, the intentional, and the self-consciously autonomous [Wilson/Schpall] |
| Full Idea: There are different levels of action, including at least: unconscious and/or involuntary behaviour, purposeful or goal-directed activity, intentional action, and the autonomous acts or actions of self-consciously active human agents. | |
| From: Wilson,G/Schpall,S (Action [2012], 1) | |
| A reaction: The fourth class is obviously designed to distinguish us from the other animals. It immediately strikes me as very optimistic to distinguish four (at least) clear categories, but you have to start somewhere. |
| 20019 | Maybe bodily movements are not actions, but only part of an agent's action of moving [Wilson/Schpall] |
| Full Idea: Some say that the movement's of agent's body are never actions. It is only the agent's direct moving of, say, his leg that constitutes a physical action; the leg movement is merely caused by and/or incorporated as part of the act of moving. | |
| From: Wilson,G/Schpall,S (Action [2012], 1.2) | |
| A reaction: [they cite Jennifer Hornsby 1980] It seems normal to deny a twitch the accolade of an 'action', so I suppose that is right. Does the continual movement of my tongue count as action? Only if I bring it under control? Does it matter? Only in forensics. |
| 20021 | Is the action the arm movement, the whole causal process, or just the trying to do it? [Wilson/Schpall] |
| Full Idea: Some philosophers have favored the overt arm movement the agent performs, some favor the extended causal process he initiates, and some prefer the relevant event of trying that precedes and 'generates' the rest. | |
| From: Wilson,G/Schpall,S (Action [2012], 1.2) | |
| A reaction: [Davidson argues for the second, Hornsby for the third] There seems no way to settle this, and a compromise looks best. Mere movement won't do, and mere trying won't do, and whole processes get out of control. |
| 20022 | To be intentional, an action must succeed in the manner in which it was planned [Wilson/Schpall] |
| Full Idea: If someone fires a bullet to kill someone, misses, and dislodges hornets that sting him to death, this implies that an intentional action must include succeeding in a manner according to the original plan. | |
| From: Wilson,G/Schpall,S (Action [2012], 2) | |
| A reaction: [their example, compressed] This resembles Gettier's problem cases for knowledge. If the shooter deliberately and maliciously brought down the hornet's nest, that would be intentional murder. Sounds right. |
| 20023 | If someone believes they can control the lottery, and then wins, the relevant skill is missing [Wilson/Schpall] |
| Full Idea: If someone enters the lottery with the bizarre belief that they can control who wins, and then wins it, that suggest that intentional actions must not depend on sheer luck, but needs competent exercise of the relevant skill. | |
| From: Wilson,G/Schpall,S (Action [2012], 2) | |
| A reaction: A nice companion to Idea 20022, which show that a mere intention is not sufficient to motivate and explain an action. |
| 20025 | We might intend two ways to acting, knowing only one of them can succeed [Wilson/Schpall] |
| Full Idea: If an agent tries to do something by two different means, only one of which can succeed, then the behaviour is rational, even though one of them is an attempt to do an action which cannot succeed. | |
| From: Wilson,G/Schpall,S (Action [2012], 2) | |
| A reaction: [a concise account of a laborious account of an example from Bratman 1984, 1987] Bratman uses this to challenge the 'Simple View', that intention leads straightforwardly to action. |
| 20031 | On one model, an intention is belief-desire states, and intentional actions relate to beliefs and desires [Wilson/Schpall] |
| Full Idea: On the simple desire-belief model, an intention is a combination of desire-belief states, and an action is intentional in virtue of standing in the appropriate relation to these simpler terms. | |
| From: Wilson,G/Schpall,S (Action [2012], 4) | |
| A reaction: This is the traditional view found in Hume, and is probably endemic to folk psychology. They cite Bratman 1987 as the main opponent of the view. |
| 20028 | Groups may act for reasons held by none of the members, so maybe groups are agents [Wilson/Schpall] |
| Full Idea: Rational group action may involve a 'collectivising of reasons', with participants acting in ways that are not rationally recommended from the individual viewpoint. This suggests that groups can be rational, intentional agents. | |
| From: Wilson,G/Schpall,S (Action [2012], 2) | |
| A reaction: [Pettit 2003] is the source for this. Gilbert says individuals can have joint commitment; Pettit says the group can be an independent agent. The matter of shared intentions is interesting, but there is no need for the ontology to go berserk. |
| 20027 | If there are shared obligations and intentions, we may need a primitive notion of 'joint commitment' [Wilson/Schpall] |
| Full Idea: An account of mutual obligation to do something may require that we give up reductive individualist accounts of shared activity and posit a primitive notion of 'joint commitment'. | |
| From: Wilson,G/Schpall,S (Action [2012], 2) | |
| A reaction: [attributed to Margaret Gilbert 2000] If 'we' are trying to do something, that seems to give an externalist picture of intentions, rather like all the other externalisms floating around these days. I don't buy any of it, me. |
| 20016 | Strong Cognitivism identifies an intention to act with a belief [Wilson/Schpall] |
| Full Idea: A Strong Cognitivist is someone who identifies an intention with a certain pertinent belief about what she is doing or about to do. | |
| From: Wilson,G/Schpall,S (Action [2012], 1.1) | |
| A reaction: (Sarah Paul 2009 makes this distinction) The belief, if so, seems to be as much counterfactual as factual. Hope seems to come into it, which isn't exactly a belief. |
| 20017 | Weak Cognitivism says intentions are only partly constituted by a belief [Wilson/Schpall] |
| Full Idea: A Weak Cognitivist holds that intentions are partly constituted by, but are not identical with, relevant beliefs about the action. Grice (1971) said an intention is willing an action, combined with a belief that this will lead to the action. | |
| From: Wilson,G/Schpall,S (Action [2012], 1.1) | |
| A reaction: [compressed] I didn't find Strong Cognitivism appealing, but it seems hard to argue with some form of the weak version. |
| 20018 | Strong Cognitivism implies a mode of 'practical' knowledge, not based on observation [Wilson/Schpall] |
| Full Idea: Strong Cognitivists say intentions/beliefs are not based on observation or evidence, and are causally reliable in leading to appropriate actions, so this is a mode of 'practical' knowledge that has not been derived from observation. | |
| From: Wilson,G/Schpall,S (Action [2012], 1.1) | |
| A reaction: [compressed - Stanford unnecessarily verbose!] I see no mention in this discussion of 'hoping' that your action will turn out OK. We are usually right to hope, but it would be foolish to say that when we reach for the salt we know we won't knock it over. |
| 20012 | Maybe the explanation of an action is in the reasons that make it intelligible to the agent [Wilson/Schpall] |
| Full Idea: Some have maintained that we explain why an agent acted as he did when we explicate how the agent's normative reasons rendered the action intelligible in his eyes. | |
| From: Wilson,G/Schpall,S (Action [2012], Intro) | |
| A reaction: Modern psychology is moving against this, by showing how hidden biases can predominate over conscious reasons (as in Kahnemann's work). I would say this mode of explanation works better for highly educated people (but you can chuckle at that). |
| 20029 | Causalists allow purposive explanations, but then reduce the purpose to the action's cause [Wilson/Schpall] |
| Full Idea: Most causalists allow that reason explanations are teleological, but say that such purposive explanations are analysable causally, where the primary reasons for the act are the guiding causes of the act. | |
| From: Wilson,G/Schpall,S (Action [2012], 3) | |
| A reaction: The authors observe that it is hard to adjudicate on this matter, and that the concept of the 'cause' of an action is unclear. |
| 20013 | It is generally assumed that reason explanations are causal [Wilson/Schpall] |
| Full Idea: The view that reason explanations are somehow causal explanations remains the dominant position. | |
| From: Wilson,G/Schpall,S (Action [2012], Intro) | |
| A reaction: I suspect that this is only because no philosopher has a better idea, and the whole issue is being slowly outflanked by psychology. |
| 6379 | A mummified heart has the teleological function of circulating blood [Polger] |
| Full Idea: A preserved heart in a jar of formaldehyde has the teleological function of circulating blood. | |
| From: Thomas W. Polger (Natural Minds [2004], §5.4) | |
| A reaction: A nice illustration. |
| 6377 | Teleological notions of function say what a thing is supposed to do [Polger] |
| Full Idea: Teleological notions of function specify not just what a thing happens to do, but what it is supposed to do. | |
| From: Thomas W. Polger (Natural Minds [2004], Ch.5.3) | |
| A reaction: This is the basis of a distinct theory of the mind. It seems to be akin to the 'dispositions' of behaviourism, so that the mind becomes once more a theoretical and abstract entity, rather than a thing of occurrent events and processes. |