46 ideas
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| Full Idea: Until the 1960s standard truth-table semantics were the only ones that there were. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.10.1) | |
| A reaction: The 1960s presumably marked the advent of possible worlds. |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| Full Idea: 'dom R' indicates the 'domain' of a relation, that is, the set of all objects that are members of ordered pairs and that have that relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| Full Idea: 'fld R' indicates the 'field' of a relation, that is, the set of all objects that are members of ordered pairs on either side of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| Full Idea: 'ran R' indicates the 'range' of a relation, that is, the set of all objects that are members of ordered pairs and that are related to by the first objects. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| Full Idea: We write F : A → B to indicate that A maps into B, that is, the domain of relating things is set A, and the things related to are all in B. If we add that F = B, then A maps 'onto' B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
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Full Idea:
F(x) is a 'function', which indicates the unique value which y takes in |
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| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| Full Idea: A relation is 'symmetric' on a set if every ordered pair in the set has the relation in both directions. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| Full Idea: A relation is 'transitive' on a set if the relation can be carried over from two ordered pairs to a third. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| Full Idea: The 'powerset' of a set is all the subsets of a given set. Thus: PA = {x : x ⊆ A}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| Full Idea: Two sets are 'disjoint' iff their intersection is empty (i.e. they have no members in common). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| Full Idea: The 'domain' of a relation is the set of all objects that are members of ordered pairs that are members of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| Full Idea: A 'relation' is a set of ordered pairs. The ordering relation on the numbers 0-3 is captured by - in fact it is - the set of ordered pairs {<0,1>,<0,2>,<0,3>,<1,2>,<1,3>,<2,3>}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) | |
| A reaction: This can't quite be a definition of order among numbers, since it relies on the notion of a 'ordered' pair. |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| Full Idea: A 'function' is a relation which is single-valued. That is, for each object, there is only one object in the function set to which that object is related. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| Full Idea: A function 'maps A into B' if the domain of relating things is set A, and the things related to are all in B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| Full Idea: A function 'maps A onto B' if the domain of relating things is set A, and the things related to are set B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| Full Idea: A relation is 'reflexive' on a set if every member of the set bears the relation to itself. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| Full Idea: A relation satisfies 'trichotomy' on a set if every ordered pair is related (in either direction), or the objects are identical. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| Full Idea: A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| Full Idea: An 'equivalence relation' is a binary relation which is reflexive, and symmetric, and transitive. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| Full Idea: Equivalence classes will 'partition' a set. That is, it will divide it into distinct subsets, according to each relation on the set. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| Full Idea: The process is dubbed 'conversational implicature' when the inference is not from the content of what has been said, but from the fact that it has been said. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7.3) |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| Full Idea: The point of logic is to give an account of the notion of validity,..in two standard ways: the semantic way says that a valid inference preserves truth (symbol |=), and the proof-theoretic way is defined in terms of purely formal procedures (symbol |-). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..) | |
| A reaction: This division can be mirrored in mathematics, where it is either to do with counting or theorising about things in the physical world, or following sets of rules from axioms. Language can discuss reality, or play word-games. |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| Full Idea: A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.3.4) | |
| A reaction: So the final column of every line of the truth table will be T. |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| Full Idea: A truth assignment 'satisfies' a formula, or set of formulae, if it evaluates as True when all of its components have been assigned truth values. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.2) | |
| A reaction: [very roughly what Enderton says!] The concept becomes most significant when a large set of wff's is pronounced 'satisfied' after a truth assignment leads to them all being true. |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| Full Idea: If every proof-theoretically valid inference is semantically valid (so that |- entails |=), the proof theory is said to be 'sound'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| Full Idea: If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| Full Idea: If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: [Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that? |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| Full Idea: A set of expressions is 'decidable' iff there exists an effective procedure (qv) that, given some expression, will decide whether or not the expression is included in the set (i.e. doesn't contradict it). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7) | |
| A reaction: This is obviously a highly desirable feature for a really reliable system of expressions to possess. All finite sets are decidable, but some infinite sets are not. |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| Full Idea: The Enumerability Theorem says that for a reasonable language, the set of valid wff's can be effectively enumerated. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: There are criteria for what makes a 'reasonable' language (probably specified to ensure enumerability!). Predicates and functions must be decidable, and the language must be finite. |
| 4986 | A weaker kind of reductionism than direct translation is the use of 'bridge laws' [Kirk,R] |
| Full Idea: If multiple realisability means that psychological terms cannot be translated into physics, one weaker kind of reductionism resorts to 'bridge laws' which link the theory to be reduced to the reducing theory. | |
| From: Robert Kirk (Mind and Body [2003], §3.8) | |
| A reaction: It seems to me that reduction is all-or-nothing, so there can't be a 'weaker' kind. If they are totally separate but linked by naturally necessary laws (e.g. low temperature and ice), they are supervenient, but not reducible to one another. |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| Full Idea: Not all sentences using 'if' are conditionals. Consider 'if you want a banana, there is one in the kitchen'. The rough test is that a conditional can be rewritten as 'that A implies that B'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.6.4) |
| 5001 | Maybe we should see intentionality and consciousness as a single problem, not two [Kirk,R] |
| Full Idea: Many philosophers today have adopted the view that we can achieve an enormous simplification by reducing the two components of the mind-body problem - intentionality and consciousness - into one; ...consciousness is no more than representations. | |
| From: Robert Kirk (Mind and Body [2003], §8.4) | |
| A reaction: One would then see subjective experience and informational content as two consequences of a single mental activity. This strikes me as the correct route to go. We do, after all, learn BY experiencing. Hence concepts are tied in with qualia. |
| 4993 | If a bird captures a worm, we could say its behaviour is 'about' the worm [Kirk,R] |
| Full Idea: When a bird pulls a worm from the ground, then swallows it piece by piece, there is a sense in which its behaviour can be said to be about the worm. | |
| From: Robert Kirk (Mind and Body [2003], §5.4) | |
| A reaction: This is preparing the ground for a possible behaviourist account of intentionality. Reply: you could say rain is about puddles, or you could say we have adopted Dennett's 'intentional stance' to birds, but it tells us nothing about their psychology. |
| 5000 | Behaviourism says intentionality is an external relation; language of thought says it's internal [Kirk,R] |
| Full Idea: The conflict over whether intentionality is a matter of behavioural relations with the rest of the world, or of the internal states of the subject, is at its most dramatic in the contrast between behaviourism and the language of thought hypothesis. | |
| From: Robert Kirk (Mind and Body [2003], §7.10) | |
| A reaction: I just don't believe any behaviourist external account of intentionality, which ducks the question of how it all works. Personally I am more drawn to maps and models than to a language of thought. I plan my actions in an imagined space-time world. |
| 4982 | Dualism implies some brain events with no physical cause, and others with no physical effect [Kirk,R] |
| Full Idea: If the mind causes brain events, then they are not caused by other brain events, and such causal gaps should be detectable by scientists; there should also be a gap of brain-events which cause no other brain events, because they are causing mind events. | |
| From: Robert Kirk (Mind and Body [2003], §2.5) | |
| A reaction: This is the double causation problem which Spinoza had spotted (Idea 4862). Expressed this way, it seems a screamingly large problem for dualism. We should be able to discover some VERY strange physical activity in the brain. |
| 4991 | Behaviourism seems a good theory for intentional states, but bad for phenomenal ones [Kirk,R] |
| Full Idea: For many kinds of mental states, notably intentional ones such as beliefs and desires, behaviourism is appealing, ..but for sensations and experiences such as pain, it seems grossly implausible. | |
| From: Robert Kirk (Mind and Body [2003], §5.1) | |
| A reaction: The theory does indeed make a bit more sense for intentional states, but it still strikes me as nonsense that there is no more to my belief that 'Whales live in the Atlantic' than a disposition to say something. WHY do I say this something? |
| 4994 | Behaviourism offers a good alternative to simplistic unitary accounts of mental relationships [Kirk,R] |
| Full Idea: There is a temptation to think that 'aboutness', and the 'contents' of thoughts, and the relation of 'reference', are single and unitary relationships, but behaviourism offers an alternative approach. | |
| From: Robert Kirk (Mind and Body [2003], §5.5) | |
| A reaction: Personally I wouldn't touch behaviourism with a barge-pole (as it ducks the question of WHY certain behaviour occurs), but a warning against simplistic accounts of intentional states is good. I am sure there cannot be a single neat theory of refererence. |
| 4992 | In 'holistic' behaviourism we say a mental state is a complex of many dispositions [Kirk,R] |
| Full Idea: There is a non-reductive version of behaviourism ( which we can call 'global' or 'holistic') which says there is no more to having mental states than having a complex of certain kinds of behavioural dispositions. | |
| From: Robert Kirk (Mind and Body [2003], §5.2) | |
| A reaction: This is designed to meet a standard objection to behaviourism - that there is no straight correlation between what I think and how I behave. The present theory is obviously untestable, because a full 'complex' of human dispositions is never repeated. |
| 4990 | The inverted spectrum idea is often regarded as an objection to behaviourism [Kirk,R] |
| Full Idea: The inverted spectrum idea is often regarded as an objection to behaviourism. | |
| From: Robert Kirk (Mind and Body [2003], §4.5) | |
| A reaction: Thus, my behaviour at traffic lights should be identical, even if I have a lifelong inversion of red and green. A good objection. Note that physicalists can believe in inverted qualia as well a dualists, as long as the brain states are also inverted. |
| 4984 | All meaningful psychological statements can be translated into physics [Kirk,R] |
| Full Idea: All psychological statements which are meaningful, that is to say, which are in principle verifiable, are translatable into propositions which do not involve psychological concepts, but only the concepts of physics. | |
| From: Robert Kirk (Mind and Body [2003], §3.8) | |
| A reaction: This shows how eliminativist behaviourism arises out of logical positivism (by only allowing what is verifiable). The simplest objection: we can't verify the mental states of others, because they are private, but they are still the best explanation. |
| 4998 | Instead of representation by sentences, it can be by a distribution of connectionist strengths [Kirk,R] |
| Full Idea: In a connectionist system, information is represented not by sentences but by the total distribution of connection strengths. | |
| From: Robert Kirk (Mind and Body [2003], §7.6) | |
| A reaction: Neither sentences (of a language of thought) NOR connection strengths strike me as very plausible ways for a brain to represent things. It must be something to do with connections, but it must also be to do with neurons, or we get bizarre counterexamples. |
| 4985 | If mental states are multiply realisable, they could not be translated into physical terms [Kirk,R] |
| Full Idea: If psychological states are multiply realisable it is hard to see how they could possibly be translated into physical terms. | |
| From: Robert Kirk (Mind and Body [2003], §3.8) | |
| A reaction: Reductive funtionalism would do it. A writing iimplement is physical and multiply realisable. Personally I prefer the strategy of saying mental states are NOT multiply realisable. If frog brains differ from ours, they probably don't feel pain like us. |
| 4997 | It seems unlikely that most concepts are innate, if a theory must be understood to grasp them [Kirk,R] |
| Full Idea: It is widely accepted that for many concepts, if not all, grasping the concept requires grasping some theory, ...which makes difficulties for the view that concepts are not learned: for 'radical concept nativism', as Fodor calls it. | |
| From: Robert Kirk (Mind and Body [2003], §7.3) | |
| A reaction: Not a problem for traditional rationalist theories, where the whole theory can be innate along with the concept, but a big objection to modern more cautious non-holistic views (such as Fodor's). Does a bird have a concept AND theory of a nest? |
| 4999 | For behaviourists language is just a special kind of behaviour [Kirk,R] |
| Full Idea: Behaviourists regard the use of language as just a special kind of behaviour. | |
| From: Robert Kirk (Mind and Body [2003], §7.9) | |
| A reaction: This is not an intuitively obvious view of language. We behave, and then we talk about behaviour. Performative utterances (like promising) have an obvious behavioural aspect, as do violent threats, but not highly theoretical language (such as maths). |
| 4995 | Behaviourists doubt whether reference is a single type of relation [Kirk,R] |
| Full Idea: To most behaviourists it seems misguided to expect there to be a single relation that connects referring expressions with their referents. | |
| From: Robert Kirk (Mind and Body [2003], §5.5) | |
| A reaction: You don't need to be a behaviourist to feel this doubt. Think about names of real people, names of fictional people, reference to misunderstood items, or imagined items, or reference in dreams, or to mathematical objects, or negations etc. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |