17 ideas
| 14187 | If logic is topic-neutral that means it delves into all subjects, rather than having a pure subject matter [Read] |
| Full Idea: The topic-neutrality of logic need not mean there is a pure subject matter for logic; rather, that the logician may need to go everywhere, into mathematics and even into metaphysics. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Logic') |
| 14188 | Not all arguments are valid because of form; validity is just true premises and false conclusion being impossible [Read] |
| Full Idea: Belief that every valid argument is valid in virtue of form is a myth. ..Validity is a question of the impossibility of true premises and false conclusion for whatever reason, and some arguments are materially valid and the reason is not purely logical. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Logic') | |
| A reaction: An example of a non-logical reason is the transitive nature of 'taller than'. Conceptual connections are the usual example, as in 'it's red so it is coloured'. This seems to be a defence of the priority of semantic consequence in logic. |
| 14182 | If the logic of 'taller of' rests just on meaning, then logic may be the study of merely formal consequence [Read] |
| Full Idea: In 'A is taller than B, and B is taller than C, so A is taller than C' this can been seen as a matter of meaning - it is part of the meaning of 'taller' that it is transitive, but not of logic. Logic is now seen as the study of formal consequence. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Reduct') | |
| A reaction: I think I find this approach quite appealing. Obviously you can reason about taller-than relations, by putting the concepts together like jigsaw pieces, but I tend to think of logic as something which is necessarily implementable on a machine. |
| 14183 | Maybe arguments are only valid when suppressed premises are all stated - but why? [Read] |
| Full Idea: Maybe some arguments are really only valid when a suppressed premise is made explicit, as when we say that 'taller than' is a transitive concept. ...But what is added by making the hidden premise explicit? It cannot alter the soundness of the argument. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Suppress') |
| 14184 | In modus ponens the 'if-then' premise contributes nothing if the conclusion follows anyway [Read] |
| Full Idea: A puzzle about modus ponens is that the major premise is either false or unnecessary: A, If A then B / so B. If the major premise is true, then B follows from A, so the major premise is redundant. So it is false or not needed, and contributes nothing. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Repres') | |
| A reaction: Not sure which is the 'major premise' here, but it seems to be saying that the 'if A then B' is redundant. If I say 'it's raining so the grass is wet', it seems pointless to slip in the middle the remark that rain implies wet grass. Good point. |
| 14186 | Logical connectives contain no information, but just record combination relations between facts [Read] |
| Full Idea: The logical connectives are useful for bundling information, that B follows from A, or that one of A or B is true. ..They import no information of their own, but serve to record combinations of other facts. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Repres') | |
| A reaction: Anyone who suggests a link between logic and 'facts' gets my vote, so this sounds a promising idea. However, logical truths have a high degree of generality, which seems somehow above the 'facts'. |
| 13373 | Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong [Priest,G] |
| Full Idea: A natural principle is the same kind of paradox will have the same kind of solution. Standardly Ramsey's first group are solved by denying the existence of some totality, and the second group are less clear. But denial of the groups sink both. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §5) | |
| A reaction: [compressed] This sums up the argument of Priest's paper, which is that it is Ramsey's division into two kinds (see Idea 13334) which is preventing us from getting to grips with the paradoxes. Priest, notoriously, just lives with them. |
| 13368 | The 'least indefinable ordinal' is defined by that very phrase [Priest,G] |
| Full Idea: König: there are indefinable ordinals, and the least indefinable ordinal has just been defined in that very phrase. (Recall that something is definable iff there is a (non-indexical) noun-phrase that refers to it). | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
| A reaction: Priest makes great subsequent use of this one, but it feels like a card trick. 'Everything indefinable has now been defined' (by the subject of this sentence)? König, of course, does manage to pick out one particular object. |
| 13370 | 'x is a natural number definable in less than 19 words' leads to contradiction [Priest,G] |
| Full Idea: Berry: if we take 'x is a natural number definable in less than 19 words', we can generate a number which is and is not one of these numbers. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
| A reaction: [not enough space to spell this one out in full] |
| 13369 | By diagonalization we can define a real number that isn't in the definable set of reals [Priest,G] |
| Full Idea: Richard: φ(x) is 'x is a definable real number between 0 and 1' and ψ(x) is 'x is definable'. We can define a real by diagonalization so that it is not in x. It is and isn't in the set of reals. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §3) | |
| A reaction: [this isn't fully clear here because it is compressed] |
| 13366 | The least ordinal greater than the set of all ordinals is both one of them and not one of them [Priest,G] |
| Full Idea: Burali-Forti: φ(x) is 'x is an ordinal', and so w is the set of all ordinals, On; δ(x) is the least ordinal greater than every member of x (abbreviation: log(x)). The contradiction is that log(On)∈On and log(On)∉On. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §2) |
| 13367 | The next set up in the hierarchy of sets seems to be both a member and not a member of it [Priest,G] |
| Full Idea: Mirimanoff: φ(x) is 'x is well founded', so that w is the cumulative hierarchy of sets, V; &delta(x) is just the power set of x, P(x). If x⊆V, then V∈V and V∉V, since δ(V) is just V itself. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §2) |
| 13371 | If you know that a sentence is not one of the known sentences, you know its truth [Priest,G] |
| Full Idea: In the family of the Liar is the Knower Paradox, where φ(x) is 'x is known to be true', and there is a set of known things, Kn. By knowing a sentence is not in the known sentences, you know its truth. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §4) | |
| A reaction: [mostly my wording] |
| 13372 | There are Liar Pairs, and Liar Chains, which fit the same pattern as the basic Liar [Priest,G] |
| Full Idea: There are liar chains which fit the pattern of Transcendence and Closure, as can be seen with the simplest case of the Liar Pair. | |
| From: Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §4) | |
| A reaction: [Priest gives full details] Priest's idea is that Closure is when a set is announced as complete, and Transcendence is when the set is forced to expand. He claims that the two keep coming into conflict. |
| 14185 | Conditionals are just a shorthand for some proof, leaving out the details [Read] |
| Full Idea: Truth enables us to carry various reports around under certain descriptions ('what Iain said') without all the bothersome detail. Similarly, conditionals enable us to transmit a record of proof without its detail. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Repres') | |
| A reaction: This is his proposed Redundancy Theory of conditionals. It grows out of the problem with Modus Ponens mentioned in Idea 14184. To say that there is always an implied 'proof' seems a large claim. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |