16 ideas
| 13838 | A decent modern definition should always imply a semantics [Hacking] |
| Full Idea: Today we expect that anything worth calling a definition should imply a semantics. | |
| From: Ian Hacking (What is Logic? [1979], §10) | |
| A reaction: He compares this with Gentzen 1935, who was attempting purely syntactic definitions of the logical connectives. |
| 13833 | 'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking] |
| Full Idea: 'Dilution' (or 'Thinning') provides an essential contrast between deductive and inductive reasoning; for the introduction of new premises may spoil an inductive inference. | |
| From: Ian Hacking (What is Logic? [1979], §06.2) | |
| A reaction: That is, inductive logic (if there is such a thing) is clearly non-monotonic, whereas classical inductive logic is monotonic. |
| 13834 | Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking] |
| Full Idea: If A |- B and B |- C, then A |- C. This generalises to: If Γ|-A,Θ and Γ,A |- Θ, then Γ |- Θ. Gentzen called this 'cut'. It is the transitivity of a deduction. | |
| From: Ian Hacking (What is Logic? [1979], §06.3) | |
| A reaction: I read the generalisation as 'If A can be either a premise or a conclusion, you can bypass it'. The first version is just transitivity (which by-passes the middle step). |
| 13835 | Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking] |
| Full Idea: Only the cut rule can have a conclusion that is less complex than its premises. Hence when cut is not used, a derivation is quite literally constructive, building up from components. Any theorem obtained by cut can be obtained without it. | |
| From: Ian Hacking (What is Logic? [1979], §08) |
| 13845 | The various logics are abstractions made from terms like 'if...then' in English [Hacking] |
| Full Idea: I don't believe English is by nature classical or intuitionistic etc. These are abstractions made by logicians. Logicians attend to numerous different objects that might be served by 'If...then', like material conditional, strict or relevant implication. | |
| From: Ian Hacking (What is Logic? [1979], §15) | |
| A reaction: The idea that they are 'abstractions' is close to my heart. Abstractions from what? Surely 'if...then' has a standard character when employed in normal conversation? |
| 13840 | First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking] |
| Full Idea: First-order logic is the strongest complete compact theory with a Löwenheim-Skolem theorem. | |
| From: Ian Hacking (What is Logic? [1979], §13) |
| 13844 | A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking] |
| Full Idea: Henkin proved that there is no first-order treatment of branching quantifiers, which do not seem to involve any idea that is fundamentally different from ordinary quantification. | |
| From: Ian Hacking (What is Logic? [1979], §13) | |
| A reaction: See Hacking for an example of branching quantifiers. Hacking is impressed by this as a real limitation of the first-order logic which he generally favours. |
| 13842 | Second-order completeness seems to need intensional entities and possible worlds [Hacking] |
| Full Idea: Second-order logic has no chance of a completeness theorem unless one ventures into intensional entities and possible worlds. | |
| From: Ian Hacking (What is Logic? [1979], §13) |
| 13837 | With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking] |
| Full Idea: My doctrine is that the peculiarity of the logical constants resides precisely in that given a certain pure notion of truth and consequence, all the desirable semantic properties of the constants are determined by their syntactic properties. | |
| From: Ian Hacking (What is Logic? [1979], §09) | |
| A reaction: He opposes this to Peacocke 1976, who claims that the logical connectives are essentially semantic in character, concerned with the preservation of truth. |
| 13839 | Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking] |
| Full Idea: For some purposes the variables of first-order logic can be regarded as prepositions and place-holders that could in principle be dispensed with, say by a system of arrows indicating what places fall in the scope of which quantifier. | |
| From: Ian Hacking (What is Logic? [1979], §11) | |
| A reaction: I tend to think of variables as either pronouns, or as definite descriptions, or as temporary names, but not as prepositions. Must address this new idea... |
| 13843 | If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking] |
| Full Idea: A Löwenheim-Skolem theorem holds for anything which, on my delineation, is a logic. | |
| From: Ian Hacking (What is Logic? [1979], §13) | |
| A reaction: I take this to be an unusually conservative view. Shapiro is the chap who can give you an alternative view of these things, or Boolos. |
| 6900 | A prior understanding of beauty is needed to assert that the Form of the Beautiful is beautiful [Westaway] |
| Full Idea: If it were asserted that the Form of the Beautiful was itself beautiful, such a statement would require a prior understanding of the concept of beauty, so would immediately lead to an infinite regress, so the Forms can't be self-predicating. | |
| From: Luke Westaway (talk [2005]), quoted by PG - Db (ideas) | |
| A reaction: This is a nice clear statement of the mess that Plato gets himself into if he wants the Forms to be self-predicating. Clearly the Form of the Beautiful can't be beautiful, but must be that which gives other things their beauty. |
| 6956 | At what point does an object become 'whole'? [Westaway] |
| Full Idea: At what point does an object become 'whole'? | |
| From: Luke Westaway (talk [2005]), quoted by PG - Db (ideas) | |
| A reaction: This nice question strikes me as the central one in mereology. It is tempting to reply that the matter is entirely conventional, but there seems an obvious fact about something missing if one piece is absent from a jigsaw, or a cube is chipped. |
| 7335 | The Chinese Room should be able to ask itself questions in Mandarin [Westaway] |
| Full Idea: If the Chinese Room is functionally equivalent to a Mandarin speaker, it ought to be able to ask itself questions in Mandarin (and it can't). | |
| From: Luke Westaway (talk [2005]), quoted by PG - Db (ideas) | |
| A reaction: Searle might triumphantly say that this proves there is no understanding in the room, but the objection won't go away, because the room is presumably functionally equivalent to a speaker, and not just a mere translator (who might use mechanical tricks). |
| 7293 | It is legitimate to do harm if it is the unintended side-effect of an effort to achieve a good [Grayling] |
| Full Idea: The doctrine of double effect says that it is legitimate to do harm if the harm is the unintended side-effect of an effort to achieve a legitimate goal. | |
| From: A.C. Grayling (Among the Dead Cities [2006], Ch.6) | |
| A reaction: I think a key principle of morality is our duty to think about possible unnoticed consequences of our actions. To neglect concern for side-effects is wicked. Beyond that, the issue must concern the particulars of the situation. |
| 7292 | War must also have a good chance of success, and be waged with moderation [Grayling] |
| Full Idea: To Aquinas's three conditions for war (Idea 7291) modern theorists have added two others: that to be just a war must have a reasonable chance of success, and that the means used to conduct it must be proportional to the ends sought. | |
| From: A.C. Grayling (Among the Dead Cities [2006], Ch.6) | |
| A reaction: These two principles strike me as being much more civilized and humane than Aquinas's original contribution, suggesting that in our theoretical thinking we might be making some progress. |