15 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. |
| 20977 | Natural rights are nonsense, and unspecified natural rights is nonsense on stilts [Bentham] |
| Full Idea: Natural rights is simple nonsense: natural and imprescriptible rights, rhetorical nonsense — nonsense upon stilts. | |
| From: Jeremy Bentham (Anarchical Fallacies: on the Declaration of Rights [1796]) | |
| A reaction: If you want your opinion to be remembered, express it memorably! I take natural rights to be the basic principles and values which are obvious to almost everyone when they come for formulate legal rights (which are the only true rights). |
| 21003 | Only laws can produce real rights; rights from 'law of nature' are imaginary [Bentham] |
| Full Idea: Right, the substantive right, is the child of law; from real laws come real rights; but from imaginary laws, from 'law of nature' can come only imaginary rights. | |
| From: Jeremy Bentham (Anarchical Fallacies: on the Declaration of Rights [1796], II.523), quoted by Amartya Sen - The Idea of Justice 17 'Ethics' | |
| A reaction: I am coming to agree with this. What are called 'natural rights' are just self-evident good reasons why someone should be allowed a right. A right can, of course, come from an informal agreement. The question is: why award that particular legal right? |
| 12719 | Clearly, force is that from which action follows, when unimpeded [Leibniz] |
| Full Idea: The notion of force is as clear as that of action and passion, because it is that from which action follows when nothing prevents it. | |
| From: Gottfried Leibniz (Letters to Paul Pellison-Fontinier [1691], A1.6.226), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 4 | |
| A reaction: For Leibniz, force seems to be a metaphysical notion, rather than a feature of the physical world. I take it to be the bottom level of explanation, and it equates with Aristotelian form and essence. |
| 12720 | Time doesn't exist, since its parts don't coexist [Leibniz] |
| Full Idea: Time never exists, since all of its parts never exist together. | |
| From: Gottfried Leibniz (Letters to Paul Pellison-Fontinier [1691], A1.6.226), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 4 | |
| A reaction: The problem here is that he seems to be admitting that time has 'parts'. Can something have parts and not exist? Events will also fail to exist by this criterion, though we could hardly deny that events (or some such) 'happen'. |