20 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. |
17954 | Essence is a thing's necessities, but what about its possibilities (which may not be realised)? [Vetter] |
Full Idea: Essence is, as it were, necessity rooted in things, ...but how about possibility rooted in things? ...Having the potential to Φ, unlike being essentially Φ, does not entail being actually Φ. | |
From: Barbara Vetter (Essence and Potentiality [2010], §2) | |
A reaction: To me this invites the question 'what is it about the entity which endows it with its rooted possibilities?' A thing has possibilities because it has a certain nature (at a given time). |
17953 | Real definition fits abstracta, but not individual concrete objects like Socrates [Vetter] |
Full Idea: I can understand the notion of real definition as applying to (some) abstact entities, but I have no idea how to apply it to a concrete object such as Socrates or myself. | |
From: Barbara Vetter (Essence and Potentiality [2010], §1) | |
A reaction: She is objecting to Kit Fine's account of essence, which is meant to be clearer than the normal account of essences based on necessities. Aristotle implies that definitions get fuzzy when you reach the level of the individual. |
17952 | Modal accounts make essence less mysterious, by basing them on the clearer necessity [Vetter] |
Full Idea: The modal account was meant, I take it, to make the notion of essence less mysterious by basing it on the supposedly better understood notion of necessity. | |
From: Barbara Vetter (Essence and Potentiality [2010], §1) |
17959 | Metaphysical necessity is even more deeply empirical than Kripke has argued [Vetter] |
Full Idea: We support the views of metaphysical modality on which metaphysical necessity is an even more deeply empirical matter than Kripke has argued. | |
From: Barbara Vetter (Essence and Potentiality [2010], p.2) | |
A reaction: [co-author E. Viebahn] This seems to pinpoint the spirit of scientific essentialism. She cites Bird and Shoemaker. If it is empirical, doesn't that make it a matter of epistemology, and hence further from absolute necessity? |
17955 | Possible worlds allow us to talk about degrees of possibility [Vetter] |
Full Idea: The apparatus of possible worlds affords greater expressive power than mere talk of possibility and necessity. In particular, possible worlds talk allows us to introduce degrees of possibility. | |
From: Barbara Vetter (Essence and Potentiality [2010], §3) | |
A reaction: A nice feature, but I'm not sure that either the proportion of possible worlds or the closeness of possible worlds captures what we actually mean by a certain degree of possibility. There is 'accidental closeness', or absence of contingency. See Vetter. |
17957 | Maybe possibility is constituted by potentiality [Vetter] |
Full Idea: We should look at the claim that possibility is constituted by potentiality. | |
From: Barbara Vetter (Essence and Potentiality [2010], §4) | |
A reaction: A problem that comes to mind is possibilities arising from coincidence. The whole of reality must be described, to capture all the possibilities for a particular thing. So potentialities of what? Nice thought, though. |
17958 | The apparently metaphysically possible may only be epistemically possible [Vetter] |
Full Idea: Some of what metaphysicians take to be metaphysically possible turns out to be only epistemically possible. | |
From: Barbara Vetter (Essence and Potentiality [2010], §4) | |
A reaction: A nice clear expression of the increasingly common view that conceivability may be a limited way to grasp possibility. |
17956 | Closeness of worlds should be determined by the intrinsic nature of relevant objects [Vetter] |
Full Idea: The closeness of possible worlds should be determined by similarity in the intrinsic constitution of whatever object it is whose potentialities are at issue. | |
From: Barbara Vetter (Essence and Potentiality [2010], §3) | |
A reaction: Nice thought. This seems to be the essentialist approach to possible worlds, but it makes the natures of the objects more fundamental than the framework of the worlds. She demurs because there are also extrinsic potentialities. |
17009 | I won't object if someone shows that gravity consistently arises from the action of matter [Newton] |
Full Idea: If someone explains gravity along with all its laws by the action of some subtle matter, and shows that the motion of the planets and comets will not be disturbed by this matter, I shall be far from objecting. | |
From: Isaac Newton (Letters to Leibniz 1 [1693], 1693.10.16) | |
A reaction: Important if you think that Newton is the hero of the descriptive regularity theory of laws. Newton probably thought laws came from God, but he wouldn't object to Leibniz's view, that God planted the laws within the matter. |