23 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. |
| 17818 | How many? must first partition an aggregate into sets, and then logic fixes its number [Yourgrau] |
| Full Idea: We want to know How many what? You must first partition an aggregate into parts relevant to the question, where no partition is privileged. How the partitioned set is to be numbered is bound up with its unique members, and follows from logic alone. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'New Problem') | |
| A reaction: [Compressed wording of Yourgrau's summary of Frege's 'relativity argument'] Concepts do the partitioning. Yourgau says this fails, because the same argument applies to the sets themselves, as well as to the original aggregates. |
| 17822 | Nothing is 'intrinsically' numbered [Yourgrau] |
| Full Idea: Nothing at all is 'intrinsically' numbered. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'What the') | |
| A reaction: Once you are faced with distinct 'objects' of some sort, they can play the role of 'unit' in counting, so his challenge is that nothing is 'intrinsically' an object, which is the nihilism explored by Unger, Van Inwagen and Merricks. Aristotle disagrees... |
| 17817 | Defining 'three' as the principle of collection or property of threes explains set theory definitions [Yourgrau] |
| Full Idea: The Frege-Maddy definition of number (as the 'property' of being-three) explains why the definitions of Von Neumann, Zermelo and others work, by giving the 'principle of collection' that ties together all threes. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'A Fregean') | |
| A reaction: [compressed two or three sentences] I am strongly in favour of the best definition being the one which explains the target, rather than just pinning it down. I take this to be Aristotle's view. |
| 17815 | We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau] |
| Full Idea: Sets could hardly serve as a foundation for number theory if we had to await detailed results in the upper reaches of the edifice before we could make our first move. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'Two') |
| 17821 | You can ask all sorts of numerical questions about any one given set [Yourgrau] |
| Full Idea: We can address a set with any question at all that admits of a numerical reply. Thus we can ask of {Carter, Reagan} 'How many feet do the members have?'. | |
| From: Palle Yourgrau (Sets, Aggregates and Numbers [1985], 'On Numbering') | |
| A reaction: This is his objection to the Fregean idea that once you have fixed the members of a set, you have thereby fixed the unique number that belongs with the set. |
| 17499 | Theoretical models can represent, by mapping onto the data-models [Portides] |
| Full Idea: The semantic approach contends that theoretical models ...are candidates for representing physical systems by virtue of the fact that they stand in mapping relations to corresponding data-models. | |
| From: Demetris Portides (Models [2008], 'Current') | |
| A reaction: Sounds like a neat and satisfying picture. |
| 17498 | In the 'received view' models are formal; the 'semantic view' emphasises representation [Portides, by PG] |
| Full Idea: The 'received view' of models is that they are Tarskian formal axiomatic calculi interpreted by meta-mathematical models. The 'semantic' view of models gives equal importance to their representational capacity. | |
| From: report of Demetris Portides (Models [2008], 'background') by PG - Db (ideas) | |
| A reaction: The Tarskian view is the one covered in my section on Model Theory. Portides favours the semantic account, and I am with him all the way. Should models primarily integrate with formal systems, or with the world? Your choice... |
| 17501 | Representational success in models depends on success of their explanations [Portides] |
| Full Idea: Models are representational, independently of the strength of their relation to theory, depending on how well they achieve the purpose of providing explanations for what occurs in physical systems. | |
| From: Demetris Portides (Models [2008], 'Current') | |
| A reaction: This doesn't sound quite right. It seems possible to have a perfect representation of a system which remains quite baffling (because too complex, or with obscure ingredients). Does the stylised London tube map explain well but represent badly? |
| 17502 | The best model of the atomic nucleus is the one which explains the most results [Portides] |
| Full Idea: The unified model can be considered a better representation of the atomic nucleus in comparison to the liquid-drop and shell models, because it explains most of the known results about the nucleus. | |
| From: Demetris Portides (Models [2008], 'Current') | |
| A reaction: The point here is that models are evaluated not just by their accuracy, but by their explanatory power. Presumably a great model is satisfying and illuminating. Do the best models capture the essence of a thing? |
| 17496 | 'Model' belongs in a family of concepts, with representation, idealisation and abstraction [Portides] |
| Full Idea: A better understanding of 'model', as used in science, could be achieved if we examine it as a member of the triad of concepts of representation, idealisation and abstraction. | |
| From: Demetris Portides (Models [2008], 'Intro') | |
| A reaction: Abstraction seems to have a bad name in philosophy, and yet when you come to discuss things like models, you can't express it any other way. |
| 17497 | Models are theory-driven, or phenomenological (more empirical and specific) [Portides] |
| Full Idea: 'Theory-driven' models are constructed in a systematic theory-regulated way by supplementing the theoretical calculus with locally operative hypotheses. 'Phenomenological' models deploy semi-empirical results, with ad hoc hypotheses, and extra concepts. | |
| From: Demetris Portides (Models [2008], 'Intro') | |
| A reaction: [compressed] I am not at all clear about this distinction, even after reading his whole article. The first type of model seems more general, while the second seems tuned to particular circumstances. He claims the second type is more explanatory. |
| 17500 | General theories may be too abstract to actually explain the mechanisms [Portides] |
| Full Idea: If theoretical models are highly abstract and idealised descriptions of phenomena, they may only represent general features, and fail to explain the specific mechanisms at work in physical systems. | |
| From: Demetris Portides (Models [2008], 'Current') | |
| A reaction: [compressed] While there may be an ideal theory that explains everything, it sounds right capturing the actual mechanism (such as the stirrup bone in the ear) is not at all theoretical. |