25 ideas
| 18951 | For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam] |
| Full Idea: For a language L there is a predicate 'true-in-L' which one can employ for all scientific purposes in place of intuitive truth, and this predicate admits of a precise definition using only the vocabulary of L itself plus set theory. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.2) | |
| A reaction: He refers, of course, to Tarski's theory. I'm unclear of the division between 'scientific purposes' and the rest of life (which is why some people embrace 'minimal' theories of ordinary truth). I'm struck by set theory being a necessary feature. |
| 18953 | Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam] |
| Full Idea: In modern notation we can consider potential logical principles that Aristotle never considered because of his general practice of looking at inferences each of whose premises involved exactly two class-names. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
| A reaction: Presumably you can build up complex inferences from a pair of terms, just as you do with pairs in set theory. |
| 18949 | The universal syllogism is now expressed as the transitivity of subclasses [Putnam] |
| Full Idea: On its modern interpretation, the validity of the inference 'All S are M; All M are P; so All S are P' just expresses the transitivity of the relation 'subclass of'. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.1) | |
| A reaction: A simple point I've never quite grasped. Since lots of syllogisms can be expressed as Venn Diagrams, in which the circles are just sets, it's kind of obvious really. So why does Sommers go back to 'terms'? See 'Term Logic'. |
| 18952 | '⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam] |
| Full Idea: The symbol '⊃' (read 'if...then') is used with the definition 'Px ⊃ Qx' ('if Px then Qx') is short for '¬(Px & ¬Qx)'. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
| A reaction: So ⊃ and → are just abbreviations, and not really a proper part of the language. Notoriously, though, this is quite a long way from what 'if...then' means in ordinary English, and it leads to paradoxical oddities. |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 18958 | In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam] |
| Full Idea: In the theory of types, 'x ∈ y' is well defined only if x and y are of the appropriate type, where individuals count as the zero type, sets of individuals as type one, sets of sets of individuals as type two. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.6) |
| 18954 | Before the late 19th century logic was trivialised by not dealing with relations [Putnam] |
| Full Idea: It was essentially the failure to develop a logic of relations that trivialised the logic studied before the end of the nineteenth century. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
| A reaction: De Morgan, Peirce and Frege were, I believe, the people who put this right. |
| 18956 | Asserting first-order validity implicitly involves second-order reference to classes [Putnam] |
| Full Idea: The natural understanding of first-order logic is that in writing down first-order schemata we are implicitly asserting their validity, that is, making second-order assertions. ...Thus even quantification theory involves reference to classes. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
| A reaction: If, as a nominalist, you totally rejected classes, presumably you would get by in first-order logic somehow. To say 'there are no classes so there is no logical validity' sounds bonkers. |
| 18962 | Unfashionably, I think logic has an empirical foundation [Putnam] |
| Full Idea: Today, the tendency among philosophers is to assume that in no sense does logic itself have an empirical foundation. I believe this tendency is wrong. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.9) | |
| A reaction: I agree, not on the basis of indispensability to science, but on the basis of psychological processes that lead from experience to logic. Russell and Quine are Putnam's allies here, and Frege is his opponent. Putnam developed a quantum logic. |
| 18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |
| Full Idea: Instead of identifying functions with certain sets, I might have identified sets with certain functions. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.9) |
| 18955 | Having a valid form doesn't ensure truth, as it may be meaningless [Putnam] |
| Full Idea: I don't think all substitution-instances of a valid schema are 'true'; some are clearly meaningless, such as 'If all boojums are snarks and all snarks are egglehumphs, then all boojums are egglehumphs'. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
| A reaction: This seems like a very good challenge to Quine's claim that it is only form which produces a logical truth. Keep deductive and semantic consequence separate, with two different types of 'logical truth'. |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 18959 | Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam] |
| Full Idea: Sets of a very high type or very high cardinality (higher than the continuum, for example) should today be investigated in an 'if-then' spirit. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.7) | |
| A reaction: This attitude goes back to Hilbert, but it fits with Quine's view of what is indispensable for science. It is hard to see a reason for the cut-off, just looking at the logic of expanding sets. |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 18957 | Nominalism only makes sense if it is materialist [Putnam] |
| Full Idea: Nominalists must at heart be materialists, or so it seems to me: otherwise their scruples are unintelligible. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.5) | |
| A reaction: This is modern nominalism - the rejection of abstract objects. I largely plead guilty to both charges. |
| 18950 | Physics is full of non-physical entities, such as space-vectors [Putnam] |
| Full Idea: Physics is full of references to such 'non-physical' entities as state-vectors, Hamiltonians, Hilbert space etc. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.2) | |
| A reaction: I take these to be concepts which are 'abstracted' from the physical facts, and so they don't strike me as being much of an ontological problem, or an objection to nominalism (which Putnam takes them to be). |
| 18960 | Most predictions are uninteresting, and are only sought in order to confirm a theory [Putnam] |
| Full Idea: Scientists want successful predictions in order to confirm their theories; they do not want theories in order to obtain the predictions, which are in some cases of not the slightest interest in themselves. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.8) | |
| A reaction: Equally, we might only care about the prediction, and have no interest at all in the theory. Farmers want weather predictions, not a PhD in meteorology. |
| 15473 | How does anything get outside itself? [Fodor, by Martin,CB] |
| Full Idea: Fodor asks the stirring and basic question 'How does anything get outside itself?' | |
| From: report of Jerry A. Fodor (works [1986]) by C.B. Martin - The Mind in Nature 03.6 | |
| A reaction: Is this one of those misconceived questions, like major issues concerning 'what's it like to be?' In what sense am I outside myself? Is a mind any more mysterious than a shadow? |
| 2981 | Is intentionality outwardly folk psychology, inwardly mentalese? [Lyons on Fodor] |
| Full Idea: For Fodor the intentionality of the propositional-attitude vocabulary of our folk psychology is the outward expression of the inward intentionality of the language of the brain. | |
| From: comment on Jerry A. Fodor (works [1986]) by William Lyons - Approaches to Intentionality p.39 | |
| A reaction: I would be very cautious about this. Folk psychology works, so it must have a genuine basis in how brains work, but it breaks down in unusual situations, and might even be a total (successful) fiction. |
| 2985 | Are beliefs brains states, but picked out at a "higher level"? [Lyons on Fodor] |
| Full Idea: Fodor holds that beliefs are brain states or processes, but picked out at a 'higher' or 'special science' level. | |
| From: comment on Jerry A. Fodor (works [1986]) by William Lyons - Approaches to Intentionality p.82 | |
| A reaction: I don't think you can argue with this. Levels of physical description exist (e.g. pure physics tells you nothing about the weather), and I think 'process' is the best word for the mind (Idea 4931). |
| 3135 | Is thought a syntactic computation using representations? [Fodor, by Rey] |
| Full Idea: The modest mentalism of the Computational/Representational Theory of Thought (CRTT), associated with Fodor, says mental processes are computational, defined over syntactically specified entities, and these entities represent the world (are also semantic). | |
| From: report of Jerry A. Fodor (works [1986]) by Georges Rey - Contemporary Philosophy of Mind Int.3 | |
| A reaction: This seems to imply that if you built a machine that did all these things, it would become conscious, which sounds unlikely. Do footprints 'represent' feet, or does representation need prior consciousness? |
| 2983 | Maybe narrow content is physical, broad content less so [Lyons on Fodor] |
| Full Idea: Fodor is concerned with producing a realist and physicalist account of 'narrow content' (i.e. wholly in-the-head content). | |
| From: comment on Jerry A. Fodor (works [1986]) by William Lyons - Approaches to Intentionality p.54 | |
| A reaction: The emergence of 'wide' content has rather shaken Fodor's game plan. We can say "Oh dear, I thought I was referring to H2O", so there must be at least some narrow aspect to reference. |