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10987 | Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism' |
Full Idea: Three traditional names for rules are 'Simplification' (P from 'P and Q'), 'Addition' ('P or Q' from P), and 'Disjunctive Syllogism' (Q from 'P or Q' and 'not-P'). | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
11004 | Necessity is provability in S4, and true in all worlds in S5 |
Full Idea: In S4 necessity is said to be informal 'provability', and in S5 it is said to be 'true in every possible world'. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.4) | |||
A reaction: It seems that the S4 version is proof-theoretic, and the S5 version is semantic. |
11018 | There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers |
Full Idea: In fuzzy logic, besides fuzzy predicates, which define fuzzy sets, there are also fuzzy quantifiers (such as 'most' and 'few') and fuzzy modifiers (such as 'usually'). | |||
From: Stephen Read (Thinking About Logic [1995], Ch.7) |
11011 | Same say there are positive, negative and neuter free logics |
Full Idea: It is normal to classify free logics into three sorts; positive free logics (some propositions with empty terms are true), negative free logics (they are false), and neuter free logics (they lack truth-value), though I find this unhelpful and superficial. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.5) |
11020 | Realisms like the full Comprehension Principle, that all good concepts determine sets |
Full Idea: Hard-headed realism tends to embrace the full Comprehension Principle, that every well-defined concept determines a set. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.8) | |||
A reaction: This sort of thing gets you into trouble with Russell's paradox (though that is presumably meant to be excluded somehow by 'well-defined'). There are lots of diluted Comprehension Principles. |
10986 | Not all validity is captured in first-order logic |
Full Idea: We must recognise that first-order classical logic is inadequate to describe all valid consequences, that is, all cases in which it is impossible for the premisses to be true and the conclusion false. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: This is despite the fact that first-order logic is 'complete', in the sense that its own truths are all provable. |
10972 | The non-emptiness of the domain is characteristic of classical logic |
Full Idea: The non-emptiness of the domain is characteristic of classical logic. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
11024 | Semantics must precede proof in higher-order logics, since they are incomplete |
Full Idea: For the realist, study of semantic structures comes before study of proofs. In higher-order logic is has to, for the logics are incomplete. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.9) | |||
A reaction: This seems to be an important general observation about any incomplete system, such as Peano arithmetic. You may dream the old rationalist dream of starting from the beginning and proving everything, but you can't. Start with truth and meaning. |
10985 | We should exclude second-order logic, precisely because it captures arithmetic |
Full Idea: Those who believe mathematics goes beyond logic use that fact to argue that classical logic is right to exclude second-order logic. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
10970 | A theory of logical consequence is a conceptual analysis, and a set of validity techniques |
Full Idea: A theory of logical consequence, while requiring a conceptual analysis of consequence, also searches for a set of techniques to determine the validity of particular arguments. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
10984 | Logical consequence isn't just a matter of form; it depends on connections like round-square |
Full Idea: If classical logic insists that logical consequence is just a matter of the form, we fail to include as valid consequences those inferences whose correctness depends on the connections between non-logical terms (such as 'round' and 'square'). | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: He suggests that an inference such as 'round, so not square' should be labelled as 'materially valid'. |
10973 | A theory is logically closed, which means infinite premisses |
Full Idea: A 'theory' is any logically closed set of propositions, ..and since any proposition has infinitely many consequences, including all the logical truths, so that theories have infinitely many premisses. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: Read is introducing this as the essential preliminary to an account of the Compactness Theorem, which relates these infinite premisses to the finite. |
11007 | Quantifiers are second-order predicates |
Full Idea: Quantifiers are second-order predicates. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.5) | |||
A reaction: [He calls this 'Frege's insight'] They seem to be second-order in Tarski's sense, that they are part of a metalanguage about the sentence, rather than being a part of the sentence. |
10978 | In second-order logic the higher-order variables range over all the properties of the objects |
Full Idea: The defining factor of second-order logic is that, while the domain of its individual variables may be arbitrary, the range of the first-order variables is all the properties of the objects in its domain (or, thinking extensionally, of the sets objects). | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: The key point is that the domain is 'all' of the properties. How many properties does an object have. You need to decide whether you believe in sparse or abundant properties (I vote for very sparse indeed). |
10971 | A logical truth is the conclusion of a valid inference with no premisses |
Full Idea: Logical truth is a degenerate, or extreme, case of consequence. A logical truth is the conclusion of a valid inference with no premisses, or a proposition in the premisses of an argument which is unnecessary or may be suppressed. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
10988 | Any first-order theory of sets is inadequate |
Full Idea: Any first-order theory of sets is inadequate because of the Löwenheim-Skolem-Tarski property, and the consequent Skolem paradox. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: The limitation is in giving an account of infinities. |
10974 | Compactness is when any consequence of infinite propositions is the consequence of a finite subset |
Full Idea: Classical logical consequence is compact, which means that any consequence of an infinite set of propositions (such as a theory) is a consequence of some finite subset of them. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
10975 | Compactness does not deny that an inference can have infinitely many premisses |
Full Idea: Compactness does not deny that an inference can have infinitely many premisses. It can; but classically, it is valid if and only if the conclusion follows from a finite subset of them. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
10977 | Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite) |
Full Idea: Compact consequence undergenerates - there are intuitively valid consequences which it marks as invalid, such as the ω-rule, that if A holds of the natural numbers, then 'for every n, A(n)', but the proof of that would be infinite, for each number. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
10976 | Compactness makes consequence manageable, but restricts expressive power |
Full Idea: Compactness is a virtue - it makes the consequence relation more manageable; but it is also a limitation - it limits the expressive power of the logic. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: The major limitation is that wholly infinite proofs are not permitted, as in Idea 10977. |
11014 | Self-reference paradoxes seem to arise only when falsity is involved |
Full Idea: It cannot be self-reference alone that is at fault. Rather, what seems to cause the problems in the paradoxes is the combination of self-reference with falsity. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.6) |
11025 | Infinite cuts and successors seems to suggest an actual infinity there waiting for us |
Full Idea: Every potential infinity seems to suggest an actual infinity - e.g. generating successors suggests they are really all there already; cutting the line suggests that the point where the cut is made is already in place. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.8) | |||
A reaction: Finding a new gambit in chess suggests it was there waiting for us, but we obviously invented chess. Daft. |
10979 | Although second-order arithmetic is incomplete, it can fully model normal arithmetic |
Full Idea: Second-order arithmetic is categorical - indeed, there is a single formula of second-order logic whose only model is the standard model ω, consisting of just the natural numbers, with all of arithmetic following. It is nevertheless incomplete. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: This is the main reason why second-order logic has a big fan club, despite the logic being incomplete (as well as the arithmetic). |
10980 | Second-order arithmetic covers all properties, ensuring categoricity |
Full Idea: Second-order arithmetic can rule out the non-standard models (with non-standard numbers). Its induction axiom crucially refers to 'any' property, which gives the needed categoricity for the models. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) |
10997 | Von Neumann numbers are helpful, but don't correctly describe numbers |
Full Idea: The Von Neumann numbers have a structural isomorphism to the natural numbers - each number is the set of all its predecessors, so 2 is the set of 0 and 1. This helps proofs, but is unacceptable. 2 is not a set with two members, or a member of 3. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.4) |
11016 | Would a language without vagueness be usable at all? |
Full Idea: We must ask whether a language without vagueness would be usable at all. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.7) | |||
A reaction: Popper makes a similar remark somewhere, with which I heartily agreed. This is the idea of 'spreading the word' over the world, which seems the right way of understanding it. |
11013 | Identities and the Indiscernibility of Identicals don't work with supervaluations |
Full Idea: In supervaluations, the Law of Identity has no value for empty names, and remains so if extended. The Indiscernibility of Identicals also fails if extending it for non-denoting terms, where Fa comes out true and Fb false. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.5) |
11012 | A 'supervaluation' gives a proposition consistent truth-value for classical assignments |
Full Idea: A 'supervaluation' says a proposition is true if it is true in all classical extensions of the original partial valuation. Thus 'A or not-A' has no valuation for an empty name, but if 'extended' to make A true or not-true, not-A always has opposite value. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.5) |
11019 | Supervaluations say there is a cut-off somewhere, but at no particular place |
Full Idea: The supervaluation approach to vagueness is to construe vague predicates not as ones with fuzzy borderlines and no cut-off, but as having a cut-off somewhere, but in no particular place. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.7) | |||
A reaction: Presumably you narrow down the gap by supervaluation, then split the difference to get a definite value. |
10995 | A haecceity is a set of individual properties, essential to each thing |
Full Idea: The haecceitist (a neologism coined by Duns Scotus, pronounced 'hex-ee-it-ist', meaning literally 'thisness') believes that each thing has an individual essence, a set of properties which are essential to it. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.4) | |||
A reaction: This seems to be a difference of opinion over whether a haecceity is a set of essential properties, or a bare particular. The key point is that it is unique to each entity. |
11001 | Equating necessity with truth in every possible world is the S5 conception of necessity |
Full Idea: The equation of 'necessity' with 'true in every possible world' is known as the S5 conception, corresponding to the strongest of C.I.Lewis's five modal systems. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.4) | |||
A reaction: Are the worlds naturally, or metaphysically, or logically possible? |
10989 | The standard view of conditionals is that they are truth-functional |
Full Idea: The standard view of conditionals is that they are truth-functional, that is, that their truth-values are determined by the truth-values of their constituents. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.3) |
10992 | The point of conditionals is to show that one will accept modus ponens |
Full Idea: The point of conditionals is to show that one will accept modus ponens. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.3) | |||
A reaction: [He attributes this idea to Frank Jackson] This makes the point, against Grice, that the implication of conditionals is not conversational but a matter of logical convention. See Idea 21396 for a very different view. |
11017 | Some people even claim that conditionals do not express propositions |
Full Idea: Some people even claim that conditionals do not express propositions. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.7) | |||
A reaction: See Idea 14283, where this appears to have been 'proved' by Lewis, and is not just a view held by some people. |
10983 | Knowledge of possible worlds is not causal, but is an ontology entailed by semantics |
Full Idea: The modal Platonist denies that knowledge always depends on a causal relation. The reality of possible worlds is an ontological requirement, to secure the truth-values of modal propositions. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: [Reply to Idea 10982] This seems to be a case of deriving your metaphyics from your semantics, of which David Lewis seems to be guilty, and which strikes me as misguided. |
10982 | How can modal Platonists know the truth of a modal proposition? |
Full Idea: If modal Platonism was true, how could we ever know the truth of a modal proposition? | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: I take this to be very important. Our knowledge of modal truths must depend on our knowledge of the actual world. The best answer seems to involve reference to the 'powers' of the actual world. A reply is in Idea 10983. |
10996 | Actualism is reductionist (to parts of actuality), or moderate realist (accepting real abstractions) |
Full Idea: There are two main forms of actualism: reductionism, which seeks to construct possible worlds out of some more mundane material; and moderate realism, in which the actual concrete world is contrasted with abstract, but none the less real, possible worlds. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.4) | |||
A reaction: I am a reductionist, as I do not take abstractions to be 'real' (precisely because they have been 'abstracted' from the things that are real). I think I will call myself a 'scientific modalist' - we build worlds from possibilities, discovered by science. |
10981 | A possible world is a determination of the truth-values of all propositions of a domain |
Full Idea: A possible world is a complete determination of the truth-values of all propositions over a certain domain. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.2) | |||
A reaction: Even if the domain is very small? Even if the world fitted the logic nicely, but was naturally impossible? |
11000 | If worlds are concrete, objects can't be present in more than one, and can only have counterparts |
Full Idea: If each possible world constitutes a concrete reality, then no object can be present in more than one world - objects may have 'counterparts', but cannot be identical with them. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.4) | |||
A reaction: This explains clearly why in Lewis's modal realist scheme he needs counterparts instead of rigid designation. Sounds like a slippery slope. If you say 'Humphrey might have won the election', who are you talking about? |
10998 | The mind abstracts ways things might be, which are nonetheless real |
Full Idea: Ways things might be are real, but only when abstracted from the actual way things are. They are brought out and distinguished by the mind, by abstraction, but are not dependent on mind for their existence. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.4) | |||
A reaction: To me this just flatly contradicts itself. The idea that the mind can 'bring something out' by its operations, with the result being then accepted as part of reality is nonsense on stilts. What is real is the powers that make the possibilities. |
11005 | Negative existentials with compositionality make the whole sentence meaningless |
Full Idea: A problem with compositionality is negative existential propositions. If some of the terms of the proposition are empty, and don't refer, then compositionality implies that the whole will lack meaning too. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.5) | |||
A reaction: I don't agree. I don't see why compositionality implies holism about sentence-meaning. If I say 'that circular square is a psychopath', you understand the predication, despite being puzzled by the singular term. |
10966 | A proposition objectifies what a sentence says, as indicative, with secure references |
Full Idea: A proposition makes an object out of what is said or expressed by the utterance of a certain sort of sentence, namely, one in the indicative mood which makes sense and doesn't fail in its references. It can then be an object of thought and belief. | |||
From: Stephen Read (Thinking About Logic [1995], Ch.1) | |||
A reaction: Nice, but two objections: I take it to be crucial to propositions that they eliminate ambiguities, and I take it that animals are capable of forming propositions. Read seems to regard them as fictions, but I take them to be brain events. |