8472 | Sentential logic is consistent (no contradictions) and complete (entirely provable) |
Full Idea: Sentential logic has been proved consistent and complete; its consistency means that no contradictions can be derived, and its completeness assures us that every one of the logical truths can be proved. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.5) | |
A reaction: The situation for quantificational logic is not quite so clear (Orenstein p.98). I do not presume that being consistent and complete makes it necessarily better as a tool in the real world. |
8476 | Axiomatization simply picks from among the true sentences a few to play a special role |
Full Idea: In axiomatizing, we are merely sorting out among the truths of a science those which will play a special role, namely, serve as axioms from which we derive the others. The sentences are already true in a non-conventional or ordinary sense. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.5) | |
A reaction: If you were starting from scratch, as Euclidean geometers may have felt they were doing, you might want to decide which are the simplest truths. Axiomatizing an established system is a more advanced activity. |
8480 | S4: 'poss that poss that p' implies 'poss that p'; S5: 'poss that nec that p' implies 'nec that p' |
Full Idea: The five systems of propositional modal logic contain successively stronger conceptions of necessity. In S4 'it is poss that it is poss that p' implies 'it is poss that p'. In S5, 'it is poss that it is nec that p' implies 'it is nec that p'. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.7) | |
A reaction: C.I. Lewis originated this stuff. Any serious student of modality is probably going to have to pick a system. E.g. Nathan Salmon says that the correct modal logic is even weaker than S4. |
8474 | Unlike elementary logic, set theory is not complete |
Full Idea: The incompleteness of set theory contrasts sharply with the completeness of elementary logic. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.5) | |
A reaction: This seems to be Quine's reason for abandoning the Frege-Russell logicist programme (quite apart from the problems raised by Gödel. |
8465 | Mereology has been exploited by some nominalists to achieve the effects of set theory |
Full Idea: The theory of mereology has had a history of being exploited by nominalists to achieve some of the effects of set theory. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.3) | |
A reaction: Some writers refer to mereology as a 'theory', and others as an area of study. This appears to be an interesting line of investigation. Orenstein says Quine and Goodman showed its limitations. |
8452 | Traditionally, universal sentences had existential import, but were later treated as conditional claims |
Full Idea: In traditional logic from Aristotle to Kant, universal sentences have existential import, but Brentano and Boole construed them as universal conditionals (such as 'for anything, if it is a man, then it is mortal'). | |
From: Alex Orenstein (W.V. Quine [2002], Ch.2) | |
A reaction: I am sympathetic to the idea that even the 'existential' quantifier should be treated as conditional, or fictional. Modern Christians may well routinely quantify over angels, without actually being committed to them. |
8475 | The substitution view of quantification says a sentence is true when there is a substitution instance |
Full Idea: The substitution view of quantification explains 'there-is-an-x-such-that x is a man' as true when it has a true substitution instance, as in the case of 'Socrates is a man', so the quantifier can be read as 'it is sometimes true that'. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.5) | |
A reaction: The word 'true' crops up twice here. The alternative (existential-referential) view cites objects, so the substitution view is a more linguistic approach. |
8454 | The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc. |
Full Idea: The 'natural' numbers are the whole numbers 1, 2, 3 and so on. The 'rational' numbers consist of the natural numbers plus the fractions. The 'real' numbers include the others, plus numbers such a pi and root-2, which cannot be expressed as fractions. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.2) | |
A reaction: The 'irrational' numbers involved entities such as root-minus-1. Philosophical discussions in ontology tend to focus on the existence of the real numbers. |
8473 | The logicists held that is-a-member-of is a logical constant, making set theory part of logic |
Full Idea: The question to be posed is whether is-a-member-of should be considered a logical constant, that is, does logic include set theory. Frege, Russell and Whitehead held that it did. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.5) | |
A reaction: This is obviously the key element in the logicist programme. The objection seems to be that while first-order logic is consistent and complete, set theory is not at all like that, and so is part of a different world. |
8458 | Just individuals in Nominalism; add sets for Extensionalism; add properties, concepts etc for Intensionalism |
Full Idea: Modest ontologies are Nominalism (Goodman), admitting only concrete individuals; and Extensionalism (Quine/Davidson) which admits individuals and sets; but Intensionalists (Frege/Carnap/Church/Marcus/Kripke) may have propositions, properties, concepts. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.3) | |
A reaction: I don't like sets, because of Idea 7035. Even the ontology of individuals could collapse dramatically (see the ideas of Merricks, e.g. 6124). The intensional items may be real enough, but needn't have a place at the ontological high table. |
8457 | The Principle of Conservatism says we should violate the minimum number of background beliefs |
Full Idea: The principle of conservatism in choosing between theories is a maxim of minimal mutilation, stating that of competing theories, all other things being equal, choose the one that violates the fewest background beliefs held. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.2) | |
A reaction: In this sense, all rational people should be conservatives. The idea is a modern variant of Hume's objection to miracles (Idea 2227). A Kuhnian 'paradigm shift' is the dramatic moment when this principle no longer seems appropriate. |
8477 | People presume meanings exist because they confuse meaning and reference |
Full Idea: A good part of the confidence people have that there are meanings rests on the confusion of meaning and reference. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.6) | |
A reaction: An important point. Everyone assumes that sentences link to the world, but Frege shows that that is not part of meaning. Words like prepositions and conjunctions ('to', 'and') don't have 'a meaning' apart from their function and use. |
8471 | Three ways for 'Socrates is human' to be true are nominalist, platonist, or Montague's way |
Full Idea: 'Socrates is human' is true if 1) subject referent is identical with a predicate referent (Nominalism), 2) subject reference member of the predicate set, or the subject has that property (Platonism), 3) predicate set a member of the subject set (Montague) | |
From: Alex Orenstein (W.V. Quine [2002], Ch.3) | |
A reaction: Orenstein offers these as alternatives to Quine's 'inscrutability of reference' thesis, which makes the sense unanalysable. |
8484 | If two people believe the same proposition, this implies the existence of propositions |
Full Idea: If we can say 'there exists a p such that John believes p and Barbara believes p', logical forms such as this are cited as evidence for our ontological commitment to propositions. | |
From: Alex Orenstein (W.V. Quine [2002], Ch.7) | |
A reaction: Opponents of propositions (such as Quine) will, of course, attempt to revise the logical form to eliminate the quantification over propositions. See Orenstein's outline on p.171. |