16 ideas
| 15163 | The interest of quantified modal logic is its metaphysical necessity and essentialism [Soames] |
| Full Idea: The chief philosophical interest in quantified modal logic lies with metaphysical necessity, essentialism, and the nontrivial modal de re. | |
| From: Scott Soames (Philosophy of Language [2010], 3.1) |
| 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) |
| 18189 | ZFC could contain a contradiction, and it can never prove its own consistency [MacLane] |
| Full Idea: We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC. | |
| From: Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics | |
| A reaction: Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30). |
| 15158 | Indefinite descriptions are quantificational in subject position, but not in predicate position [Soames] |
| Full Idea: The indefinite description in 'A man will meet you' is naturally treated as quantificational, but an occurrence in predicative position, in 'Jones is not a philosopher', doesn't have a natural quantificational counterpart. | |
| From: Scott Soames (Philosophy of Language [2010], 1.23) |
| 15157 | Recognising the definite description 'the man' as a quantifier phrase, not a singular term, is a real insight [Soames] |
| Full Idea: Recognising the definite description 'the man' as a quantifier phrase, rather than a singular term, is a real insight. | |
| From: Scott Soames (Philosophy of Language [2010], 1.22) | |
| A reaction: 'Would the man who threw the stone come forward' seems like a different usage from 'would the man in the black hat come forward'. |
| 15156 | The universal and existential quantifiers were chosen to suit mathematics [Soames] |
| Full Idea: Since Frege and Russell were mainly interested in formalizing mathematics, the only quantifiers they needed were the universal and existential one. | |
| From: Scott Soames (Philosophy of Language [2010], 1.22) |
| 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) |
| 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) |
| 15161 | There are more metaphysically than logically necessary truths [Soames] |
| Full Idea: The set of metaphysically necessary truths is larger than the set of logically necessary truths. | |
| From: Scott Soames (Philosophy of Language [2010], 3.1) | |
| A reaction: Likewise, the set of logically possible truths is much larger than the set of metaphysically possible truths. If a truth is logically necessary, it will clearly be metaphysically necessary. Er, unless it is necessitated by daft logic... |
| 15162 | We understand metaphysical necessity intuitively, from ordinary life [Soames] |
| Full Idea: Our understanding of metaphysical necessity is intuitive - drawn from our ordinary thought and talk. | |
| From: Scott Soames (Philosophy of Language [2010], 3.1) | |
| A reaction: This, of course, is a good reason for analytic philosophers to dislike metaphysical necessity. |
| 15152 | To study meaning, study truth conditions, on the basis of syntax, and representation by the parts [Soames] |
| Full Idea: The systematic study of meaning requires a framework for specifying the truth conditions of sentences on the basis of their syntactic structure, and the representational contents of their parts. | |
| From: Scott Soames (Philosophy of Language [2010], Intro) | |
| A reaction: Soames presents this as common sense, on the first page of his book, and it is hard to disagree. Representation will shade off into studying the workings of the mind. Fodor seems a good person to start with. |
| 15153 | Tarski's account of truth-conditions is too weak to determine meanings [Soames] |
| Full Idea: The truth conditions provided by Tarski's theories (based on references of subsentential constituents) are too weak to determine meanings, because they lacked context-sensitivity and various forms of intensionality. | |
| From: Scott Soames (Philosophy of Language [2010], Intro) | |
| A reaction: Interesting. This suggests that stronger modern axiomatic theories of truth might give a sufficient basis for a truth conditions theory of meaning. Soames says possible worlds semantics was an attempt to improve things. |
| 15154 | We should use cognitive states to explain representational propositions, not vice versa [Soames] |
| Full Idea: Instead of explaining the representationality of sentences and cognitive states in terms of propositions, we must explain the representationality of propositions in terms of the representationality of the relevant cognitive states. | |
| From: Scott Soames (Philosophy of Language [2010], Intro) | |
| A reaction: Music to my ears. I am bewildered by this Russellian notion of a 'proposition' as some abstract entity floating around in the world waiting to be expressed. The vaguer word 'facts' (and false facts?) will cover that. It's Frege's fault. |