16 ideas
| 10751 | Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg] |
| Full Idea: Second-order logic raises doubts because of its ontological commitment to the set-theoretic hierarchy, and the allegedly problematic epistemic status of the second-order consequence relation. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1) | |
| A reaction: The 'epistemic' problem is whether you can know the truths, given that the logic is incomplete, and so they cannot all be proved. Rossberg defends second-order logic against the second problem. A third problem is that it may be mathematics. |
| 10757 | Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg] |
| Full Idea: Henkin semantics (for second-order logic) specifies a second domain of predicates and relations for the upper case constants and variables. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: This second domain is restricted to predicates and relations which are actually instantiated in the model. Second-order logic is complete with this semantics. Cf. Idea 10756. |
| 10759 | There are at least seven possible systems of semantics for second-order logic [Rossberg] |
| Full Idea: In addition to standard and Henkin semantics for second-order logic, one might also employ substitutional or game-theoretical or topological semantics, or Boolos's plural interpretation, or even a semantics inspired by Lesniewski. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: This is helpful in seeing the full picture of what is going on in these logical systems. |
| 10753 | Logical consequence is intuitively semantic, and captured by model theory [Rossberg] |
| Full Idea: Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
| A reaction: If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation. |
| 10752 | Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg] |
| Full Idea: Deductive consequence, written Γ|-S, is loosely read as 'the sentence S can be deduced from the sentences Γ', and semantic consequence Γ|=S says 'all models that make Γ true make S true as well'. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
| A reaction: We might read |= as 'true in the same model as'. What is the relation, though, between the LHS and the RHS? They seem to be mutually related to some model, but not directly to one another. |
| 10754 | In proof-theory, logical form is shown by the logical constants [Rossberg] |
| Full Idea: A proof-theorist could insist that the logical form of a sentence is exhibited by the logical constants that it contains. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
| A reaction: You have to first get to the formal logical constants, rather than the natural language ones. E.g. what is the truth table for 'but'? There is also the matter of the quantifiers and the domain, and distinguishing real objects and predicates from bogus. |
| 10756 | A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg] |
| Full Idea: A standard model is a set of objects called the 'domain', and an interpretation function, assigning objects in the domain to names, subsets to predicate letters, subsets of the Cartesian product of the domain with itself to binary relation symbols etc. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: The model actually specifies which objects have which predicates, and which objects are in which relations. Tarski's account of truth in terms of 'satisfaction' seems to be just a description of those pre-decided facts. |
| 10758 | If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg] |
| Full Idea: A mathematical theory is 'categorical' if, and only if, all of its models are isomorphic. Such a theory then essentially has just one model, the standard one. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: So the term 'categorical' is gradually replacing the much-used phrase 'up to isomorphism'. |
| 10761 | Completeness can always be achieved by cunning model-design [Rossberg] |
| Full Idea: All that should be required to get a semantics relative to which a given deductive system is complete is a sufficiently cunning model-theorist. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §5) |
| 10755 | A deductive system is only incomplete with respect to a formal semantics [Rossberg] |
| Full Idea: No deductive system is semantically incomplete in and of itself; rather a deductive system is incomplete with respect to a specified formal semantics. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: This important point indicates that a system might be complete with one semantics and incomplete with another. E.g. second-order logic can be made complete by employing a 'Henkin semantics'. |
| 14348 | An 'antidote' allows a manifestation to begin, but then blocks it [Corry] |
| Full Idea: An 'antidote' (or 'mask') to a disposition (unlike a 'finkish' one) leaves the disposition intact, but interferes with the causal chain between the disposition and its manifestation so that the manifestation doesn't come about. | |
| From: Richard Corry (Dispositional Essentialism Grounds Laws of Nature? [2010], 2) | |
| A reaction: [He cites Bird 1997] Thus the disposition of the poison at least begins to manifest, but its disposition to kill is blocked. So what was the disposition of the poison? |
| 14347 | A 'finkish' disposition is one that is lost immediately after the appropriate stimulus [Corry] |
| Full Idea: An object's disposition is said to be 'finkish' if the object loses the disposition after the occurrence of the appropriate stimulus, but before the manifestation has had time to come about. | |
| From: Richard Corry (Dispositional Essentialism Grounds Laws of Nature? [2010], 2) | |
| A reaction: [He cites Lewis 1997] An example would be some sort of safety device which only cuts in if the disposition seems about to operate (e.g. turns off electricity). It seems to block analyses of dispositions simply in terms of their outcomes. |
| 14350 | If a disposition is never instantiated, it shouldn't be part of our theory of nature [Corry] |
| Full Idea: If we have no good reason to believe that a disposition is instantiated, then the disposition should play no role in our theorizing about the world. | |
| From: Richard Corry (Dispositional Essentialism Grounds Laws of Nature? [2010], 3) | |
| A reaction: It is part of our theory that a substantial lump of uranium will explode, but also that a galaxy-sized lump of uranium would explode. Surely we are committed to the latter, even though it never happens? |
| 14351 | Maybe an experiment unmasks an essential disposition, and reveals its regularities [Corry] |
| Full Idea: The dispositional essentialist can argue that what happens in laboratory conditions is that, by controlling external influences, we effectively 'unmask' the relevant dispositions, and thus observe the regularities to which those dispositions give rise. | |
| From: Richard Corry (Dispositional Essentialism Grounds Laws of Nature? [2010], 5) | |
| A reaction: That seems to me to be exactly right, though Corry dislikes it, and even suggests that dispositional essentialist might not like it. |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 14346 | Dispositional essentialism says fundamental laws of nature are strict, not ceteris paribus [Corry] |
| Full Idea: Dispositional essentialism implies that the fundamental laws of nature must be strict, not ceteris paribus. | |
| From: Richard Corry (Dispositional Essentialism Grounds Laws of Nature? [2010], 1) | |
| A reaction: I am not keen on the 'laws' of nature, but since essentialism seems to make them necessary, you can't get stricter than that. |