18 ideas
| 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. |
| 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. |
| 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'. |
| 17488 | Empiricist theories are sets of laws, which give explanations and reductions [Glennan] |
| Full Idea: In the empiricist tradition theories were understood to be deductive closures of sets of laws, explanations were understood as arguments from covering laws, and reduction was understood as a deductive relationship between laws of different theories. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: A lovely crisp summary of the whole tradition of philosophy of science from Comte through to Hempel. Mechanism and essentialism are the new players in the game. |
| 17493 | Modern mechanism need parts with spatial, temporal and function facts, and diagrams [Glennan] |
| Full Idea: Modern champions of mechanisms say models should identify both the parts and their spatial, temporal and functional organisation, ...and the practical importance of diagrams in addition to or in place of linguistic representations of mechanisms. | |
| From: Stuart Glennan (Mechanisms [2008], 'Discover') | |
| A reaction: Apparently chemists obtain much more refined models by using mathematics than they did by diagrams or 3D models (let alone verbal descriptions). For that reason, I'm thinking that 'model' might be a better term than 'mechanism'. |
| 17487 | Mechanistic philosophy of science is an alternative to the empiricist law-based tradition [Glennan] |
| Full Idea: To a significant degree, a mechanistic philosophy of science can be seen as an alternative to an earlier logical empiricist tradition in philosophy of science that gave pride of place to laws of nature. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: Lovely! Someone who actually spells out what's going on here. Most philosophers are far too coy about explaining what their real game is. Mechanism is fine in chemistry and biology. How about in 'mathematical' physics, or sociology? |
| 17489 | Mechanisms are either systems of parts or sequences of activities [Glennan] |
| Full Idea: There are two sorts of mechanisms: systems consist of collections of parts that interact to produce some behaviour, and processes are sequences of activities which produce some outcome. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: [compressed] The second one is important because it is more generic, and under that account all kinds the features of the world that need to be explained can be subsumed. E.g. hyperinflation in an economy is a 'mechanism'. |
| 17490 | 17th century mechanists explained everything by the kinetic physical fundamentals [Glennan] |
| Full Idea: 17th century mechanists said that interactions governed by chemical, electrical or gravitational forces would have to be explicable in terms of the operation of some atomistic (or corpuscular) kinetic mechanism. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: Glennan says science has rejected this, so modern mechanists do not reduce mechanisms to anything in particular. |
| 17491 | Unlike the lawlike approach, mechanistic explanation can allow for exceptions [Glennan] |
| Full Idea: One of the advantages of the move from nomological to mechanistic modes of explanation is that the latter allows for explanations involving exception-ridden generalizations. | |
| From: Stuart Glennan (Mechanisms [2008], 'regular') | |
| A reaction: The lawlike approach has endless problems with 'ceteris paribus' ('all things being equal') laws, where specifying all the other 'things' seems a bit tricky. |
| 19941 | Thou shalt love thy neighbour as thyself [Anon (Leviticus)] |
| Full Idea: Thou shalt love thy neighbour as thyself. | |
| From: Anon (Lev) (03: Book of Leviticus [c.700 BCE], 19.18) | |
| A reaction: Most Christians think Jesus originated this thought. Interestingly, this precedes Socrates, who taught a similar idea. |
| 17494 | Since causal events are related by mechanisms, causation can be analysed in that way [Glennan] |
| Full Idea: Causation can be analyzed in terms of mechanisms because (except for fundamental causal interactions) causally related events will be connected by intervening mechanisms. | |
| From: Stuart Glennan (Mechanisms [2008], 'causation') | |
| A reaction: This won't give us the metaphysics of causation (which concerns the fundamentals), but this strikes me as a very coherent and interesting proposal. He mentions electron interaction as non-mechanistic causation. |