13 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'. |
21125 | Liberals must respect family freedom - but families are the great oppressors of women [Nussbaum] |
Full Idea: A liberal society should give people considerable latitude to form families as they choose. …On the other hand the family …is one of the most notorious homes of sex hierarchy, denial of sexual opportunity, and sex-based violence and humiliation. | |
From: Martha Nussbaum (Rawls and Feminism [2003], 03), quoted by Andrew Shorten - Contemporary Political Theory | |
A reaction: The question of how the state might intervene in the family rarely seems to turn up in standard political theory. This idea shows why that is a mistake. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |