22 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. |
| 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. |
| 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'. |
| 2176 | There is only a problem of free will if you think the notion of 'voluntary' can be metaphysically deepened [Williams,B] |
| Full Idea: There is a problem of free will only for those who think that the notion of voluntary can be metaphysically deepened. | |
| From: Bernard Williams (Shame and Necessity [1993], III - p.68) | |
| A reaction: Years later, I now see that his refers to a pet hate of mine in discussions of free will, which is the idea that a person can have something called 'ultimate' responsibility for an action (which is the 'deep' version of 'you did it'). |
| 2181 | It is an absurd Kantian idea that at the limit rationality and freedom coincide [Williams,B] |
| Full Idea: There is a deluded Kantian idea that at the limit rationality and freedom will totally coincide. | |
| From: Bernard Williams (Shame and Necessity [1993], VI - p.158) |
| 4317 | We judge weakness of will by an assessment after the event is concluded [Williams,B, by Cottingham] |
| Full Idea: Williams has shown that whether an action was weakness of will depends on an evaluation after the event, as in the question of whether Gauguin was right to abandon his family to pursue his art. | |
| From: report of Bernard Williams (Shame and Necessity [1993]) by John Cottingham - Reason, Emotions and Good Life p.1 | |
| A reaction: The 'Gauguin Problem' is that Gauguin's actions only become weakness of will if the pictures are no good, and he can't know that till he's painted them. Good point. |
| 2174 | Responsibility involves cause, intention, state of mind, and response after the event [Williams,B] |
| Full Idea: The four elements of any conception of responsibility are cause, intention, state of mind, and response after the event. | |
| From: Bernard Williams (Shame and Necessity [1993], III - p.53) |
| 2178 | In bad actions, guilt points towards victims, and shame to the agent [Williams,B] |
| Full Idea: In what I have done, the guilt points in one direction towards what has happened to others, and the shame in another direction to what I am. | |
| From: Bernard Williams (Shame and Necessity [1993], IV - p.92) | |
| A reaction: Not convinced. I think shame has the fear of being observed as an inescapable component. Even when alone shame involves imagining what others might think. |
| 2169 | Greek moral progress came when 'virtue' was freed from social status [Williams,B] |
| Full Idea: There was moral progress in the ancient Greek world, notably to the extent that the idea of areté, human excellence, was freed to some extent from determination by social position. | |
| From: Bernard Williams (Shame and Necessity [1993], I - p.7) |
| 2172 | The modern idea of duty is unknown in archaic Greece [Williams,B] |
| Full Idea: Duty in some abstract modern sense is largely unknown to the Greeks, in particular to archaic Greeks. | |
| From: Bernard Williams (Shame and Necessity [1993], II - p.41) |
| 2179 | If the moral self is seen as characterless, then other people have a very limited role in our moral lives [Williams,B] |
| Full Idea: The conception of the moral self as characterless leaves only a limited positive role to other people in one's moral life. | |
| From: Bernard Williams (Shame and Necessity [1993], IV - p.95) |
| 2180 | If reason cannot lead people to good, we must hope they have an internal voice [Williams,B] |
| Full Idea: If we think the power of reason is not enough by itself to distinguish good and bad, then we would hope that people have limited autonomy, that there is an internalised other in them that carries some social weight. | |
| From: Bernard Williams (Shame and Necessity [1993], IV - p.100) |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |
| 2175 | There is a problem of evil only if you expect the world to be good [Williams,B] |
| Full Idea: There is a "problem of evil" only for those who expect the world to be good. | |
| From: Bernard Williams (Shame and Necessity [1993], III - p.68) |