21 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'. |
| 13267 | Temporal parts is a crazy doctrine, because it entails constantly creating stuff ex nihilo [Thomson, by Koslicki] |
| Full Idea: Thomson famously objects that the doctrine of temporal parts is 'a crazy metaphysic - obviously false', since it entails that material objects are constantly being generated ex nihilo (or, at least, the stuff of which they are composed is). | |
| From: report of Judith (Jarvis) Thomson (Parthood and Identity across Time [1983], p.210) by Kathrin Koslicki - The Structure of Objects 2.2 | |
| A reaction: The related objections are to ask what the temporal 'width' of a part is, and whether the joins are visible. |
| 16209 | How can point-duration slices of people have beliefs or desires? [Thomson] |
| Full Idea: Can one really think that point-duration temporal slices of bodies believe things or want things? | |
| From: Judith (Jarvis) Thomson (People and Their Bodies [1997], p.211), quoted by Katherine Hawley - How Things Persist 2.9 n21 | |
| A reaction: There is a problem with a slice doing anything long-term. The bottom line is that things are said to 'endure', but that is precisely what time-slices are unable to do. Hawley rejects this idea. |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 4058 | Is someone's right to life diminished if they were conceived by a rape? [Thomson] |
| Full Idea: Can we say that a person has a right to life only if they didn't come into existence through rape, or that the latter have less right to life? | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.126) | |
| A reaction: This would clearly be an inconsistency for some opponents of abortion who allow rape as an exception. |
| 4060 | The right to life does not bestow the right to use someone else's body to support that life [Thomson] |
| Full Idea: Having a right to life does not guarantee having either a right to be given the use of or a right to be allowed continued use of another person's body. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.131) | |
| A reaction: A very nice point. You have a right to your life once you are the sole owner of it. |
| 4062 | No one is morally required to make huge sacrifices to keep someone else alive for nine months [Thomson] |
| Full Idea: No one is morally required to make large sacrifices, of health, and other interests and commitments, for nine months, in order to keep another person alive. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.135) | |
| A reaction: It is a trade-off. It might become a duty if society (or even a husband) urgently needed the baby. |
| 4061 | The right to life is not a right not to be killed, but not to be killed unjustly [Thomson] |
| Full Idea: Maybe the right to life consists not in the right not to be killed, but in the right not to be killed unjustly. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.131) | |
| A reaction: Sounds tautological. There is no right to life, then, but just the requirement that people behave justly? |
| 4057 | A newly fertilized ovum is no more a person than an acorn is an oak tree [Thomson] |
| Full Idea: A newly fertilized ovum, a newly implanted clump of cells, is no more a person than an acorn is an oak tree. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.125) | |
| A reaction: This relies heavily on the philosopher's concept of a 'person', but it seems right to me. |
| 4695 | Maybe abortion can be justified despite the foetus having full human rights [Thomson, by Foot] |
| Full Idea: Thomson suggests that abortion can be justified without the need to deny that the foetus has the moral rights of a human person. | |
| From: report of Judith (Jarvis) Thomson (A Defense of Abortion [1971]) by Philippa Foot - Killing and Letting Die p.86 | |
| A reaction: Thomson uses a dubious analogy between pregnancy and being hooked up to someone for life-support. Presumably killing an innocent person is occasionally justifiable, but the situation would normally be more abnormal than pregnancy. |
| 4059 | It can't be murder for a mother to perform an abortion on herself to save her own life [Thomson] |
| Full Idea: It cannot seriously be thought to be murder if a mother performs an abortion on herself to save her own life (if, say, she had a serious heart condition). | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.127) | |
| A reaction: An extreme view might condemn such an action, but it can hardly be based on the 'sanctity of life'. |
| 4696 | The foetus is safe in the womb, so abortion initiates its death, with the mother as the agent. [Foot on Thomson] |
| Full Idea: A fetus is not in jeopardy because it is in the womb, so an abortion originates the fatal sequence, and the mother is the agent. Hence Thomson's argument is invalid, and we must return to question of the moral status of the foetus. | |
| From: comment on Judith (Jarvis) Thomson (A Defense of Abortion [1971]) by Philippa Foot - Killing and Letting Die p.86 | |
| A reaction: The problem would be if a 'person' was safe, but only if I continue some sustained effort which is not required of me by normal duties. |