20 ideas
| 21918 | Sufficient Reason can't be proved, because all proof presupposes it [Schopenhauer, by Lewis,PB] |
| Full Idea: Schopenhauer said the principle of sufficient reason is not susceptible to proof for the simple reason that it is presupposed in any argument or proof. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §14 p.32-3) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: I would have thought it might be disproved by a counterexample, such as the Gödel sentence of his incompleteness proof, or quantum effects which seem to elude causation. Personally I believe the principle, which I see as the first axiom of philosophy. |
| 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'. |
| 21920 | No need for a priori categories, since sufficient reason shows the interrelations [Schopenhauer, by Lewis,PB] |
| Full Idea: Schopenhauer dispenses with Kant's a priori categories, since all interrelations between representations are given through the principle of sufficient reason. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: I'm not sure how Schopenhauer manages this move. Is it the stoic idea that reality has a logical structure, which can be inferred? Sounds good to me. Further investigation required. |
| 16669 | Everything that exists is either a being, or some mode of a being [Malebranche] |
| Full Idea: It is absolutely necessary that everything in the world be either a being or a mode [maničre] of a being. | |
| From: Nicolas Malebranche (The Search After Truth [1675], III.2.8.ii), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.4 |
| 21362 | Necessity is physical, logical, mathematical or moral [Schopenhauer, by Janaway] |
| Full Idea: For Schopenauer there are physical necessity, logical necessity, mathematical necessity and moral necessity. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
| A reaction: These derive from four modes of explanation, by causes, by grounding in truths or facts, by mathematical reality, and by motives. Not clear why mathematics gets its own necessity. I like metaphysics derived from explanations, though. Necessity makers. |
| 21361 | For Schopenhauer, material things would not exist without the mind [Schopenhauer, by Janaway] |
| Full Idea: Schopenhauer is not a realist about material things, but an idealist: that is, material things would not exist, for him, without the mind. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
| A reaction: Janaway places his views as close to Kant's, but it is not clear that Kant would agree that no mind means no world. Did Schopenhauer believe in the noumenon? |
| 21919 | Object for a subject and representation are the same thing [Schopenhauer] |
| Full Idea: To be object for a subject and to be representation is to be one and the same thing. All representations are objects for a subject, all objects for a subject are representations. | |
| From: Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §16 p.41-2), quoted by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: This is pure idealism in early Schopenhauer, derived from Kant. Are being 'an object for a subject' and being an object 'in itself' two different things? Compare Idea 21914, written later. I think Nietzsche's 'perspective' representations helps here. |
| 21917 | The four explanations: objects by causes, concepts by ground, maths by spacetime, ethics by motive [Schopenhauer, by Lewis,PB] |
| Full Idea: There are four forms of explanation, depending on their topic. Causes explain objects. Grounding explains concepts, Points and moments explain mathematics. Motives explain ethics. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §43 p.214) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: [My compression of Lewis's summary. I'm particularly pleased with this. I have done Schopenhauer a huge favour, should anyone ever visit this website]. The quirky account of mathematics derives from Kant. I greatly admire this whole idea. |
| 21921 | Concepts are abstracted from perceptions [Schopenhauer, by Lewis,PB] |
| Full Idea: For Schopenhauer concepts are abstractions from perception, what he calls 'representations of representations', and are linked to the creation of language. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: This is a traditional view which dates back to Aristotle, and which I personally think is entirely correct. These days I am in minority on that. This idea means that (contrary to Kant) perception is not conceptual. |
| 21363 | Motivation is causality seen from within [Schopenhauer] |
| Full Idea: Motivation is causality seen from within. | |
| From: Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], p.214), quoted by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
| A reaction: This is more illuminating about causation than about motivation, since we can be motivated without actually doing anything. |
| 12726 | In a true cause we see a necessary connection [Malebranche] |
| Full Idea: A true cause is one in which the mind perceives a necessary connection between the cause and its effect. | |
| From: Nicolas Malebranche (The Search After Truth [1675], 1.649 (450)), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 5 | |
| A reaction: Presumably Hume was ignorant of 'true' causes, since he says he never saw this connection. But then is the perception done by the mind, or by the senses? |