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. |
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'. |
21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
Full Idea: "I see nobody on the road," said Alice. - "I only wish I had such eyes," the King remarked. ..."To be able to see Nobody! ...Why, it's as much as I can do to see real people." | |
From: Lewis Carroll (C.Dodgson) (Through the Looking Glass [1886], p.189), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.7 | |
A reaction: [Moore quotes this, inevitably, in a chapter on Hegel] This may be a better candidate for the birth of philosophy of language than Frege's Groundwork. |
21094 | There are two kinds of right - to power, and to property [Hume] |
Full Idea: Right is of two kinds: right to power and right to property. | |
From: David Hume (Of the First Principles of Government [1750], p.25) | |
A reaction: These seem to be positive rights. No mention of the right not be to unjustly abused. It is hard to find any sort of radical political thinking in Hume. His empirical scepticism extends to his politics. He approves of modern consitutional monarchy. |
21095 | It is an exaggeration to say that property is the foundation of all government [Hume] |
Full Idea: A noted author has made property the foundation of all government; and most of our political writers seem inclined to follow him in that particular. This is carrying the matter too far. | |
From: David Hume (Of the First Principles of Government [1750], p.25) | |
A reaction: This obviously refers to John Locke. Locke's idea strikes me as hideous. It says the foundation of government is the right of property owners to protect what they have against non-owners. It implies social exclusion in the constitution. |