Combining Texts

All the ideas for 'The Roots of Reference', 'The Principles of Chemistry' and 'First-order Logic, 2nd-order, Completeness'

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15 ideas

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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.
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.
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.
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
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.
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
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.
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
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.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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.
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
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'.
5. Theory of Logic / K. Features of Logics / 4. Completeness
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)
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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'.
8. Modes of Existence / C. Powers and Dispositions / 3. Powers as Derived
Dispositions are physical states of mechanism; when known, these replace the old disposition term [Quine]
     Full Idea: Each disposition, in my view, is a physical state or mechanism. ...In some cases nowadays we understand the physical details and set them forth explicitly in terms of the arrangement and interaction of small bodies. This replaces the old disposition.
     From: Willard Quine (The Roots of Reference [1990], p.11), quoted by Stephen Mumford - Dispositions 01.3
     A reaction: A challenge to the dispositions and powers view of nature, one which rests on the 'categorical' structural properties, rather than the 'hypothetical' dispositions. But can we define a mechanism without mentioning its powers?
27. Natural Reality / B. Modern Physics / 4. Standard Model / a. Concept of matter
Mendeleev saw three principles in nature: matter, force and spirit (where the latter seems to be essence) [Mendeleev, by Scerri]
     Full Idea: Mendeleev rejected one unifying principles in favour of three basic components of nature: matter (substance), force (energy), and spirit (soul). 'Spirit' is said to be what we now mean by essentialism - what is irreducibly peculiar to the object.
     From: report of Dmitri Mendeleev (The Principles of Chemistry [1870]) by Eric R. Scerri - The Periodic Table 04 'Making'
27. Natural Reality / F. Chemistry / 2. Modern Elements
Elements don't survive in compounds, but the 'substance' of the element does [Mendeleev]
     Full Idea: Neither mercury as a metal nor oxygen as a gas is contained in mercury oxide; it only contains the substance of the elements, just as steam only contains the substance of ice.
     From: Dmitri Mendeleev (The Principles of Chemistry [1870], I:23), quoted by Eric R. Scerri - The Periodic Table 04 'Nature'
     A reaction: [1889 edn] Scerri glosses the word 'substance' as meaning essence.
27. Natural Reality / F. Chemistry / 3. Periodic Table
Mendeleev focused on abstract elements, not simple substances, so he got to their essence [Mendeleev, by Scerri]
     Full Idea: Because he was attempting to classify abstract elements, not simple substances, Mendeleev was not misled by nonessential chemical properties.
     From: report of Dmitri Mendeleev (The Principles of Chemistry [1870]) by Eric R. Scerri - The Periodic Table 04 'Making'
     A reaction: I'm not fully clear about this, but I take it that Mendeleev stood back from the messy observations, and tried to see the underlying simpler principles. 'Simple substances' were ones that had not so far been decomposed.
Mendeleev had a view of elements which allowed him to overlook some conflicting observations [Mendeleev]
     Full Idea: His view of elements allowed Mendeleev to maintain the validity of the periodic table even in instances where observational evidence seemed to point against it.
     From: Dmitri Mendeleev (The Principles of Chemistry [1870]), quoted by Eric R. Scerri - The Periodic Table 04 'Making'
     A reaction: Mendeleev seems to have focused on abstract essences of elements, rather than on the simplest substances they had so far managed to isolate.