28 ideas
9470  Modal logic is not an extensional language [Parsons,C] 
Full Idea: Modal logic is not an extensional language.  
From: Charles Parsons (A Plea for Substitutional Quantification [1971], p.159 n8)  
A reaction: [I record this for investigation. Possible worlds seem to contain objects] 
10888  Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo] 
Full Idea: We can define a set by 'enumeration' (by listing the items, within curly brackets), or by 'abstraction' (by specifying the elements as instances of a property), pretending that they form a determinate totality. The latter is written {x  x is P}.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) 
10889  The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo] 
Full Idea:
The 'Cartesian Product' of two sets, written A x B, is the relation which pairs every element of A with every element of B. So A x B = { 

From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) 
10890  A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo] 
Full Idea: A binary relation in a set is a 'partial ordering' just in case it is reflexive, antisymmetric and transitive.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) 
10886  Determinacy: an object is either in a set, or it isn't [Zalabardo] 
Full Idea: Principle of Determinacy: For every object a and every set S, either a is an element of S or a is not an element of S.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.2) 
13418  The old problems with the axiom of choice are probably better ascribed to the law of excluded middle [Parsons,C] 
Full Idea: The difficulties historically attributed to the axiom of choice are probably better ascribed to the law of excluded middle.  
From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], §2)  
A reaction: The law of excluded middle was a target for the intuitionists, so presumably the debate went off in that direction. 
10887  Specification: Determinate totals of objects always make a set [Zalabardo] 
Full Idea: Principle of Specification: Whenever we can specify a determinate totality of objects, we shall say that there is a set whose elements are precisely the objects that we have specified.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3)  
A reaction: Compare the Axiom of Specification. Zalabardo says we may wish to consider sets of which we cannot specify the members. 
10897  A firstorder 'sentence' is a formula with no free variables [Zalabardo] 
Full Idea: A formula of a firstorder language is a 'sentence' just in case it has no free variables.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.2) 
10893  Γ = φ for sentences if φ is true when all of Γ is true [Zalabardo] 
Full Idea: A propositional logic sentence is a 'logical consequence' of a set of sentences (written Γ = φ) if for every admissible truthassignment all the sentences in the set Γ are true, then φ is true.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4)  
A reaction: The definition is similar for predicate logic. 
10899  Γ = φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo] 
Full Idea: A formula is the 'logical consequence' of a set of formulas (Γ = φ) if for every structure in the language and every variable interpretation of the structure, if all the formulas within the set are true and the formula itself is true.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) 
10896  Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo] 
Full Idea: In propositional logic, any set containing ¬ and at least one of ∧, ∨ and → is expressively complete.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.8) 
9469  Substitutional existential quantifier may explain the existence of linguistic entities [Parsons,C] 
Full Idea: I argue (against Quine) that the existential quantifier substitutionally interpreted has a genuine claim to express a concept of existence, which may give the best account of linguistic abstract entities such as propositions, attributes, and classes.  
From: Charles Parsons (A Plea for Substitutional Quantification [1971], p.156)  
A reaction: Intuitively I have my doubts about this, since the whole thing sounds like a verbal and conventional game, rather than anything with a proper ontology. Ruth Marcus and Quine disagree over this one. 
9468  On the substitutional interpretation, '(∃x) Fx' is true iff a closed term 't' makes Ft true [Parsons,C] 
Full Idea: For the substitutional interpretation of quantifiers, a sentence of the form '(∃x) Fx' is true iff there is some closed term 't' of the language such that 'Ft' is true. For the objectual interpretation some object x must exist such that Fx is true.  
From: Charles Parsons (A Plea for Substitutional Quantification [1971], p.156)  
A reaction: How could you decide if it was true for 't' if you didn't know what object 't' referred to? 
10898  The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo] 
Full Idea: The semantic pattern of a firstorder language is the ways in which truth values depend on which individuals instantiate the properties and relations which figure in them. ..So we pair a truth value with each combination of individuals, sets etc.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.3)  
A reaction: So truth reduces to a combination of 'instantiations', which is rather like 'satisfaction'. 
10902  We can do semantics by looking at given propositions, or by building new ones [Zalabardo] 
Full Idea: We can look at semantics from the point of view of how truth values are determined by instantiations of properties and relations, or by asking how we can build, using the resources of the language, a proposition corresponding to a given semantic pattern.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6)  
A reaction: The second version of semantics is model theory. 
10892  We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo] 
Full Idea: A truth assignment is a function from propositions to the set {T,F}. We will think of T and F as the truth values true and false, but for our purposes all we need to assume about the identity of these objects is that they are different from each other.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4)  
A reaction: Note that T and F are 'objects'. This remark is important in understanding modern logical semantics. T and F can be equated to 1 and 0 in the language of a computer. They just mean as much as you want them to mean. 
10895  'Logically true' (= φ) is true for every truthassignment [Zalabardo] 
Full Idea: A propositional logic sentence is 'logically true', written = φ, if it is true for every admissible truthassignment.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) 
10900  Logically true sentences are true in all structures [Zalabardo] 
Full Idea: In firstorder languages, logically true sentences are true in all structures.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) 
10901  Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo] 
Full Idea: A set of formulas of a firstorder language is 'satisfiable' if there is a structure and a variable interpretation in that structure such that all the formulas of the set are true.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) 
10894  A sentenceset is 'satisfiable' if at least one truthassignment makes them all true [Zalabardo] 
Full Idea: A propositional logic set of sentences Γ is 'satisfiable' if there is at least one admissible truthassignment that makes all of its sentences true.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) 
10903  A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo] 
Full Idea: A structure is a model of a sentence if the sentence is true in the model; a structure is a model of a set of sentences if they are all true in the structure.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) 
17447  Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] 
Full Idea: In Parsons's demonstrative model of counting, '1' means the first, and counting says 'the first, the second, the third', where one is supposed to 'tag' each object exactly once, and report how many by converting the last ordinal into a cardinal.  
From: report of Charles Parsons (Frege's Theory of Numbers [1965]) by Richard G. Heck  Cardinality, Counting and Equinumerosity 3  
A reaction: This sounds good. Counting seems to rely on that fact that numbers can be both ordinals and cardinals. You don't 'convert' at the end, though, because all the way you mean 'this cardinality in this order'. 
10891  If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] 
Full Idea: Defining a set by induction enables us to use the method of proof by induction to establish that all the elements of the set have a certain property.  
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.3) 
18201  General principles can be obvious in mathematics, but bold speculations in empirical science [Parsons,C] 
Full Idea: The existence of very general principles in mathematics are universally regarded as obvious, where on an empiricist view one would expect them to be bold hypotheses, about which a prudent scientist would maintain reserve.  
From: Charles Parsons (Mathematical Intuition [1980], p.152), quoted by Penelope Maddy  Naturalism in Mathematics  
A reaction: This is mainly aimed at Quine's and Putnam's indispensability (to science) argument about mathematics. 
13419  If functions are transfinite objects, finitists can have no conception of them [Parsons,C] 
Full Idea: The finitist may have no conception of function, because functions are transfinite objects.  
From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], §4)  
A reaction: He is offering a view of Tait's. Above my pay scale, but it sounds like a powerful objection to the finitist view. Maybe there is a finitist account of functions that could be given? 
13417  If a mathematical structure is rejected from a physical theory, it retains its mathematical status [Parsons,C] 
Full Idea: If experience shows that some aspect of the physical world fails to instantiate a certain mathematical structure, one will modify the theory by sustituting a different structure, while the original structure doesn't lose its status as part of mathematics.  
From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], §2)  
A reaction: This seems to be a beautifully simple and powerful objection to the Quinean idea that mathematics somehow only gets its authority from physics. It looked like a daft view to begin with, of course. 
21912  Fichte, Schelling and Hegel rejected transcendental idealism [Lewis,PB] 
Full Idea: Fichte, Schelling and Hegel were united in their opposition to Kant's Transcendental Idealism.  
From: Peter B. Lewis (Schopenhauer [2012], 3)  
A reaction: That is, they preferred genuine idealism, to the mere idealist attitude Kant felt that we are forced to adopt. 
21911  Fichte, Hegel and Schelling developed versions of Absolute Idealism [Lewis,PB] 
Full Idea: At the University of Jena, Fichte, Hegel and Schelling critically developed aspects of Kant's philosophy, each in his own way, thereby giving rise to the movement known as Absolute Idealism, see reality as universal Godlike selfconsciousness.  
From: Peter B. Lewis (Schopenhauer [2012], 2)  
A reaction: Is asking how anyone can possibly have believed such a bizarre and ridiculous idea a) uneducated, b) stupid, c) unimaginative, or d) very sensible? It sounds awfully like Spinoza's concept of God. Also Anaxagoras. 