18 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] |
17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
18739 | Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-) [Horsten/Pettigrew] |
Full Idea: Three periods can be distinguished in philosophical logic: the syntactic stage, from Russell's definite descriptions to the 1950s, the dominance of possible world semantics from the 50s to 80s, and a current widening of the subject. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 1) | |
A reaction: [compressed] I've read elsewhere that the arrival of Tarski's account of truth in 1933, taking things beyond the syntactic, was also a landmark. |
18741 | Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew] |
Full Idea: Logical formalization forces the investigator to make the central philosophical concepts precise. It can also show how some philosophical concepts and objects can be defined in terms of others. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 2) | |
A reaction: This is the main rationale of the highly formal and mathematical approach to such things. The downside is when you impose 'precision' on language that was never intended to be precise. |
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? |
18744 | Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew] |
Full Idea: A (logical) model is a set with functions and relations defined on it that specify the denotation of the non-logical vocabulary. A series of recursive clauses explicate how truth values of complex sentences are compositionally determined from the parts. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 3) | |
A reaction: See the ideas on 'Functions in logic' and 'Relations in logic' (in the alphabetical list) to expand this important idea. |
17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
A reaction: Each expansion brings a limitation, but then you can expand again. |
17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
18740 | If 'exist' doesn't express a property, we can hardly ask for its essence [Horsten/Pettigrew] |
Full Idea: If there is indeed no property of existence that is expressed by the word 'exist', then it makes no sense to ask for its essence. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 2) | |
A reaction: As far as I can tell, this was exactly Aristotle's conclusion, so he skirted round the question of 'being qua being', and focused on the nature of objects instead. Grand continental talk of 'Being' doesn't sound very interesting. |
18745 | A Tarskian model can be seen as a possible state of affairs [Horsten/Pettigrew] |
Full Idea: A Tarskian model can in a sense be seen as a model of a possible state of affairs. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 3) | |
A reaction: I include this remark to show how possible worlds semantics built on the arrival of model theory. |
18747 | The 'spheres model' was added to possible worlds, to cope with counterfactuals [Horsten/Pettigrew] |
Full Idea: The notion of a possible worlds model was extended (resulting in the concept of a 'spheres model') in order to obtain a satisfactory logical treatment of counterfactual conditional sentences. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 4) | |
A reaction: Thus we add 'centred' worlds, and an 'actual' world, to the loose original model. It is important to remember when we discuss 'close' worlds that we are then committed to these presuppositions. |
18748 | Epistemic logic introduced impossible worlds [Horsten/Pettigrew] |
Full Idea: The idea of 'impossible worlds' was introduced into epistemic logic. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 4) | |
A reaction: Nathan Salmon seems interested in their role in metaphysics (presumably in relation to Meinongian impossible objects, like circular squares, which must necessarily be circular). |
18746 | Possible worlds models contain sets of possible worlds; this is a large metaphysical commitment [Horsten/Pettigrew] |
Full Idea: Each possible worlds model contains a set of possible worlds. For this reason, possible worlds semantics is often charged with smuggling in heavy metaphysical commitments. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 3) | |
A reaction: To a beginner it looks very odd that you should try to explain possibility by constructing a model of it in terms of 'possible' worlds. |
18750 | Using possible worlds for knowledge and morality may be a step too far [Horsten/Pettigrew] |
Full Idea: When the possible worlds semantics were further extended to model notions of knowledge and of moral obligation, the application was beginning to look distinctly forced and artificial. | |
From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 5) | |
A reaction: They accept lots of successes in modelling necessity and time. |