Ideas from 'Mathematical Methods in Philosophy' by Horsten,L/Pettigrew,R [2014], by Theme Structure

[found in 'Bloomsbury Companion to Philosophical Logic' (ed/tr Horsten,L/Pettigrew,R) [Bloomsbury 2014,978-1-4725-2303-0]].

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5. Theory of Logic / A. Overview of Logic / 9. Philosophical Logic
Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-)
                        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.
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
Logical formalization makes concepts precise, and also shows their interrelation
                        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.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Models are sets with functions and relations, and truth built up from the components
                        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.
7. Existence / A. Nature of Existence / 1. Nature of Existence
If 'exist' doesn't express a property, we can hardly ask for its essence
                        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.
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
A Tarskian model can be seen as a possible state of affairs
                        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.
The 'spheres model' was added to possible worlds, to cope with counterfactuals
                        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.
10. Modality / E. Possible worlds / 1. Possible Worlds / b. Impossible worlds
Epistemic logic introduced impossible worlds
                        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).
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
Possible worlds models contain sets of possible worlds; this is a large metaphysical commitment
                        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.
Using possible worlds for knowledge and morality may be a step too far
                        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.