Studia Logica Issue on Psychologism in Logic

Hannes Leitgeb has edited an interesting special issue of Studia Logica on “Psychologism in Logic?”. From the introduction:

There is no doubt that Frege’s and Husserl’s famous attack on Psychologism in logic had a significant influence on the emergence of logic as a separate discipline. Now that this battle can be safely regarded won, it is time to reconsider psychologism from a modern point of view. Logic has taken a cognitive turn in the meantime: formal representations of agents are used as parts of logical models, cognitive concepts are treated as logical constants in much the same way as the negation sign or the quantifiers, the logic of commonsense reasoning has become the joint interest of theoretical computer scientists and psychologists, and naturalistic accounts of logic and mathematics aim to reduce the gap between the apriori and the empirical. Does this necessitate a reassessment of psychologism in logic? That is the question to be addressed by this special issue.

Philosophy of Math at NYU

NYU Philosophy is hosting a conference on the philosophy of mathematics, October April 10-12, 2008 2009. The speakers are John Burgess, Haim Gaifman, Joel Hamkins, Kai Hauser, Peter Koellner, Stewart Shapiro, Stephen Simpson, Bill Tait, Neil Tennant, and Hugh Woodin.

Towards a New Epistemology of Mathematics

There’s a very interesting issue of Erkenntnis just out. It’s the proceedings of PhiPMSAP 1. PhiMSAP is the Network on Philosophy of Mathematics: Sociological Aspects and Mathematical Practice of the DFG, run mostly by Benedikt Löwe and Thomas Müller, the third workshop of which I just had the pleasure of attending. Contents of the issue:

Experimental Mathematics by Alan Baker
Visualizations in Mathematics Kajsa Bråting and Johanna Pejlare
A Mathematician Reflects on the Useful and Reliable Illusion of Reality in Mathematics by Keith Devlin
The Role of Axioms in Mathematics by Kenny Easwaran
What can the Philosophy of Mathematics Learn from the History of Mathematics? by Brendan Larvor
On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan have Proved the Jordan-Hölder Theorem? by Dirk Schlimm
Pi on Earth, or Mathematics in the Real World by Bart Van Kerkhove and Jean Paul Van Bendegem

First Issue of Review of Symbolic Logic Out Soon!

The first issue of the Review of Symbolic Logic will be out soon (Cambridge UP page here). The Review, like the Journal and Bulletin of Symbolic Logic will be mailed to all members of the Association for Symbolic Logic. So if you’re not a member (and you probably should be, if you’re reading this!), join now! (Only 76$ per year–38$ for students– for three of the most important journals in logic, plus all the other benefits of membership)

Table of contents:

A Cut-Free Simple Sequent Calculus for Modal Logic S5
FRANCESCA POGGIOLESI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
How Applied Mathematics Became Pure
PENELOPE MADDY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
The Closing of the Mind: How the Particular Quantifier Became Existentially Loaded Behind Our Backs
GRAHAM PRIEST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Recognizing Strong Random Reals
DANIEL OSHERSON AND SCOTT WEINSTEIN. . . . . . . . . . . . . . . . 56
BENJAMIN SCHNIEDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Probabilistic Conditionals are Almost Monotonic
MATTHEW P. JOHNSON AND ROHIT PARIKH . . . . . . . . . . . . . . . . 73
On Adopting Kripke Semantics in Set Theory
LUCA INCURVATI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
The Iterative Conception of Set
THOMAS FORSTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A Decision Procedure for Probability Calculus with Applications
BRANDEN FITELSON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Ultimate Truth vis-à-vis Stable Truth
P.D. WELCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Philosophical Logic and Mathematical Logic in the PGR

Last week, Brian Leiter posted about possibly re-drawing the dividing lines between the specialty areas ranked in the Philosophical Gourmet Report

In a comment, Cian asks:

Philosophical Logic and Mathematical Logic. While there is a fair amount of divergence between the two rankings, I also see, for example, that NYU gets ranked at 4.5 in mathematical logic even though as far as I can see, almost everything the relevant people have written about logic is more naturally classified as philosophical logic than as mathematical logic. Is that a sign that the borderline between these two areas is too wide and fuzzy for the distinction to be worth making by the PGR?

If one thinks of mathematical logic as “formal logic motivated by mathematical concerns”—roughly, this is the conception according to which mathematical logic consists of model theory, set theory, recursion theory, and proof theory–then it is indeed puzzling why NYU gets ranked at the top of group 2 in the mathematical logic ranking, in addition to the top in the philosophical logic ranking. But one might also think of mathematical logic the way it’s defined in the AMS Mathematics Subject Classification (03). Now how do we take “philosophical logic”? But if we take the now-standard (at least in North America) definition of “philosophical logic”, then it’s that part of formal logic that paradigmatically includes: “various versions of modal, temporal, epistemic, and deontic logic; constructive logics; relevance and other sub-classical logics; many-valued logics; logics of conditionals; quantum logic; decision theory, inductive logic, logics of belief change, and formal epistemology; defeasible and nonmonotonic logics; formal philosophy of language; vagueness; and theories of truth and validity” (from editorial description of the Journal of Philosophical Logic). Almost all of that is included in MSC 03Bxx! Clearly, for the purpose of the PGR at least, it would be better to define “mathematical logic” as “formal logic, but not philosophical logic”.

I don’t know if the evaluators for the PGR “mathematical logic” and “philosophical logic” categories get instructions on the intended scope of the category. It probably also makes a big difference–bigger than in other specialty areas–if faculty with appointments in the mathematics (or computer science) departments get included in the PGR faculty lists. As the note at the bottom of the ranking says, “much work in mathematical logic goes on in Mathematics and Computer Science departments.”

Then there’s also the confusion between the definition of “philosophical logic” as “formal logic motivated by philosophy” and the older (British) use of “philosophical logic” to mean “philosophy motivated by logic” (and including philosophical study of notions such as reference, necessity, truth, analyticity, etc.) Maybe a better term for that is “philosophy of logic”. Leiter’s proposed restructuring would have a category “philosophy of language & logic” (but no philosophical logic category).

I don’t think that “philosophical logic” and “mathematical logic” should be combined in the PGR ranking. But as it currently stands–with the scope of these categories so unclear–the rankings aren’t particularly informative. I’m not sure what would be more informative, but getting clarity on the definitions would be one step. Maybe it wouldn’t be such a bad idea to include “philosophy of logic” in the “philosophy of language” category, and reserve “philosophical logic” for the formal work you find in the JPL or the Review of Symbolic Logic. Maybe it wouldn’t even be a bad idea to merge mathematical logic into the philosophy of mathematics category. What do others think?

Kohlenbach’s Applied Proof Theory is Out!

What more do we know about a theorem if we have a proof (by restricted means) than merely that it is true? That’s an old question of Kreisel’s that motivated his “unwinding program”: extract additional information from proofs of theorems in constructive theories, such as bounds on y in theorems of the form ∀xy A(x, y).
Ulrich Kohlenbach has worked on problems like this more than anyone else, and his book Applied Proof Theory: Proof Interpretations and Their Use in Mathematics is now out from Springer.

Ulrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as independence of solutions from certain parameters, generalizations of proofs by elimination of premises.

The book first develops the necessary logical machinery emphasizing novel forms of Goedel’s famous functional (Dialectica) interpretation. It then establishes general logical metatheorems that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics.

1 Introduction

2 Unwinding proofs (‘Proof Mining’)
2.1 Introductory remark
2.2 Informal treatment of ineffective proofs
2.3 Herbrand’s theorem and the no-counterexample interpretation
2.4 Exercises, historical comments and suggested further reading

3 Intuitionistic and classical arithmetic in all finite types
3.1 Intuitionistic and classical predicate logic
3.2 Intuitionistic (‘Heyting’) arithmetic HA and Peano arithmetic PA
3.3 Extensional intuitionistic (‘Heyting’) and classical (‘Peano’)
arithmetic in all finite types
3.4 Fragments of (W)E-HA^ω and (W)E-PA^ω
3.5 Fragments corresponding to the Grzegorczyk hierarchy
3.6 Models of E-PA^ω
3.7 Exercises, historical comments and suggested further reading

4 RepresentationofPolish metric spaces
4.1 Representation of real numbers
4.2 Representation of complete separable metric (‘Polish’) spaces
4.3 Special representation of compact metric spaces
4.4 Fragments, exercises, historical comments and suggested further

5 Modified realizability
5.1 The soundness and program extraction theorems
5.2 Remarks on fragments of E-HA^ω
5.3 Exercises, historical comments and suggested further reading

6 Majorizability and the fan rule
6.1 A syntactic treatment of majorization and the fan rule
6.2 Exercises, historical comments and suggested further reading

7 Semi-intuitionistic systems and monotone modified realizability
7.1 The soundness and bound extraction theorems
7.2 Fragments, exercises, historical comments and suggested further

8 Gödel’s functional (‘Dialectica’) interpretation
8.1 Introduction
8.2 The soundness and program extraction theorems
8.3 Fragments, exercises, historical comments and suggested further

9 Semi-intuitionistic systems and monotone functional interpretation
9.1 The soundness and bound extraction theorems
9.2 Applications of monotone functional interpretation
9.3 Examples of axioms Δ : Weak König’s lemmaWKL
9.4 WKL as a universal sentence Δ
9.5 Fragments, exercises, historical comments and suggested further

10 Systems based on classical logic and functional interpretation
10.1 The negative translation
10.2 Combination of negative translation and functional interpretation
10.3 Application: Uniform weak König’s lemma UWKL
10.4 Elimination of extensionality
10.5 Fragments of (W)E-PA^ω
10.6 The computational strength of full extensionality
10.7 Exercises, historical comments and suggested further reading

11 Functional interpretation of full classical analysis
11.1 Functional interpretation of full comprehension
11.2 Functional interpretation of dependent choice
11.3 Functional interpretation of arithmetical comprehension
11.4 Functional interpretation of (IPP) by finite bar recursion
11.5 Models of bar recursion
11.6 Exercises, historical comments and suggested further reading

12 A non-standard principle of uniform boundedness
12.1 The Σ^0_1 -boundedness principle
12.2 Applications of Σ^0_1 -boundedness
12.3 Remarks on the fragments E-G_nA^ω
12.4 Exercises, historical comments and suggested further reading

13 Elimination of monotone Skolem functions
13.1 Skolem functions of type degree 1 in fragments of finite type
13.2 Elimination of Skolem functions for monotone formulas
13.3 The principle of convergence for bounded monotone sequences
of real numbers (PCM)
13.4 Π^0_1 -CA and Π^0_1 -AC
13.5 The Bolzano-Weierstraß property for bounded sequences in R^d
13.6 Exercises, historical comments and suggested further reading

14 The Friedman A-translation
14.1 The A-translation
14.2 Historical comments and suggested further reading

15 Applications to analysis: general metatheorems I
15.1 A general metatheorem for Polish spaces
15.2 Applications to uniqueness proofs
15.3 Applications to monotone convergence theorems
15.4 Applications to proofs of contractivity
15.5 Remarks on fragments of T^ω
15.6 Historical comments and suggested further reading

16 Case study I: Uniqueness proofs in approximation theory
16.1 Uniqueness proofs in best approximation theory
16.2 Best Chebycheff approximation I
16.3 Best Chebycheff approximation II
16.4 Best L_1-approximation
16.5 Exercises, historical comments and suggested further reading

17 Applications to analysis: general metatheorems II
17.1 Introduction
17.2 Main results in the metric and hyperbolic case
17.3 The case of normed spaces
17.4 Proofs of theorems 17.35, 17.52 and 17.69
17.5 Further variations
17.6 Treatment of several metric or normed spaces X_1 . . . , X_n
17.7 A generalized uniform boundedness principle ∃-UB^X
17.8 Applications of ∃-UB^X
17.9 Fragments of A^ω [. . .]
17.10 Exercises, historical comments and suggested further reading

18 Case study II: Applications to the fixed point theory of nonexpansive
18.1 General facts
18.2 Applications of the metatheorems from chapter 17
18.3 Logical analysis of the proof of the Borwein-Reich-Shafrir
18.4 Asymptotically nonexpansive mappings
18.5 Applications of proof mining in ergodic theory
18.6 Exercises, historical comments and suggested further reading

19 Final comments