Wolfgang Schwarz

Restrict the Gamma Rule?

The following restriction might be a way out of the problems I mentioned in my last posting:

The gamma rule must not be applied if the result of its previous application has not yet been replaced by the Closure rule.

(The gamma rule deals with and formulae; the Closure rule is the rule that allows to replace dummy constants by real constants iff that leads to the closure of at least one branch.)

Counter-Models and Free-Variable Tableau Systems

I'm half-way into programming a more efficient tree prover, based on free-variable tableaux. But now I'm not sure any more if this is really what I want.

The basic idea in free-variable tableaux is that you use dummy constants to instantiate universally quantified formulas, and only replace these dummy constants by real constants if this allows you to close a branch. In automatised tableaux, this dramatically decreases the steps required for certain proofs. For example, my old tree prover internally creates an 860-node tree to prove xy(Fxy zFzz) x y Fxy x Fxx, whereas a free-variable system only needs 12 nodes.

Oh Dear

I just noticed that my tree prover fails on this simple formula! This is a good opportunity to rewrite the part of the script that does the proving and implement some shortcuts, and maybe some "loop detection".

Keyboard Commands in Postbote

I've added keyboard commands to Postbote: If the focus is on the frame with the mail listing, press "R" to refresh, "A" to select all mails, and "G" to quickly change the listing offset (this is for Hermann, who has 1600 mails in his mailbox...).

How To Define Theoretical Predicates: The Problem

Suppose some theory T(F) implicitly defines the predicate F. If we want to apply the Ramsey-Carnap-Lewis account of theoretical expressions, we first of all have to replace F by an individual constant f, and accordingly change every occurrance of "Fx" in T by "x has f" etc. The empirical content of the resulting theory T'(f) can then be captured by something like its Ramsey sentence f T'(f), and the definition of f by the stipulation that 'f' denote the only x such that T'(x), or nothing if there is no such (unique) x.

Implicit Definitions, Part 4: Summing Up (And a Partial Defence of Implicit Definition)

In the previous three entries, I've tried to argue that there are no genuinely implicit definitions: Whenever a new expression is introduced via an alleged implicit definition, either there is no question of definition at all, as in the case of new expressions used as bound variables in mathematics, or there is an explicit definition nearby.

This latter fact, that sometimes explicit definitions are only nearby, provides a partial vindication of implicit definitions. For example, let's assume that folk psychology implicitly defines "pain". But folk psychology itself is not equivalent to the nearby explicit definition. To get an explicit definition, we have to turn folk psychology into something like its Carnap sentence. So the theory itself could be called a genuinely implicit definition.

Implicit Definitions, Part 3: Contextual Definition

I've said that an explicit definition introduces a new expression by stipulating that it be semantically equivalent to an old expression. If there are no non-explicit definitions, this means that you can only define expressions that are in principle redundant. Aren't there counterexamples to this claim?

Consider the definition of the propositional connectives. We can explicitly define some of them with the help of others, but what if we want to define all of them from scratch? The common strategy here is to recursively provide necessary and sufficient conditions for the truth of a sentence governed by the connective: A B is true iff A is true and B is true.

Implicit Definitions, Part 2: Theoretical Terms

Scientific theories are often said to implicitly define their theoretical terms: phlogiston theory implicitly defines "phlogiston", quantum mechanics implicitly defines "spin". This is easily extended to non-scientific theories: ectoplasm theory implicitly defines "ectoplasm", folk psychology implicitly defines "pain".

The first problem from the mathematical case applies here too: Since all these theories make substantial claims about reality, their truth is not a matter of stipulation. For example, no stipulation can make phlogiston theory true. That's why, according to the standard Ramsey-Carnap-Lewis account, what defines a term (or several terms) t occurring in a theory T(t) is not really the stipulation of T(t) itself, but rather the stipulation of something like its 'Carnap sentence' x T(x) T(t). All substantial claims in T(t) are here cancelled out by the antecedent.

Implicit Definitions, part 1: Mathematics

I vaguely believe that there are no implicit definitions. So I've decided to write a couple of entries to defend this belief. The defence may well lead me to give it up, though. Anyway, here is part 1.

Explicit definitions introduce a new expression by stipulating that it be in some sense synonymous or semantically equivalent to an old expression. For ordinary purposes this can be done without the use of semantic vocabulary by stipulations of the form

Luxury Flat

This weekend, I've moved into my new flat, which has both a bath room and a fridge, and also lot's of funny records from the 1970s.