Wednesday, September 12, 2012

Avoiding Ceasar by BLV? Unlikely!

If you look at Bob Hale and Crispin Wright's Logicism in the twenty-first century, in this book (esp. pp. 168-169) (also: at Frege's works themselves and at other explications of Frege), the way things are supposed to have gone is this:
  • Frege used Hume's Principle (see my previous post) to derive second-order Peano Arithmetic.
  • Yet, he was unhappy with the explanatory role of Hume's Principle: "we can never—to take a crude example—decide by means of our definitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or not."
  • So, he introduced extensions by means of Basic Law V (the extension of F is the same as the extension of G iff exactly the same objects are Fs and Gs), and defined numbers in terms of extensions.
While people usually think that Frege moved to BLV to avoid the Ceasar problem, I haven't found anywhere, neither in Frege, nor in secondary literature, a clear explanation of how BLV is supposed to help. Sure, once you have a theory of extensions in place, you have an explicit definition of what a number is. BUT, if you're Frege, you've introduced your extensions by means of an abstraction principle (okay, let's pretend for a while it's consistent with the rest of your theory), and this only moves your problem one level up:

We can never decide by means of our definitions whether any concept has the extension JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is an extension or not.

Am I missing something?

Hume's Principle in Hume

Hume's Principle (HP), as used nowadays, states that the number of Fs is the same as the number of Gs iff there is a 1-1 correspondence between Fs and Gs. While this sounds pretty obvious, with second-order logic in the background you can use this to derive second-order Peano Arithmetic (PA). (all this is well known, just like the role of HP in a fairly fashionable stream in phil of math called neologicism - check out this or this if you haven't heard of this stuff). Anyway, the person who really used HP to obtain PA was Frege, so just in case you wondered why the principle is called Hume's Principle, I dug up the passage where Hume formulates it (Treatise 1.3.1). It's only moderately interesting, but if you're geeky enough to be still reading this, you might be just geeky enough to be interested in the quote:
We might proceed, after the same manner, in fixing the proportions of quantity or number, and might at one view observe a superiority or inferiority betwixt any numbers, or figures; especially where the difference is very great and remarkable. As to equality or any exact proportion, we can only guess at it from a single consideration; except in very short numbers, or very limited portions of extension; which are comprehended in an instant, and where we perceive an impossibility of falling into any considerable error. In all other cases we must settle the proportions with some liberty, or proceed in a more artificial manner.
There remain, therefore, algebra and arithmetic as the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty. We are possest of a precise standard, by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard, we determine their relations, without any possibility of error. When two numbers are so combined, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and it is for want of such a standard of equality in extension, that geometry can scarce be esteemed a perfect and infallible science.

Monday, September 10, 2012

The Godel-Yablo paper online

The paper on an arithmetization of Yablo's paradox with provability instead of truth, written jointly with Cezary Cieśliński is now available (open access) in its final form published in the Journal of Philosophical Logic.

Abstract: We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser’s provability predicate.

Saturday, September 8, 2012

Reading Łukowski's "Paradoxes" (Springer 2011)

I've just finished reading the book. Here are some remarks (copy-pasted from LaTeX, apologies for resulting infelicities).

Piotr Łukowski's Paradoxes is an abridged, revised and translated version of a book written in Polish (the English version has 194 pages and the original habilitation is 535 pages long). It divides into four parts: Paradoxes of Wrong Intuition, Paradoxes of Ambiguity, Paradoxes of Self-Reference and Ontological Paradoxes. The first part deals with a selection of pseudo-paradoxical arguments which Łukowski  attempts to explain away by indicating why the intuitions involved in them are mistaken.  The second part is devoted to pseudo-paradoxical arguments which can be accounted for by pointing out ambiguities. The third part discusses the classics: some variants of the Liar paradox, and Richard's, Berry's and Grelling antinomies, among others. The part devoted to ontological paradoxes revolves around vagueness, change and identity through time.

The book is not only a collection of paradoxes. Łukowski, more often than not, develops his own assessment of the arguments involved and provides his own explanation of how a given paradox arises and hints as to how it is to be avoided. As a source of paradoxes and arguments related to them, the book is  valuable. (By the way, the book suffers from some minor typographic and linguistic errors, but they are harmless.)

The depth  of  discussion is not evenly distributed. Some paradoxes are only described in passing, some of them are discussed at length (most notably, the paradox of Holy Trinity, Protagoras' paradox, the paradox of the stone, the Liar and the heap paradox). Similarly, the quality of the discussion tends to vary. In some cases, Łukowski seems to hit the nail on the head. In some other cases, his judgment and arguments are rather hasty.  A few examples of what I take to be cases of the latter kind:
  • The paradox of Holy Trinity is supposed to arise when we consider the claim that there is only one God who somehow divides into three distinct divine persons. Łukowski (p. 31-32) argues that there is no inconsistency involved in this, because it is possible to define three different sequences of natural numbers such that each can be obtained from another by elimination of some elements of the sequence (he gives an example of such sequences):     
``Thus each sequence is a proper subsequence of every other one, including itself. This means that in some sense three completely different sequences are indeed one and the same sequence... we have demonstrated that the concept of Trinity can be conceived of not only in theology but also in the most precise of sciences that is available for man, i.e., mathematics.'' (p. 31-2)
To my mind, this is very hasty. The existence of such mathematical sequences has no bearing on the original problem. On the mathematical side we have the relation of ''being obtainable by elimination of some elements''. Just because it can occur between three sequences, Łukowski insists that ``in some sense three completely different sequences are indeed one and the same sequence''. In what sense? In the sense that this relation occurs between them? Łukowski fails to explain and fails to indicate that this relation and the resulting ``identity in some sense'' have anything to do with theological relations between divine persons and with God's unity.

  • Łukowski briefly considers the Church-Fitch paradox of knowability. If ``$K$'' stands for ``it is known that'' and ``$\Diamond$'' for ``it is possible that'', the argument (The presentation is mine, it slightly differs from Łukowski's.) starts with the claim that there are some unknown truths:
  •           \begin{equation} \label{F1} \exists p [p \wedge \neg Kp]\end{equation} and shows that it is inconsistent with the assumption that all truths are knowable: \begin{equation} \label{F2}\forall p[p \rightarrow  \Diamond K p]\end{equation}The first move is to apply  existential instantiation to the first premiss  to obtain:\begin{equation}\label{F3} q  \wedge   \neg  Kq \end{equation}Then, the whole thing is substituted for $p$ when we apply universal instantiation to second premiss:  \begin{equation}(q  \wedge   \neg  Kq) \rightarrow \Diamond K (q  \wedge \neg Kq)\end{equation}  Modus ponens  yields:\begin{equation}\label{F5}\Diamond K (q  \wedge   \neg  Kq) \end{equation}Distribution of $K$ over conjunction gives:\begin{equation}\Diamond (Kq \wedge K   \neg  K q)\end{equation}and the factivity of $K$ in the second conjunct results in:\begin{equation}  \Diamond (Kq \wedge   \neg  K q)\end{equation}which is quite problematic and with sufficient modal theory in the background leads to straightforward contradiction. Now, to all this  Łukowski says:
        ``It could seem that no matter whether we analyse the above reasoning in a more or less formalized form we cannot help noticing that it contains an elementary mistake, i.e., it gives a concrete proposition $q$ in the assumptions \dots even though out of principle it should be unknown, so it should not be represented by any concrete symbol. For using symbol $q$ means here that we speak of a concrete proposition identified as $q$. (pp. 33-4)''
        So Łukowski resolves the paradox by abolishing existential instantiation, a fairly standard move in natural deductions in classical logic. The problem with this move is that when we eliminate the existential quantifier, we \emph{freeze} the variable without saying that we know what proposition $q$ is. Similarly we can argue:

        $\exists xP(x)$     Assumption
        $\forall x (P(x) \rightarrow R(x))$     Assumption
        $P(y)$     $\exists$I
        $P(y) \rightarrow R(y)$     $\forall$I
        $R(y) $   MP
        $\exists x R(x)$  $\exists$U

        The moves are classically innocent, and as long as the final conclusion does not involve the frozen variable (in other words, as long as we discharge the assumption that $P(y)$), there is no reason to worry.
        • When talking about the dialetheist response to the Liar paradox, Łukowski complains:
        ``Liar Antinomy has a special importance for dialetheists, because the liar sentence is apparently the only example of a sentence that is true and false at the same time\dots From this point of view, dialetheism seems to go too far. What is worse, it is difficult to find another, truly philosophical justification of it apart from the liar sentence.'' (p. 96)
          While dialetheism is a controversial position, claiming that their only argument is the existence of the liar paradox is setting up a straw man. It is enough to look at any of the main works of Graham Priest to see that he thinks that there are many other reasons to think there are true contradictions. Whether Priest is right is a different issue, but Łukowski's dismissal is definitely too hasty.
          • Łukowski briefly discusses Tarski's theory of truth (pp. 83-86). He takes the T-schema to be Tarski's definition of truth, and in general somewhat misleadingly conflates Tarski's definition of truth, Tarski's convention T, T-schema and instances of T-schema.
            Apart from a few stumbles of the above kind, Łukowski often develops his analyses carefully. In his book he not only gathers the paradoxes (some of them I haven't heard about before reading the book) but also presents a rather unified and interesting view on them. It is definitely a good read.

            [UPDATE: the review is forthcoming in Studia Logica]