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]