Poizat is a little idiosyncratic (just look at the Introduction!)

Here is the last paragraph, which you can read via Google Books:

Indeed, I wrote the outline of this book while wandering across India, so that, in my mind, Henkin's method is inextricably linked to the droves of wild elephants that I met while crawling among the swamp plants of the preserves of Kerala; the elimination of imaginaries, to the gliding of the vultures above the high Himalayan peaks; and the theorem of the bound, to the naked bodies of the Mauryan women that the traveler saw on the bends of a jungle trail, before they had time to cover themselves. I dare hope only that this book will evoke similarly pleasant images in my reader; I wish only that it will be a pleasant companion for you, as it was for me.

The first paragraph of the preface to the English translation (which I found via Amazon's "look inside" feature) says this:

It was written in a dialect of Latin that is spoken as a native language in some parts of Europe, Canada, the U.S.A., the West Indies, and is used as a language of communication between several countries in Africa.

which sounds extremely weird until it transpires that he means it was written in French. (It appears that the book was less well received than the author hoped, at least partly on account of its having been written in French, and he is still cross about it, which I think is why he expresses himself in such a peculiar way: he's both parodying and criticizing the idea that there's something obscure about French.)

So, yeah, "idiosyncratic" is right. As for the mathematical approach, here's another quotation from the Introduction:

[...] I have deliberately chosen an unusual approach to the foundations of logic: The idea that I take as primitive is that of the back-and-forth construction in the style of Fraissé, rather than that of satisfaction of a formula.

(Gosh!)