## Getting Things Right

I used to much enjoy contributing to the admirable Ask Philosophers site (you can check out my efforts here!) Rather sadly I had to give that up. But it seems that I can’t resist the pedagogic imperative. So my energies got  diverted instead to another  admirable question-and-answer site,  math.stackexchange; I occasionally do my bit there, when the spirit moves, to answer some elementary logic questions, trying to  Get Things Right.

A lot of my answers are to questions of no lasting interest. But  I’ve linked to an assorted 140 answers which are of perhaps more than ephemeral interest. There should be something here to amuse, even instruct, students at various levels.

Is contributing to a question-and-answer site like that worth doing? Well, there are far worse ways of procrastinating on the internet! But anyway, I’ve just noticed the estimate for the number of readers for my answers. And while we all know that idle browsing doesn’t in general mean that we are paying much attention, someone is unlikely to be visiting math.se and clicking on the link to an answer without some level of interest. I hope. Anyway, the stats are that approximately one million people have now viewed my answers there. Heavens!

Does that count as “impact”? Who cares! It is enough encouragement to continue. (But excuse me, must dash,  someone is wrong on the internet, confusing entailment and the material conditional yet again …!)

## The Consistency of Arithmetic

There’s a nice new piece on the Consistency of Arithmetic by Timothy Chow in the Mathematical Intelligencer, which the author has made freely available. As he says in a FOM posting, he has put extra effort  into trying to make Gentzen’s proof accessible to the “mathematician in the street”. Of course, this kind of expository effort will always strike some readers as requiring too much of them to follow, and strike other readers as not going far enough into the details they want. But it seems an admirable effort to me, that students in particular could find pretty helpful.

## What Frege didn’t say about functions and quantification

Frege’s conception of a function is closely related to his discovery that quantifiers like $\forall$ (“for all”) and $\exists$ (“for some”) operate on what are now called open expressions — expressions containing free variables.

Say we’re interested in a series of calculations like this:

(A) $\quad 3^2 + 6\cdot 3 + 1$ and $4^2 + 6\cdot 4 + 1$ and $5^2 + 6\cdot 5 + 1$.

We soon begin to realize a pattern here; we are taking the square of a number, adding that to the result of multiplying the number by 6, and then adding 1. Following mathematical practice, we depict the pattern by replacing ‘3’ in first example in (A) by ‘x’:

(B) $\quad x^2 + 6 \cdot x + 1$

This example pictures a function. Contemporary logicians think of such examples as having a variable reference; when the variable ‘x’ is assigned a number, this will refer to the result of applying the function to that number. Frege thought of (B) as having an indefinite reference. It corresponds to a function, which is something incomplete or unsaturated. Saturation is accomplished, and reference — say, reference to the number 28 — is achieved when a referring expression like ‘3’ is substituted for ‘x’  in (B).

You wouldn’t, I hope, be particularly happy about this as an account of Frege’s thought from a student. Leave aside the fact that dots aren’t yet joined up (to tell us how, for Frege, quantifiers do apply to expressions for functions mapping to truth-values). For a start, you’d want to point out that what express functions for Frege are expressions with gaps not expressions with free variables. So, for example, rather than (B) he would use

(C) $\quad \xi^2 + 6 \cdot \xi + 1$

where the Greek letter is very clearly explained as a gap-marker, indicating that the two gaps are to be filled in the same way; and the Greek letters do not strictly belong to the concept-script, but are a convenient device in our metalinguistic commentary. And of course, Frege didn’t think that the likes of the gappy (C) as having indefinite reference. They have a definite reference to a function!

Now, it is true that — as well as the Gothic letters he uses as bound variables in his concept script, and the informal Greek gap markers  — Frege also uses italic Roman variables in his concept script. But Frege wouldn’t use them in an expression for a function comparable to (C) — for they are only to appear in expressions for assertible contents that can follow a judgement stroke.

Moreover, Frege’s Roman letters never occur in the scope of a corresponding explicit quantifier (in fact, they approximately function like parametric letters in natural deduction). For Frege, what quantifier expressions are applied to is — of course — open expressions in the sense of expressions with gaps, not to sentences with free variables. And — in modernized notation — we should think of the Fregean quantifier expression in e.g.

(D) $(\forall x)(Fx \to Gx)$

not as simply ‘$\forall$’  nor as ‘$(\forall x)$’ (neither does any gap filling!) but rather as something we might represent as ‘$(\forall x) \ldots x \ldots x \ldots$’  which is applied to the gappy ‘$(F\xi \to G\xi)$’.

And so it goes. It is a bit depressing, then, to report that the quotation above is lifted with only minor (and irrelevant) changes and omissions from p. 23 of a newly published CUP book aimed at linguistics students, Philosophy of Language, by Zoltán Gendler Szabó and Richmond H. Thomason. OK, I if anyone should know how difficult it is to write introductory logical stuff without corrupting the youth! But there is surely a boundary to how rough and ready you are allowed to be, and by my lights the authors overstep it here, given it would have been pretty easy to have been significantly more accurate without confusing the reader. And this sort of thing must make you wonder how trustworthy the authors are as guides elsewhere …

Posted in This and that | 1 Comment

## Pretty serious people …

Posted by Wigmore Hall on Thursday, November 8, 2018

## Forgetting Turgenev?

Isaiah Berlin writes so well in his Russian Thinkers about Ivan Turgenev. Yes, Turgenev is — there on the surface — “a writer of beautiful lyrical prose, … the elegiac poet of the last enchantments of decaying country houses and of their ineffective but irresistibly attractive inhabitants, the incomparable story-teller with a marvellous gift for describing nuances of mood and feeling, the poetry of nature and of love”. We can indeed love him for that. But there is so much more. It is not for nothing that Turgenev’s books were so controversial when they appeared and (at least for Russians) long remained so. Yes, as Berlin puts it, “unlike his great contemporaries, Tolstoy and Dostoevsky, he was not a preacher and did not wish to thunder at his generation.” But he was deeply engaged from the beginning “with the controversies, moral and political, social and personal, which divide the educated Russians of his day; in particular, the profound and bitter conflicts between Slavophil nationalists and admirers of the west, conservatives and liberals, liberals and radicals, moderates and fanatics, realists and visionaries, above all between old and young. He tried to stand aside and see the scene objectively. He did not always succeed. But because he was an acute and responsive observer, self-critical and self-effacing both as a man and as a writer, and, above all, because he was not anxious to bind his vision upon the reader, to preach, to convert, he proved a better prophet than the two self-centred, angry literary giants with whom he is usually compared, and discerned the birth of social issues which have grown world-wide since his day.” (For “Slavophil nationalists vs admirers of the west” read “English nostalgists vs those who hanker after an ideal of ‘Europe’”: the bitter conflicts are still with us!)

I wonder though how much Turgenev is read now, at least here in the England he visited a number of times and which used to much admire him. Is even the great Fathers and Sons still on people’s ‘must read’ lists (their real lists, I mean, recording live intentions!)?  Or has cultural amnesia set in here too?

Turgenev was born on November 9th, in 1818 (October 28th, old style). So it is his bicentenary today — an occasion which seems to be passing here almost entirely unremarked. But I for one will be raising a glass of wine in his memory, as I sit down to reread once more his so evocative, so sad, novella First Love (wonderfully translated, in itself a work of love, by Isaiah Berlin).

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## Lorenzo Lotto

We went to the Members’ preview day at the National Gallery to see the new exhibition, now open (and free!), of portraits by Lorenzo Lotto. A fascinating curatorial talk, and a very engaging show which we enjoyed a great deal. Lotto is interestingly original, and there is real psychological depth to be found here. Definitely worth visiting.

I came away with a copy of the exhibition catalogue — and yes, in a house already overflowing with books, this is an indulgence (as didn’t go unremarked). But it is fascinating reading too. And I’m struck yet again how terrific so many art exhibition catalogues can be: works of art in their own right, beautifully produced, put together with enthusiasm and great learning, and (when we are used to academic book prices) too-temptingly inexpensive.

If only getting round London wasn’t such a depressing business …

## Tarski on truth — a thumbnail sketch

A couple of days ago, someone wanting brief headline-news-for-non-experts asked me about Tarski’s views on truth — in “The Concept of Truth in Formalized Languages” — and about how these ideas related to what was going on in the Vienna Circle at the time when Tarski presented those views in 1935. By coincidence, I was thinking a bit about some related Tarskian issues in redrafting sections of IFL; so I was happy to pause and to try to gather together some (very) brief thoughts. I’m certainly no expert myself; but for whatever it is worth, here is a version of what I said in reply, in case anyone else might find the rather arm-waving remarks interesting/helpful.

The Tractatus — which of course much influenced central figures in the Circle —  offers what looks like a correspondence theory of truth, with true propositions picturing facts. However, the very notion of “facts” seems metaphysically loaded in a way that the positivists were deeply unhappy about; so Neurath, for example, argues that sentences can only be compared with other sentences, coming close (it might seem) to an outright coherence theory of truth. A very interesting and accessible thing to read here, revealing something of the state-of-play pre-Tarski in debates between Neurath, Carnap and Schlick, is Carl Hempel’s short paper “On the Logical Positivist’s Theory of Truth” in Analysis 1935.  So yes, an obvious question is now: how does Tarski — presenting his ideas in 1935 —  fit into the story?

In fact, Tarski is coming from outside these inner-Circle debates. His problematic and his approach arise from inside the Polish philosophy/logic tradition.  Three themes worth remarking on:

1. From Leśniewski, Tarski acquires a scepticism about naive treatments of semantic ideas, and the idea that the notion of truth applied without restraint in natural language contexts will inevitably entangle us with the Liar paradox.
2. From Kotarbiński, Tarski gets the two thoughts (a) that the classic idea that to be true is to correspond with reality can come to no more than this:  the Copernican theory is true — or  the assertion that the earth goes round the sun is true — just in case the earth goes round the sun (K’s example), and so on through other cases.  But also (b) this weak understanding of what correspondence amounts to doesn’t mean that the notion of truth is redundant.
3. From Leśniewski (3) again, Tarski gets a conception of how we should go about pinning down a troublesome concept — not by attempting to give a snappy “analysis”, but by setting out in a tidily organized deductive system or “theory” systematizing a whole body of intuitive truths in which that concept is deployed, in enough detail to thereby fix what that concept has to be, in order to play the displayed role in the theory. In this sense, the theory as a whole can be said to define the concept in question. (If it is to be elucidatory, we’ll need to be able to grasp this deductively organized theory, which means that, inter alia, it will need to be finite!)

So (1), (2) and (3) set the scene for Tarski’s pivotal paper.  Do note by the way, it is about truth for “formalized” — not “formal” — languages. That’s rather crucial. If by “formal” one means more or less de-interpreted, then Tarski explicitly rules out formal languages from  consideration in his 1935 paper. Now, many mathematical languages are at least partially de-interpreted: thus a group theorist might use “+” for a group operation, without giving any particular intended interpretation. But it is not until much later that Tarski explicitly deals with the model theory of such partially de-interpreted language. In 1935, then, Tarski is concerned with some fully interpreted languages. They are formalized in the sense of having a tidy syntax, lacking structural or lexical ambiguities etc. etc. so we can easily theorise about them. (But it is  a matter of taste and convenience whether these languages introduce symbolic shorthand too — the use of symbols isn’t really of the essence at all, although Tarski’s examples are languages deploying symbolic shorthand.)

As I say, (1), (2) and (3) set the scene. What we find, then, is Tarski continuing to endorse (1), arguing at the beginning of his paper that accepting (2a) and applying it across the board to all ordinary language, where self-reference is possible, leads us inevitably to the liar paradox. If we accept all T-biconditionals, for the moment in the form “p” is true if and only if p, but allow suitably self-referential instances of p, then trouble ensues! So we have to rein in our ambition.

Tarski’s aim then will be to show how a notion of truth can consistently be applied to limited fragments of language which lack troublesome self-reference and lack a truth-predicate. So now we will have to distinguish the limited fragment we are (at any point) theorising about from the ambient language in which we are doing the theorizing. Hence the introduction of the distinction between the language which is the current object of investigation and the metalanguage in which we are going to be doing the investigation.

So how is Tarski going to do the investigation? Exactly in accord with (3). To pin down the notion of truth for sentences in the language L under investigation Tarski is not (he explicitly says he is not) going to try for an analysis or philosopher’s-style definition. He aims for a tidily organized deductive system or “theory” (in a suitable metalanguage) systematizing a whole body of intuitive truths in which the metalanguistic concept of true-sentence-of-L is deployed. Which intuitive truths? As in (2a) — i.e. what we now call T-sentences of the shape “S is true if and only if p” where “S” denotes a sentence of L, and “p” stands in for a sentence of our metalanguage which gives the content of S (except that Tarski symbolizes the theory in the metalanguage too, but that’s not of the essence either). Note, what makes giving a tidy deductive theory of this kind feasible is that L is itself sufficiently tidily organized — formalized, as we said. In this way, we get to define true-in-L — define, that is, in the rather distinctive sense of (3).

For any interesting L, there will be infinitely many T-sentences for our finite truth-theory/truth-definition to entail, so the theory will have to proceed by paying attention to the structure of L-sentences (it will proceed by recursion, as we say, on the structure). The novel work done by Tarski is to show us how to do the recursion in the case of languages L with quantificational structure — with the recursion done not in terms of the notion of truth, but in terms of a notion of true-of (or rather its converse, satisfaction). Note though that it isn’t that Tarski is helping himself to the notion of satisfaction and taking that as already understood: rather the whole theory is now to be taken as defining satisfaction-in-L (of which truth-in-L is a special case).

So how did all this go down with the Vienna Circle? Some, e.g. Neurath, thought it just didn’t help with what they were worried about — for in the end satisfaction is a word/world relation which must be just as suspect as the classical notion of truth we started off with. But others thought this was point-missing. Tarski showed how we can happily talk about truth in what we would now think of as a deflationist spirit (at least for tamely well-behaved languages L), telling us about truth without getting embroiled in facts, structural correspondence with the facts, etc., coherence, or ideas of that ilk. It’s not so much a rival theory of truth, but a pinning down of what the notion of truth does for us which shows — as the redundancy theory in a cruder way aims to show — that we don’t  need a theory of the traditional sort. That debate about the significance of theories like Tarski’s continues …

Posted in Logic | 1 Comment

## IFL2: Propositional truth trees

My favourite new encounter from among my recent late-night reading has been with Jane Gardam’s 1985 novel Crusoe’s Daughter.

Its metaphorically cast away heroine Polly Flint is talking of novels (she is devoted to Defoe) when she remarks “Form is determined by hard secret work — in a notebook and in the subconscious and in the head.” What applies to writing novels applies to logic books too — including the bit about the subconscious. Even when your head says that the notebook drafts of some section is fine, your subconscious can remained troubled, unable to settle and remain content with what you’ve written. And then, from somewhere — without it seems conscious reflection — you are struck by how to resolve the nagging worries, and can move on.

Anyway, whatever the processes involved, I have been re-revising the revisions of the chapters on propositional trees in An Introduction to Formal Logic. (Ignore my recent wobbles about whether to drop trees in the new edition — my subconscious just couldn’t rest hppily with that!) So, after an introductory Interlude, there are now three short chapters, significantly shorter than the four chapters. But I think the result is still a very clear introduction to the truth-tree method.

I should add that tree-rules for biconditionals and examples with biconditionals are a topic for the end-of-chapter Exercises (not in this version).

All comments and/or corrections (either here or to the email address in the watermarked header) are as always most welcome.

## IFL2: the introductory chapters yet again

I’ve been again tidying the first tranche of chapters of the second edition of my Introduction to Formal Logic. This time, in particular, I’ve removed some careless talk about ‘predicates’ which — as was rightly pointed out to me in a discussion of another post — just didn’t cohere with what I said more officially about predicates later. So here’s the latest version:

IFL2, Chapters 1 to 7

For new readers, as I’ve said here before, the headline news is that these really are introductory chapters (general scene-setting before we start work in earnest on formal propositional logic).  So I introduce ideas like: validity, deduction vs induction, showing validity by ‘proofs’, showing invalidity by ‘counterexamples’. I also briefly discuss logical validity in a narrow sense vs deductive validity more generally. A quick look at the Table of Contents should give you a better idea of what these chapters are about.

Hopefully, the presentation is accessible and reasonably user-friendly without talking down to the reader. So this first part of IFL2 should be of interest and of use to any philosophy student about to start a logic course this next term/semester (indeed, they should be of use to any beginning philosopher). Do please spread the word, and do point prospective students to the link!