Those among textbook writers calling the quantifier-symbols ’, ‘’ by themselves the quantifiers include Barwise/Etchemendy (I’m not sure how rigorously they stick to this, though). Tennant also calls e.g. ‘’ a quantifier, and refers to ‘’ as a quantifier prefix.

Those calling the quantifier-symbol-plus-variable the quantifier include Bergmann/Moor/Nelson, Chiswell/Hodges, Guttenplan, Jeffrey, Lemmon, Quine, Simpson, N. Smith, P. Smith, Teller, and Thomason. (Lemmon and Quine of course use the old notation ‘’ for the universal quantifier.) Van Dalen starts by referring to ‘’ as the quantifier, but slips later into referring to ‘’ as the quantifiers.

It’s clear what the majority practice is. Why not just go with it?

Modern practice is to parse ‘’ as ‘’ applied to ‘’. However Frege (I’m reading through Dummett’s eyes, but I think this is right, mutating the necessary mutanda to allow for differences in notation) parses this as the operator we might temporarily symbolize ‘’ applied to ‘’.

To explain: Frege discerns in ‘’ the complex predicate ‘’ (what you get by starting from ‘’ and removing the name). Generalizing involves applying an operator to this complex predicate (it really is an ‘open sentence’ not in the sense of containing free variables but in containing gaps — it is unsaturated). Another way of putting it: for a Fregean, quantifying in is a single operation of taking something of syntactic category *s/n*, and forming a sentence by applying a single operation of category *s/(s/n)*. This quantifying operator is expressed by filling-the-gaps-with-a-variable-and-prefixing-by- ‘’ in one go, so to speak. The semantically significally widget here is thus ‘’. Yes, within that, ‘’ is a semantically significant part (it tells us which kind of quantification is being done). But — the Fregean story will go — ‘’ is *not *a semantically significant unit.

So, whether you think ‘’ is worthy of being called the universal quantifier is actually *not* such a trivial matter after all. For is ‘’ a semantically significant unit? You might think that the true believing Fregean protests about this sort of thing too much. I’d disagree — but at any rate the underlying issue is surely not *just* to be waved away by unargued terminological fiat.

So to make up for that, I bought a copy of her new Chopin CD, which I have been listening to repeatedly over the last few days. Now, I should say that I’m not the greatest Chopin devotee, and indeed the only recording of his music that I have returned to at all frequently in recent years has been Maria João Pires’s utterly wonderful performances of the Nocturnes. So I’m hardly in a position to give a very nuanced response to this new disc. But I am loving it.

Ivana Gavrić plays groups of Chopin’s earlier Mazurkas, seperated by a couple of Preludes, a Nocturne and the Berceuse, which makes for something like a concert programme (you should listen up to the Berceuse — which is quite hauntingly played, her left hand rocking the cradle in a way that somehow catches at the heart — and then take an interval!). Some of the Mazurkas are very familiar, but many are (as good as) new to me. The Gramophone reviewer wanted Gavrić to perhaps play with more abandon — but no, her unshowy, undeclamatory, playing seems just entirely appropriate to the scale and atmosphere of the pieces, often tinged with melacholy as they are. She is across the room from a group of you, friends and family perhaps, rather than performing to a concert hall. And repeated listenings reveal the subtle gestures and changes in tone she uses to shape the dances; these are wonderfully thought-through performances. Listen to the opening Mazurka in C sharp minor, Op. 6 no.2 – whole CD is available on Apple Music – and you will be captivated.

]]>Take the most trite of examples. It’s agreed that if *P* then *R*. I need to convince someone that if *P* and *Q* then *R* (I’m dealing with a dullard here!). I patiently spell it out:

We are given if

PthenR. So just suppose thatPandQ. Then of coursePwill be true, and hence (given what we agreed)R. So our supposition leads toR, OK? – hence as I said, ifPandQthenR.

You want that set out more formally? Here goes, indenting sub-proofs Fitch-style!

Job done!

Except that in many official versions of Fitch-style proofs, this derivation is illegitimate. We can’t appeal to the initial premiss at (1) in order to draw the modus ponens inference inside the subproof: rather, we have to first re-iterate the premiss in the subproof, and then appeal to this reiterated conditional in applying modus ponens.

This requirement for explicit reiteration seems to go back to Jaśkowski’s first method of laying out proofs; and it is endorsed by Fitch’s crucial *Symbolic Logic* (1952). Here are some other texts using the same graphical layout as introduced by Fitch, and also requiring inferences in a subproof to appeal only to what’s in that subproof, so that we have to use a reiteration rule in order to pull in anything we need from outside the subproof: Thomason (1970), Simpson (1987/1999), Teller (1989).

Now, such a reiteration rule has to say *what* you can reiterate. For example: you can reiterate items that occur earlier outside the current subproof, so long as they do not appear in subproofs that have been closed off (and then, but only then, these reiterated items can be *cited* within the current subproof). But it is less complicated and seems more natural not to require wffs actually to be rewritten. The rule can then simply be: you can cite “from within a subproof, items that occur earlier outside the current subproof, so long as they do not appear in subproofs that have been closed off” — which is exactly how Barwise and Etchemendy (1991) put it. Others not requiring reiteration include Hurley (1982), Klenk (1983) and Gamut (1991). (Bergmann, Moor and Nelson (1980) have a rule they call reiteration, but they don’t require applying a reiteration rule before you can cite earlier wffs from outside a subproof.)

So: why require that you *mus*t, Fitch-style, reiterate-into-subproofs before you can cite e.g. an initial premiss? What do we gain by insisting that we repeat wffs within a subproof before using them in that subproof?

It might be said that, with the repetitions, the subproofs are *proofs* (if we treat the reiterated wffs as new premisses). But that’s not quite right as in a Fitch-style proof it is required that the premisses be listed up front, not entered ambulando. So we’d have to re-arrange subproofs with the new supposition and the reiterated wffs all up front: and no one insists that we do that.

Anyway: at the moment I’m not seeing any compelling advantage for going with Fitch/Thomason/Simpson/Teller rather than with Barwise and Etchemendy and many others. Going the second way keeps things snappier yet doesn’t seem any more likely to engender student mistakes, does it? But has anyone any arguments for sticking with a reiteration requirement?

]]>In the second post, I gave a Fregean reason for inclining to syntactically marking parameters (the thought being we should syntactically mark important differences in semantic role).

I certainly don’t want to go with van Dalen and use free variables for this role, for the very basic reason that I simply want to avoid using free variables in an elementary text. Back to Begriffsschrift! By my Fregean lights, a formal language quantifier should be thought as ‘’, the whole being a slot-filler operating on a gappy predicate. The artifice of parsing a quantified wff as ‘’ followed by a complete wff which has a free variable ‘’ when stand-alone just obscures Frege’s insight into the nature of quantification. (And before you object that Frege has italic letter variables that look like free variables, remember that they aren’t: the italic letter wffs are just introduced as abbreviations for corresponding (universally) quantified wffs where the quantifier is given maximal scope — so for Frege, the ‘free’ variable wffs are explained in terms of quantified wffs, rather than vice versa.)

It is at least less misleading to use name-expressions from our formal language in the parametric role — relying on the similarities between, so to speak, permanent names and temporary names. But now note that there can be a clash here with the line we take on the first choice-point. If we say that there is one big language of FOL with an indefinite supply of names (only some of which get a fixed interpretation) then fine, that leaves us plenty of names to use a parameters. But suppose we take the many-language line, and think in terms of a language as having a particular fixed number of names (maybe zero, as in the basic language of first-order set theory!). Then there may not be enough names to recruit for parametric use in arbitrarily complex arguments. Compare: at least we don’t run out of variables, which is why van Dalen, and Chiswell/Hodges, who take the many-language line, have to recruit variables to use as parameters.

So I think that many-language logicians who want to use natural deduction with its parametric reasoning have very good reasons for introducing a distinguished class of symbols for parametric use. For on most many-language stories there is a fixed number, maybe zero, of proper names in the language, so we won’t have enough parameters if we stick to names. While the Fregean precept about marking important differences counts strongly against re-using variables-for-quantifiers in a quite different use as temporary names.

(What about Barwise and Etchemendy who I have down in my first post as both taking a many-language view and as using names as parameters? Well, to be honest, I find their position not entirely clear, but I *think* they conceive of a language as having some distinguished names, with an intended interpretation, though perhaps zero as in the language of set theory, but also having a supply of further names that *can* be used ad hoc, e.g. to name particular sets as on p. 16, such names also being available for use as parameters.)

But should we go for many languages or one all-purpose-language? It is notable that in my list of texts in the first of these posts, the more mathematical authors go for many languages. That is no accident surely; when in more advanced logical work we regiment mathematical theories, it is very natural to think of these various theories having their own languages, the language of set theory, the language of first-order arithmetic, the language of category theory, and so on. This conforms with how mathematicians tend to speak and think. So this gives us a serious reason to start as we mean to go on, thinking in terms of their being many first-order languages, with their distinct signatures and particular non-logical vocabularies. Another reason for taking the many-languages line from the outset is the sheer inelegance of the one all-purpose-language picture. We build up a language with infinite non-logical vocabularies of every possible arity; we then throw away again almost all the complexity either by making interpretations partial or by in principle interpreting everything and then showing almost the interpretative work is redundant. It is difficult to find that very aesthetically pleasing.

OK. None of all that is decisive. None of the logic books which make different choices are thereby bad books! Still, I think there are reasonably weighty reasons – rather more than mere considerations of taste – to go for many languages, and (as we’ve just seen, not unconnectedly) for giving a language a class of symbols to serve as parameters (‘arbitrary names’ or whatever your favourite label is) which is syntactically distinct from names (proper names, individual constants) and variables. So that’s what I’ll be doing in the newly added natural deduction chapters in the second edition of *IFL*.

*To be continued*

‘Distinguishers’ are perhaps slightly over-represented in our sample: in Pelletier and Hazen’s table of 50 texts, only 11 are reported as syntactically distinguishing “arbitrary names”. In a way, this low proportion of distinguishers is a bit surprising, for two reasons. First, there’s the weight of authority: Gentzen himself gives distinguished symbols for bound variables and variables occurring free in the role of parameters. (True, Gentzen was not the only begetter of natural deduction, and Jaśkowski does not distinguish real variables from symbols used as parameters.) But second, and much more importantly, Gentzen’s symbolic distinction marks a real difference in the role of symbols: and where we can easily do so we should surely follow (as I put it before) the Fregean precept that important differences in semantic role should be perspicuously marked in our symbolism.

True, we can recycle ordinary names or ordinary variables to play the new role of parameters (or ‘temporary names’ or whatever you want to call them). This gives us economy of notation. But Frege would insist that the desire for economy in one’s primitive symbolism should be trumped by the desire for perspicuity (on marking another distinction, he talks of “[his] endeavour to have every objective distinction reflected in symbolism”). I think we should aim, within reason, to be with Frege here.

An oddity to remark on. There are some who mark only *one* of the special use of symbols in (UI) inferences and in (EE) inferences. Thus Teller, we noted, puts hats on what he calls ‘arbitrary occurrences’ of a name in the course of a (UI) proof; but he doesn’t syntactically distinguish the use of a name as a temporary name instantiating an existential quantifier in the course of a (EE) proof. Suppes in his much used *Introduction to Logic* (1957) does things the other way about. The same variables that appear bound appear free in his (UI) inferences; but instantiations of existential quantifiers are done using special symbols as what he calls ‘ambiguous names’. Of course, both proof systems work fine! – but I’m not sure why you would want to leave it to context to mark one distinction in the use of symbols and syntactically mark the other. (Lemmon, whose *Beginning Logic *(1965) taught generations of UK students natural deduction in a Suppes-style layout, uses distinguished symbols as what he calls ‘arbitrary names’ in both (UI) and (EE) inferences.)

*To be continued.*

I’m interested here, and in the next couple of posts, about a subset of issues concerning the *language(s)* we are going to be using:

- Do we talk of
*one*language or*many*languages (different languages for different interesting first-order theories, and different temporary ad hoc languages for use when we are regimenting different arguments)? - In explaining the semantics of quantifiers, do we give a Tarski-style account in terms of satisfaction where we assign sequences of objects to sequences of
*variables*? Or do we make use of some new kind of*supplementary names*for objects in the domain, so that e.g. ‘’ comes out true on an interpretation if ‘’ is true for some extra name ‘*a’*on some perhaps extended interpretation? - What do we use as
*parameters*in natural deduction? Names? Variables? A new, syntactically distinguished, sort of symbol?

Here, the first issue is to some extent a matter of presentational style, but (as we will note in a subsequent post) the details of what you say can make a difference on more substantive points. The second issue is probably familiar enough and we needn’t delay over it just now. To elucidate the last issue – the only one of the three dealt with explicitly by Pelletier – consider a natural deduction proof to warrant the inference ‘Everyone loves pizza. Anyone who loves pizza loves ice-cream. So everyone loves ice-cream’. In skeletal form the proof goes like this, ending with a step of universal quantifier introduction (add to taste extra bells and whistles, such as Fitch-style indentation, and even rearrange into a tree):

Here, I’ve used the intentionally unusual ‘*q’*. But what would you prefer to write in its place? Some would use e.g. ‘*x’*, picked from the list of official variables for the relevant language in play. Some would use e.g. ‘*a’*, picked from the list of official individual constants (names) for the relevant language. However, we might note that ‘*q’* is neither being used as a variable tied to a prefixed quantifier, nor used as a genuine constant naming a determinate thing, but in a third way. It is, as they say, being used as a parameter. So we might wonder whether we should somehow mark syntactically when a symbol is being used as a parameter (following the Fregean precept that important semantic differences should be perspicuously marked in our symbolism).

It is interesting to see how different authors of logic textbooks for students which cover natural deduction handle our three choice-points. So let’s take a number of texts in turn, in the chronological order of editions that I can reach from my shelves. (I’ll return to mention some other books, including that by Suppes, in the next post – but for the moment I am considering just a handful of books, ones that I have learnt from over the years, and/or ones that that might still be recommended to students at least as supplementary reading.)

*Richmond Thomason, Symbolic Logic (Macmillan, 1970) *is, looking back, a strange mixture of a book, extremely lucid in some respects, but making pretty heavy weather of some of the more technical presentations. The main reason I am mentioning it here is that Thomason is emphatic about distinguishing parameters from names and variables. But otherwise, this is a many-language, almost-Tarski book (almost, because it considers valuations of all-the-parameters-at-once, rather than assigning values to all-the-variables-at-once).

*Neil Tennant, Natural Logic (Edinburgh UP, 1978/1990) *talks of *the *language of first-order logic. In giving a model, distinguished names are assigned objects, but later “undistinguished names” are employed as names of arbitrary objects considered in the course of a proof — i.e. they are employed as parameters. Free variables feature not in proofs but in the syntactic story about how wffs are built up, and the associated Tarski-style semantic story for wffs.

*Paul Teller, A Modern Formal Logic Primer (Prentice-Hall 1989) *seems to be a one-language book. At least, formation rules for a single language are introduced for sentential logic in Vol I, and Vol II carries on in the same spirit. The semantics is initially given by assuming that every object has a name. Then, late on, that assumption is dropped, but the resulting semantics is of the non-Tarskian kind I’ve indicated. Names are used in natural deduction proofs using the (UI) rule as above, but are marked with a ‘hat’ in this role, so these parameters are syntactically marked. But names used in their (admittedly slightly different) use as parameters in existential quantifier elimination (EE) arguments are *not* marked.

*Jon Barwise and John Etchemendy, The Language of First-Order Logic* (CSLI, 2nd ed. 1991) talks initially of FOL, *the* language of first-order logic, but soon starts talking of first-order languages, plural. On semantics, there is initially a version of the idea that we extend a first-order language with additional names, and ‘*’* is true on interpretation *I* so long as ‘*’* is true on some extension of *I *which is just like *I* except in what it assigns to the new name ‘*a’*. But later, an official Tarskian story is given. Wffs with free variables are allowed, but don’t feature in proofs where parameters are all constants. But when parametric, the constants are first introduced in a ‘box’ (boxing a constant *c* is “the formal analog of the English phrase ‘Let *c* denote an arbitrary object’”). But this syntactic marking of a parameter does not – unlike Teller’s ‘hats’ – persist.

*Merrie Bergmann, James Moor, Jack Nelson, The Logic Book (McGraw-Hill, 3rd ed. 1998) *introduces a single, catch-all language PL. Initially, just a very informal semantics is given, enough for most purposes. But when a formal account is eventually given, it is Tarskian. The authors allow free variables in wffs. But their natural deduction system is again presented as a system for deriving sentences from sentences, and in the course of proofs it is names (not free variables) which are used parametrically.

*R.L. Simpson, Essentials of Symbolic Logic (Broadview 1999/2008) *is perhaps the most elementary of the books mentioned in this list – it talks of *the* language of symbolic logic but we can’t answer the question of what style of formal semantics Simpson prefers as none is given. The book is worthy of mention here, however, because of the particular clarity of its ND system, and the way it handles the quantifier rules. It uses distinguished parameters in both (UI) and (EE) proofs.

*Dirk van Dalen, Logic and Structure (Springer, 4th ed. 2004) *allows for many languages, different languages of different signatures. The book’s semantics is non-Tarskian, and goes via adding extra constant symbols, one for every element of the domain, and then ‘*’* is true on interpretation *I* so long as some ‘*’* is true (where ‘*’* is one of the extra constants). The natural deduction system uses free variables in a parametric role.

*Ian Chiswell and Wilfrid Hodges, Mathematical Logic *(OUP, 2007) allows many languages with different signatures. The authors give a straight Tarskian semantics. Their natural deduction system allows wffs with free variables, and allows both variables and constants to be used parametrically (what matters, of course, is that the parametrically used terms don’t appear in premisses which the reasoning relies on, etc.).

*Nick Smith, Logic: The Laws of Truth (Princeton, 2012) *goes for one all-encompassing language of FOL, with unlimited numbers of predicates of every arity etc. When we actually use the language, we’ll only be interested in a tiny fragment of the language, and interpretations need only be partial. Semantically, Smith is non-Tarskian: ‘*’* is true on interpretation *I* so long as ‘*’* is true on some extension of *I *which is just like *I* except in what it assigns to the new name ‘*a’*. Smith’s main proof system is a tableau system; but he also describes a Fitch-style system and other ND systems. In these cases, names are used as parameters. (The one all-encompassing language of FOL will give us an unending supply.) Smith allows free variables in wffs in his syntax — but these wffs don’t get a starring role in his proof systems.

So in summary we have, I think, the following (correct me if I have mis-assigned anyone, and let me know about other comparable-level texts which cover natural deduction if they are interestingly different):

Book | One/ Many? | Tarksi/non-Tarski? | Parameters? |
---|---|---|---|

Thomason | Many | Tarski | Distinguished |

Tennant | One | Tarski | Names |

Teller | One | Non-Tarski | Semi-Distinguished |

Barwise & Etchemendy | Many | Both | Names |

Bergmann, et al. | One | Tarski | Names |

Simpson | One | — | Distinguished |

van Dalen | Many | Non-Tarski | Variables |

Chiswell and Hodges | Many | Tarski | Both |

Smith | One | Non-Tarski | Names |

The obvious question is: is there a principled reason (or at least a reason of accessibility-to-students) for preferring one selection of options over any other, or is it a matter of taste which way we go?

*Revised to take account of first two comments. To be continued.*

I previously went from T/F-signed trees to T-signed trees to unsigned trees in a messy way, and that’s gone (so we now have just one sort of tree, initially T-signed and then the Ts are dropped). I previously did trees first for the connectives and/or/not and then added the material conditional and biconditional in a separate chapter; now I just treat the first four connectives in one chapter and the biconditional is relegated to exercises. There are other tidyings, and I’ve dropped the stuff about saturated sets (destined to become an online extra).

So I hope, in short, that the result reads more cleanly and (even) more accessibly. Here it is. All comments most gratefully received!

]]>In the last few days, I’ve found been tinkering with them yet again. But this time, I have found myself feeling happier and wanting to re-write considerably less than on previous iterations of the process. So I hope that these chapters are now, at long last, in a decently stable state. But I would, it goes without saying, still *very* much welcome feedback — from colleagues but also especially from students (or their TAs) commenting on readability/clarity.

~~Here then are the chapters~~, just 25 pages. Comments to the address in the header on each page very gratefully received. (Bear in mind, though, that this is an intro book aimed at beginning philosophers, which is why it goes more slowly and more discursively than would be appropriate e.g. for a more advanced and/or more mathsy text.)

[Added] The previously linked chapters have now been superseded. Let me know if you want to see/comment on the latest updated version.

]]>- Robert Harris
*An Officer and a Spy*. I was, to be honest, pretty disappointed by Harris’s latest,*Conclave,*whose flat-footed plot twist I found so implausible as to even make me a bit annoyed I’d spent the time reading the book. A strange lapse of form. By contrast, this fictional recounting of the Dreyfus affair is both un-put-downable and satisfying. - Sarah Dunant,
*In the Company of the Courtesan*. Mrs Logic Matters has been recommending for ages that I try Dunant’s books set in Renaissance Italy: and she’s absolutely right. This is the first in the series, and my first, and just is a terrific read. - Tim Parks,
*Italian Ways.*A serendipitous find in a charity shop, this is notionally about the vagaries of various railway journeys round Italy that Tim Parks took. There’s a lot of sharp though mostly affectionate observation from a long-time resident in Italy, written with a novelist’s way with words. As you rattle through the book, you learn some history, and (if you know the country at all) will nod with pleased recognition at Italian foibles. An unexpected delight.

Any suggested holiday reads you’d like to share?

*Added *The day after I posted this, the Sunday Times printed its recommendations for summer holiday reading. They mention three other books I’ve particularly admired in recent months and which you certainly should read sometime, Rose Tremain’s *The Gustav Sonata*, Julian Barnes’s *The Noise of Time *and* *Sarah Perry’s *The Essex Serpent. *But I’m not sure that either of the first two have quite the page-turning lightness of touch you want for holiday relaxation — and while *The Essex Serpent** *is absolutely gripping, surely you should keep it to a more Dickensian time of year, to read round the fire as the nights draw in …

But then, despite my all best resolutions, the lead up to the General Election here and its aftermath have all been quite ridiculously distracting. No wonder that the writing for the second edition of my intro logic text has been going corresponding slowly. Though in some ways the slowness has been rather enjoyable: for it gives me space to I feel I am still learning and getting clearer about various issues as I go along.

I suppose you might think that, having taught the stuff for forty years, I really would have achieved settled views by now on the logical basics and on how best to expound them. But you know how it is: in the hurly burly of teaching, especially when your mind is focused on more challenging courses, you let yourself get away with conventional almost-truths, or park concerns for when you have time to return to think harder about them. And then somehow another teaching year comes round before you have had the time to settle down to do the re-thinking, and the extensive re-writing of overheads, that you promised yourself. You tinker and satisfice for another year. Well, retirement with no more teaching duties brings me the time to try to do better. We’ll have to see whether the time is being well-spent!

But just today, I got CUP’s happy approval of my suggested cover image for the second edition. You’ll have to wait to find out what I’ve chosen: I’ll only say that it’s an abstract painting by a well-known early-twentieth century artist. I now merely have to get the following four hundred pages into good enough shape not to disgrace the cover …

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