### How to mention expressions

Philosophers sometime endorse strict conventions for referring to linguistic expressions, then go ahead and knowingly violate those conventions. We'll talk about that, but first, let's learn some of the conventions.

Earlier I invited you to think about a collection of primitives that we could call "letters" or "words", and we talked about how we might build up strings or finite sequences from those primitives.

The string that we've been writing as $$\varepsilon\smalltriright\mathcal{A}\smalltriright\mathcal{B}$$ might also be written, more conveniently, as $$\mathcal{A}\mathcal{B}$$. We just should keep track of when we're talking about the letter $$\mathcal{A}$$, and when we're talking about the length-1 string $$\mathcal{A}$$ (more officially known as $$\varepsilon\smalltriright\mathcal{A}$$). And we should keep track of when we're talking about a string $$\mathcal{A}\mathcal{B}$$ that's made up of two letters, and when we're instead talking about a string made up of a single letter written $$\mathcal{A}\mathcal{B}$$. (And when we're talking about that single letter itself.) Perhaps it seems funny to think of $$\mathcal{A}\mathcal{B}$$ as a single letter, but remember that this term "letter" means just "atomic unit used to build up these sequences," and might in some contexts better be called a "word."

You can tell the difference between when I'm talking about the string or letter $$\mathcal{A}$$, and when I'm instead starting a sentence with the capitalized English word "A", because I'm using this special font for the $$\mathcal{A}$$. Instead of a special font, some authors instead use italics. Oftentimes, authors won't use either of those conventions, but will instead use quotation marks, like this:

A letter I've often discussed is the letter "A".

With that convention, it would be complete mistake to write:

"A" letter I've often discussed is the letter "A".

or:

"A" letter I've often discussed is the letter A.

The idea here is that there will be one expression, such as the one I'll display here in italics: $$Astoria$$. Then there will be a second expression, which is made by surrounding the first expression with quotation marks. I'll also display that here in italics: “$$Astoria$$”. As a word of English, the first expression designates a region in Queens where a number of people live. The second expression on the other hand, doesn't designate any region in Queens. Instead it designates the first expression. As it's sometimes put, surrounding an expression in quotation marks creates a term that refers to that very expression. (The preceding italics was for emphasis rather than naming a long English expression. I'll trust you can tell the difference from context. If you want to look closely, you should also notice that there's a small difference in the typeface used there and the typeface used for $$Astoria$$. At least, it comes out different on my browser.)

Now sometimes we're invited to believe that this is how quotation marks really work in English, or that it's a good idealization or first approximation to that. If that were correct, then sentences like this would be ungrammatical:

Quine said that "quotation has a certain anomalous feature."

She did say that she "decorated" their house.

Because on the story about quotation marks we've just heard, the expression “$$decorated$$” is a singular term that designates linguistic expressions, so it's of the same syntactic or grammatical category as a name would be. Let's agree to use the name $$Doug$$ for the English expression $$decorated$$. That is, I can correctly say:

Doug is an English expression starting with the letter "d".

just as I can say:

"Astoria" is not a region in Queens; it's an English expression starting with the letter "A".

Alright, then, so we should be able to substitute $$Doug$$ in the sentence we saw before and get:

She did say that she Doug their house.

Arguably this should have the same meaning as the original, but that doesn't matter. Even if this replacement changes the meaning in some subtle way, it shouldn't affect the grammaticality. We just replaced one expression with another expression that's supposed to belong to the same syntactic or grammatical category. And yet we can see that the sentence we end up with is ill-formed. It doesn't mean anything. It can't mean anything; it's not a grammatical sentence of English. Yet the original sentence:

She did say that she "decorated" their house.

seems both to be grammatical and to have a clear meaning. This is all evidence that quotation marks in English don't work the way the theory we described proposes.

How then do they work? This is a hard and complex question. Here are just a few ideas to help organize your thoughts. If you want to learn more about this, you can go read some articles. The 2001 paper by Recanati might be a good recent paper to start with.

One linguistic activity we engage in is using an expression with its ordinary meaning, as I use the expression "Astoria" (or $$Astoria$$) to refer to a region in Queens. Usually we'll be talking about completely extralinguistic matters when we do this, because most expressions aren't about language. (There are of course some exceptions, like the expressions "English" and "verb".)

Another linguistic activity we engage in is mentioning an expression --- that is, in some way communicating something about the expression itself, instead of or in addition to whatever the expression ordinarily means.

Here's an example of this contrast from Wikipedia:

Cheese is derived from milk.

"Cheese" is derived from the Old English word "cyse."

Sometimes both activities can take place at once, as when I say:

DUMBO was so-called in the hope that it would keep the yuppies from buying up all the lofts.

Here I'm using the expression "DUMBO" (or $$DUMBO$$) to refer to a region in Brooklyn, but at the same time I'm saying something about the expression "DUMBO" itself, and why it was selected as a name for that region.

Quotation marks in natural language are part of a body of conventions about how to shift between merely using an expression and mentioning that expression (which may or may not involve suspending the ordinary use). Exactly how those conventions work is a matter that seems to be complex, and is anyway much disputed. Moreover, it is possible to mention expressions without using quotation marks, as I did in the sentence $$DUMBO~was~so\textit{-}called\dots$$

This is relevant to our studies because we will very much need to mention expressions, and we should have some shared conventions for doing so. We don't necessarily need to use quotation marks though. If we do use quotation marks, we should think of it as a specific formal convention, and bracket the question of how well it matches quotation in ordinary English.

When we're just dealing with strings built out of letters I can write with a single character, I will sometimes continue using the special font, like this: $$\mathcal{A}\mathcal{B}$$, that being the length-2 word more officially written as $$\varepsilon\smalltriright\mathcal{A}\smalltriright\mathcal{B}$$. I think that has some advantages over quotation marks. If we're dealing with strings built out of "letters" that are really better thought of as "words", I will always use a different typeface: $$Astoria~is~in~Queens$$. One might propose a convention where that's to be understood as:

$$\varepsilon\smalltriright{A}\smalltriright{s}\smalltriright{t}\smalltriright{o}\smalltriright{r}\smalltriright{i}\smalltriright{o}\smalltriright\latin{space}\smalltriright{i}\smalltriright{s}\smalltriright\dots$$

But that's not how I want to think of it. I'm instead thinking of $$Astoria~is~in~Queens$$ as a length-4 string:

$$\varepsilon\smalltriright{Astoria}\smalltriright{is}\smalltriright{in}\smalltriright{Queens}$$

The spaces aren't themselves included; they're just part of the notation I'm using to designate that length-4 string.

The example from Wikipedia that we saw before:

"Cheese" is derived from the Old English word "cyse."

would with my convention be written as:

$$Cheese$$ is derived from the Old English word $$cyse$$.

At some points I'll write things like this:

$$Fa \horseshoe \neg Gab$$

And that string contains seven words: $$F, a, \horseshoe, \neg, G, a,\latin{ and }b$$. I'll rely on your good judgment to figure out that $$Fa$$ is a string containing two words, a predicate constant and a term constant, rather than a single word.

If I ever want to talk, not about a specific string, but instead about a schema or template that various specific strings might match, then I'll use Greek letters to represent the "holes" that should be filled in with some specific smaller string to give us a particular instance of the schema. So for example, here is a string-schema:

$$Astoria~is~\phi~Queens$$

and here are some particular instances of that schema:

$$Astoria~is~in~Queens$$

$$Astoria~is~much~smaller~than~Queens$$

Sometimes philosophers do the same thing with quotation marks. They'll say that:

"Astoria is $$\phi$$ Queens."

is a string-schema, not a string that contains a Greek letter in it. Some philosophers don't like that; they think they might sometimes want to refer to strings with Greek letters in them, or they think they've explained quotation marks in a way that's not compatible with this usage. So they'll instead express the string-schema using "corner-quotes" like this:

⌜Astoria is $$\phi$$ Queens.⌝

When a corner-quoted expression contains no Greek letters, it's supposed to mean the same as the regular-quoted expression would mean. But with the Greek letters included, as here, it's supposed to mean the type of expression you can get by filling in the blank _____ in the following with some expression designating a string:

"Astoria is "$$\curl$$ _____ $$\curl$$" Queens."

This is the same string-schema I meant with my:

$$Astoria~is~\phi~Queens$$

Some philosophers who use italics in the way I do use bold where I've used Greek. With that convention:

$$Astoria~is~prep~Queens.$$

would be a specific ungrammatical English sentence containing the expression $$prep$$, but:

$$Astoria~is~\boldsymbol{prep}~Queens.$$

would instead be a string-schema, specific instances of which are formed by filling in some specific smaller string in place of $$\boldsymbol{prep}$$.

HOMEWORK EXERCISES:

1. Read pp. 5-6 of Sider's book, and then do the exercises he provides at the end of that passage.

### Why would anyone violate such clear, rigorous conventions?

Well, have you ever written anything like this?

Although John hopes that P, P implies Q, and John doesn't hope that Q.

If we look at the first occurrence of the capital letter $$P$$, presumably you're using it as a schematic variable, like I was using $$\phi$$ earlier. For the clause beginning with $$Although$$ to be grammatical, instances of this schema would need to fill in some sentence for $$P$$, like this:

Although John hopes that the thief is caught, P implies Q, and John doesn't hope that Q.

But our substitution should be consistent: we have to fill in for the other occurrence of $$P$$, too, getting:

Although John hopes that the thief is caught, the thief is caught implies Q, and John doesn't hope that Q.

and this doesn't sound that great. For the second occurrence of $$P$$, it'd be better to fill in a singular term rather than a sentence, like this:

Although John hopes that P, the sentence (or proposition) that the thief is caught implies Q, and John doesn't hope that Q.

Here, we are essentially treating $$P$$ as a name of a sentence (or proposition), rather than a schematic variable for a sentence.

But then, again, we should fill in consistently, getting:

Although John hopes that the sentence that the thief is caught, the sentence that the thief is caught implies Q, and John doesn't hope that Q.

and that doesn't sound too good either. (Nor does substituting $$proposition$$ for $$sentence$$ help.)

Perhaps, if you're sensitive to these difficulties, you might have written, not:

Although John hopes that P, P implies Q, and John doesn't hope that Q.

Although John hopes that P, the proposition that P implies the proposition that Q, and John doesn't hope that Q.

That's a bit verbose, but not too bad. Then you could treat $$P$$ and $$Q$$ consistently as schematic sentences (like my Greek letters).

Still, you'll find if you try to write like that consistently, it'll prove to be exhausting and annoying, and the results will be harder to read as well as harder to write. This is why you find even logic books resorting to various finesses, or explaining that they'll be "sloppy" about use-versus-mention or quotation when their intended meaning is clear from context.

In my view, what is important is not for you to mechanically follow any one precise set of rules. What is important is for you to learn the skill of keeping track of what's going on. When are we (i) talking about some expressions, in abstraction from any meaning they may have; when are we (ii) talking about them being interpreted in certain ways (in the sense of their having or being assigned some meaning, not in the sense of some agent trying to figure out what their meaning is); and when are we (iii) using some expressions in the language we theorize with --- our metalanguage, rather than the object language we're theorizing about.

The readings for this week include some selections from Bostock's Intermediate Logic.

I included this in part to present some of the points he was explaining; though some of the topics, like the notion of validity, we will bracket discussion of for a few weeks.

In large part, though, I included it for "meta" reasons. There are some idiosyncracies, but on the whole the selection is very typical for the introductory sections of a logic text. Sider does some of the same things that Bostock does, just in less detail. I want to call your attention to some bits of this: sometimes because I understand things differently, but most often just to observe the conventions that are being introduced. The whole body of conventions: including both the official stipulations and the agreement to often be "sloppy" and not stick to the official stipulations.

• Bostock, Sider, and I all agree to use capital Latin letters as sentence (and later, predicate) constants in the object languages we're studying. We also agree that Greek letters will be schematic variables for (possibly complex) formulas. So this is a schema for one kind of formula in our object language:

$$\phi \vee Q$$

and these are some particular instances of that schema:

$$R \vee Q$$

$$(P \horseshoe Q) \vee Q$$

I also announced a convention to use Greek letters from the start of the alphabet ($$\alpha, \beta, \gamma$$) as schemas for terms rather than formula. In some of the languages we look at, terms might be complex (like the term $$a + b$$). But terms can also be simple, like the term constant $$a$$ or the variable $$x$$. I think Bostock and Sider use slightly different conventions than me here. If I remember right, they tend to use $$\alpha$$ specifically as a schema for term constants (like $$a$$), and Bostock at least uses the Greek letters $$\xi$$ and $$\zeta$$ specifically as schemas for variables. In class, I proposed just using small Latin letters like $$a$$ and $$x$$, but writing them in green when they were schematic, and blue when they were meant to be those specific symbols from our object-language alphabet. Hopefully I can get something like that to work on these web pages too, but if it's too much trouble, maybe we'll have to find a different convention.

• One place where I depart from Bostock (and Sider) is in how to understand the sentence constants themselves (the $$P$$s and $$Q$$s). Bostock treats these not as being real sentences in a formal language, but rather as themselves being schemas for sentences of English, or something like that. See his pp. 4, 6, and 8. Or maybe Bostock is saying that's what it is to be a sentence constant in a formal language. See also Sider p. 3, and middle of his p. 33, and p. 44 near the bottom.

I don't think of it like that. I think of our language of $$P$$s and $$Q$$s as being a language in its own right. It's an artificially-created language, like Esperanto. And it has a much simpler syntax and semantics than most natural languages. We are using this language to mathematically model some structural relations (most notably, "logical consequence") that we think are also had between sentences in English, and also between some of our thoughts. But how exactly the correspondence between English and our thoughts on the one hand, and the mathematical model, on the other, should go is complex and not something we're trying to settle here, or make any assumptions about.

As I said in the first class, I also think it's a complex issue what connection there is between the logical relations we're modeling and notions like good reasoning, epistemic impossibility, and so on.

• Note Bostock's and Sider's relaxed attitude towards formulas like $$P \wedge Q \wedge R$$ and $$P \horseshoe Q$$ --- versus their officially-required $$((P \wedge Q) \wedge R)$$ and $$(P \horseshoe Q)$$.

Also note Bostock's proposal that we assume certain precedence rules, so that $$\wedge$$ gets to consume neighboring formulas before $$\horseshoe$$ does, so that expressions like $$P \wedge Q \horseshoe R$$ are always understood as $$((P \wedge Q) \horseshoe R)$$.

There's nothing wrong with these conventions, but I think there is something unsatisfying about the justifications we're given for them. In fact, it would be possible to "bake" these conventions into our official syntax for the language; so why didn't we do so? Why do we instead get one official syntax and then immediately an informal agreement to routinely violate it in practice (in systematic ways)?

Here's how I'd recommend you think of this. The "informal" conventions that Sider and Bostock propose to follow are in fact very systematic and robust; if they weren't, they probably wouldn't work so well in practice. They could be "baked" into the syntax for the language, and thus elevated to the status of being the official notation. One way to "bake" them in would be to mandate that, say, outermost parentheses are always dropped. Another way would be to permit but not mandate that. It's possible to write formal grammars or syntactic rules for the language which do either of those things. But those grammars would be substantially harder to write, and much more difficult to read, than the grammars that Bostock and Sider actually present. And for most readers, the extra complexity wouldn't have much pedagogical value. It's much easier for you the reader to take in a simpler, fictional or "official" syntax for the language, which doesn't match what we're actually going to use, and then for the author to explain informally how the language we're actually going to use diverges from that simple grammar.

Anyway, that's how I think of it. If you find Bostock's and Sider's explanation of what they're doing clearer or more helpful, that's fine too.

• Bostock introduces the symbol $$\models$$, called the "double turnstile." Perhaps you've seen it before. We will discuss this in detail in a few weeks. There are also the single turnstile $$\proves$$ and a symbol used in the "sequent calculus" deduction system, which is written in a few different ways. We'll write it as $$\Rightarrow$$ (as Sider and Bostock also do). These three symbols have different, but connected, meanings. We'll sort out the differences later. For now we'll just talk about the notational form. As Bostock mentions, on the left hand side of $$\models$$ we should inscribe, in our metalanguage, the name of a set of sentences, like so:

$$\Gamma \models \dots$$

(The $$\dots$$ aren't part of the symbolism. I'm just omitting to specify what comes on the right-hand side.)

However, instead of writing things like:

$$\Gamma \union \Delta \models \dots$$

$$\Gamma \union \set{P} \models \dots$$

$$\set{} \models \dots$$

$$\Gamma, \Delta \models \dots$$

$$\Gamma, P \models \dots$$

$$\models \dots$$

And the latter are in fact the preferred forms (though the earlier forms are also legal).

Bostock also proposes that we let:

$$\dots \models$$

--- that is, with an empty right-hand side --- be shorthand for what I'd write as:

$$\dots \models \bot$$

meaning that the stuff on the left stands in the $$\models$$ relation to the sentence letter literal $$\bot$$ which is always interpreted as being false. Bostock's proposal has some virtues, but I'll stick to using $$\bot$$ instead. What each of these is saying is that there is no interpretation of the language on which all the formulas on the left-hand side of the $$\models$$ are true. We will talk about that later.

• In general, Bostock writes “$$P$$” rather than just relying on the italics of $$P$$ to signal that he's mentioning rather than using the letter $$P$$. But as he says, he often drops the quotation marks, even though on his view they're officially required. For example, I think the correct way to understand the $$\models$$ symbol is one where, using his conventions for how to mention sentences, he'd officially have to write:

$$\dots \models$$$$P$$

He seems to agree (see the bottom of his p. 9). Whereas I will just rely on the italics to indicate that we're mentioning the formula $$P$$, so that we get:

$$\dots \models P$$.

In fact we will almost never use the expressions in the languages we're creating and studying. At one stage in discussion, we'll suppose these expressions do have specific meanings, but we will then go on to think about how to mathematically model variations in their meanings. At that point, we could in principle still agree to let them have such-and-such specific meanings, but in practice, we'll have lost hold of any shared conventions about that. And we have more effective ways to communicate anyway. So as I said, we will almost never use these expressions, and almost always be mentioning them instead. --- Or mentioning schematic forms that they instantiate. Thus, I can also write:

$$\dots \models \phi$$

And this is not itself any specific (metalanguage) claim about the semantic properties of my object language. Instead it's a schematic template for such. Instances of the template will substitute actual object-language formulas in for the $$\phi$$. But then those formulas are themselves being mentioned, not used.

Bostock also wrestles with another way to understand $$\models$$, where the formulas that flank it are understood to be used not mentioned, and $$\models$$ expresses something like the English connective "therefore." I think this is a bad idea for several reasons, and don't recommend it.

• Bostock uses the notation $$\abs{\dots}$$ to mean "the semantic value of $$\dots$$" Sometimes he explicitly indicates which interpretation he's talking about as assigning that semantic value, with a subscript like this: $$\abs{\dots}_I$$. In the particular case where we have a sentential logic, semantic values will just be truth-values.

We'll see a variety of ways to express this idea.

My notation, also used by Partee and more common in linguistics, will be $$\interp{\dots}$$, with subscripts whose details we'll discuss later. Gamut uses that notation too, but only for the semantic values of terms; for formulas he/they use $$\operatorname{V}(\dots)$$. Sider also uses $$\operatorname{V}(\dots)$$ for formulas, and uses $$[\dots]$$ for terms.

In the branch of mathematical logic known as model theory, $$I \models \dots$$ is sometimes used to mean "the semantic value of $$\dots$$ as assigned by interpretation $$I$$." Burgess and Kaplan use this notation. This is a different use of $$\models$$ than is common in other logic settings, and which is used by Bostock, Sider, and me. Sometimes model theorists also use $$\Gamma \models \dots$$ in the same way that Bostock, Sider, and I do, where $$\Gamma$$ is a set of formulas. The different uses are distinguised by what appears on the lhs of the $$\models$$. Very confusing.

• Lastly, Bostock has some remarks on pp. 16-17 about nullary or 0-place functions. In the case of functions into the truth-values (the "Booleans"), there are only two such functions. But in the case of functions into $$\N$$, say, there would be more. We will discuss this idea when we meet.