Wednesday, 29 August 2012

U(1) and Nominalism

In 1987, I accidentally bought the The Joshua Tree, by U2, out of curiosity. A few weeks later, I gave it away!

Anyway, this isn't about U2, it's about U(1) and nominalism. U(1) is the Lie group of rotations $R_{\theta}$ about an axis, parametrized by some angle $\theta \in [0, 2\pi]$. So, you might rotate a coffee cup around a vertical axis through the middle of cup, by a certain angle $\theta$. Indeed, if the cup is quite "symmetrical", then the result always "looks the same": roughly, this is because the set $X$ of points occupied by the material of the coffee cup remains invariant when every point $x \in X$ is rotated by the rotation $R_{\theta}$: in symbols, $R_{\theta}[X] = X$. Each of these rotations is parametrized by an angle $\theta$; the rotations can be composed, there is an "identity" rotation (i.e., angle $= 0$) and each rotation has an inverse. So, the set of rotations forms a group.

Well, that's physicsese. In mathematicsese, U(n) is the unitary group of $n \times n$ matrices, and U(1) is the case where $n = 1$. The elements of U(1) are identified with the complex numbers $e^{i \theta}$, and group multiplication is simply complex multiplication. The identity is $1$ and the inverse of $e^{i \theta}$ is $e^{-i \theta}$. U(1) is Abelian because multiplication of complex numbers, $z_1, z_2$, is commuative: in this case, $e^{i \theta_1}e^{i \theta_2} = e^{i (\theta_1 + \theta_2)} = e^{i \theta_2}e^{i \theta_1}$. The connection between rotations and with complex numbers comes from the 2-dimensional representation of $\mathbb{C}$ as the plane. If $z \in \mathbb{C}$, then $e^{i \theta} z$ is the result of "rotating" $z$ by angle $\theta$ (about the origin).

Nominalism is the philosophical doctrine that there are no abstract entities, and, a fortiori, no numbers, sets, functions, groups, manifolds, Hilbert spaces and so on. Consequently, as frequently pointed out by Quine and Putnam, nominalism is inconsistent with science. For example, the following is a true statement of physics:
U(1) is the gauge group of the electromagnetic field $A_{\mu}$.
If nominalism is true, then there are no groups. If there are no groups, then this statement of physics is false.

A very brief explanation of gauge theory: Particles of matter are described by some field $\phi$; and, if they are charged, then $\phi$ is a complex field, so $\phi(x) \in \mathbb{C}$, at each point $x$ in spacetime. Associated with the field is a quantity called the Lagrangian (written $\mathcal{L}(\phi, \partial_{\mu} \phi)$) which, via an "action principle", implies the equations of motion for the field: e.g., a certain Lagrangian for the field $\phi$ implies that $\phi$ satisfies the Klein-Gordon equation:
$(\partial^{\mu} \partial_{\mu} + m^2)\phi = 0$.
Suppose one multiplies the field $\phi$ by a constant phase factor $e^{i \theta}$ (which is a kind of rotation of each complex number $\phi(x)$ specifying the field). That is,
$\phi^{\prime}(x) = e^{i \theta}\phi(x)$.
One can check that leaves the Lagrangian $\mathcal{L}$ invariant. We say that the Lagrangian $\mathcal{L}$ has a global U(1) symmetry (= a global gauge invariance; = a global gauge symmetry).

However, what if $\phi$ is subjected to a local gauge transformation, a local phase rotation, i.e.,
$\phi^{\prime}(x) = e^{i \theta(x)}\phi(x)$,
where the parameter $\theta$ can be non-constant function, varying from point to point? It turns out that the Lagrangian is not invariant.

However, this invariance can be restored by introducing a new compensating gauge field, $A_{\mu}$, and a modified Lagrangian $\mathcal{L}^{\ast}$, so that under a gauge transformation $e^{i \theta}$, the field $A_{\mu}$ transforms to
$A_{\mu}^{\prime}(x) = A_{\mu}(x) + (\partial_{\mu} \theta)(x)$.
The resulting field $A_{\mu}$ is called a gauge field: the U(1) gauge field, which "couples" with the (current of the) original charged matter field $\phi$ exactly as the electromagnetic field does: $A_{\mu}$ gives rise to the electromagnetic field. More exactly, $A_{\mu}$ is the electromagnetic potential: the electromagnetic field $F_{\mu \nu}$ itself is the exterior derivative, $\frac{\partial A_{\mu}}{\partial x^{\nu}} - \frac{\partial A_{\nu}}{\partial x^{\mu}}$. And the field $F_{\mu \nu}$ is gauge-invariant.

Here is a really brilliant exposition of the ideas of gauge theories, and much more (co-ordinate systems, fibre bundles and whatnot), by Terence Tao.

[Update (Aug 30th), minor changes.]

Saturday, 25 August 2012

Meaning, Use and Modality: M-Facts and U-Facts

I don't have a precise definition of either "M-fact" or "U-fact", but roughly, the idea is that an M-fact is a meaning-fact, while a U-fact is a usage-fact.
Examples of M-facts might be things like:
1. "Schnee" refers-in-German to snow.
2. "sensible" is true-in-Spanish of x iff x is sensitive.
3. "Schnee ist weiss" is true-in-German iff snow is white.
4. "I" refers-in-English, relative to context $C$, to the agent of the speech act in $C$.
5. "kai" means-in-Greek logical conjunction.
In each case, the language relativity has been made explicit. I think that ignoring language relativity is a major fallacy in much writing about the foundations of linguistics and philosophy of language. Tarskian T-sentences are examples of M-facts. The bearers of the semantic (and syntactic) properties are types, not tokens. Again, I think that it's major mistake to be confused about this.

Canonical examples of U-facts might be things like:
6. Speaker X uses the string "Schnee" to refer to snow.
7. Speaker Y has a disposition to utter the string "gavagai" when there are rabbits nearby.
8. Speaker Z tends to assert the string $\sigma_1 \ast$ "kai" $\ast \sigma_2$ just when Z is prepared to assert both $\sigma_1$ and $\sigma_2$.

One might initially think that U-facts explain M-facts. Or that U-facts provide evidence for M-facts. Roughly:
U-facts: evidence/data for linguistic theories.
M-facts: theoretical content of linguistic theories.
This is, I think, roughly right, but with a very important caveat, which is that M-facts cannot be explained by U-facts. The argument is this:
(i) M-Facts are Necessities.
An M-fact, such as the fact that "Schnee" refers-in-German to snow couldn't have been otherwise. The argument for this is a counterfactual thought experiment. Suppose $L$ is an interpreted language such that "Schnee" refers-in-L to sugar. Then it seems clear to me that $L$ isn't German. If one changes the meanings of a language, the language is simply a different one. Languages are very finely individuated.
(ii) U-Facts are Contigencies
A U-fact, such as the fact that Y has a disposition to utter "gavagai" when there are rabbits nearby, is contingent. It Y needn't have had that disposition. A U-fact is connected to properties of the speaker's cognitive system.

If the previous two claims, (i) and (ii), are correct, then
(iii) U-facts cannot explain, or provide evidence for, M-facts.
This conclusion follows because contingencies cannot explain necessities.

I've given this argument is several talks since 2008 (originally in a talk "Meaning, Use and Modality" in at Universidad Complutense, Madrid). The audience frequently responds with considerable surprise!

If U-facts do not explain M-facts, then what explains M-facts? I say, "Nothing". Nothing explains why "Schnee" refers in L to sugar. It is simply an intrinsic property of the language L. There is some sense in which M-facts, along with syntactic and phonological and pragmatic facts, about a language $L$ are mathematical facts. Languages are complicated (mixed) mathematical objects. For example, suppose that
The string $\phi$ is a logical consequence in $L$ of the set $\Delta$ of $L$-strings
Then this fact, about $L$, is a necessity.

If U-facts do not explain M-facts, then what do they explain or provide evidence for? I think the answer to this is,
U-facts explain, or provide evidence for, what language the speaker/agent cognizes.
So, let me call these C-facts, and these have the form:
(C) Speaker X speaks/cognizes L
So, for example,
The (contingent) U-fact that X has a disposition to utter "gavagai" when there are rabbits nearby is evidence for the (contingent) C-fact that X speaks/cognizes a language L for which the following M-fact holds of necessity: "gavagai" denotes-in-L rabbits.

Friday, 24 August 2012

Logical Consequence 2

Continuing on the same theme, consider the alphabet $A = \{0, 1, 2\}$. Consider the strings:
$\sigma_1 = (0,1,2)$
$\sigma_2 = (0,1,1)$.
Consider the question:
Is $\sigma_2$ a logical consequence of $\sigma_1$?
Clearly the question doesn't make sense.

Next, we turn this alphabet $A$ into a formalized language $L$. $String(L)$ is the set of finite sequences drawn from $A$. We define $Sent(L)$--i.e., the sentences of $L$--inductively as follows:
$Sent(L)$ is smallest subset $X \subseteq String(L)$ such that:
(i) $(1) \in X$;
(ii) $(2) \in X$;
(iii) if $\sigma_1, \sigma_2 \in X$, then $(0) \ast \sigma_1 \ast \sigma_2 \in X$.
(Here $\ast$ is sequence concatenation.) So, we get:
$(1) \in Sent(L)$;
$(2) \in Sent(L)$;
$(0,1,1) \in Sent(L)$.
$(0,1,2) \in Sent(L)$.
etc.
So, these strings are sentences in $L$.
For brevity, I now write
$012$
to mean the sequence
$(0,1,2)$
So,
$1 \in Sent(L)$;
$2 \in Sent(L)$;
$011 \in Sent(L)$.
$012 \in Sent(L)$.
etc.
One can enumerate the $L$-sentences by partioning them by length:
1. There are no $L$-sentences of length $k$, where $k$ is even.
2. The $L$-sentences of length 1 are $1$ and $2$.
3. The $L$-sentences of length 3 are $011, 012, 021$ and $022$.
4. etc.
So far, there is just the alphabet $A$, the $L$-strings and the $L$-sentences. One has no notion of logical consequence.
Let $B_2$ be the two-element Boolean algebra, $\{\top, \bot\}$.
An $L$-interpretation is a function $I : Sent(L) \rightarrow \{\top, \bot\}$ such that, for any $\phi, \theta \in Sent(L)$,
$I(0 \ast \phi \ast \theta) = \top$ iff $I(\phi) = \top$ and $I(\theta) = \top$.
Write
$I \models \phi$
for
$I(\phi) = \top$.
Let $\Sigma(L)$ be the class of such interpretations.
Finally, logical consequence is defined by:
$\theta$ is a logical consequence in $L$ of $\phi$ iff $\phi, \theta \in Sent(L)$ and, for all $I \in \Sigma(L)$, if $I \models \phi$, the $I \models \theta$.
We abbreviate this relationship as.
$\phi \vDash_L \theta$
Then we have:
$1 \vDash_L 1$.
$1 \nvDash_L 2$.
$2 \nvDash_L 1$.
$2 \vDash_L 2$.
$011 \vDash_L 1$.
$011 \nvDash_L 2$.
$012 \vDash_L 1$.
$012 \vDash_L 2$.
$021 \vDash_L 1$.
$021 \vDash_L 2$.
$022 \vDash_L 2$.
$022 \nvDash_L 1$.
This all looks quite hard to follow. Let me abbreviate a bit more. Let's write "$P$" for "$1$", and "$Q$" for "$2$" and "$\wedge$" for "$0$". (Not that this makes any difference. Strings are just strings. They have no "intrinsic" meaning.) Furthermore, instead of
$0 \phi \theta$
we write
$\phi 0 \theta$.
(I.e., infix notation.)
Then the above becomes:
$P \vDash_L P$.
$P \nvDash_L Q$.
$Q \nvDash_L P$.
$Q \vDash_L Q$.
$P \wedge P \vDash_L P$.
$P \wedge P \nvDash_L Q$.
$P \wedge Q \vDash_L P$.
$P \wedge Q \vDash_L Q$.
$Q \wedge P \vDash_L P$.
$Q \wedge P \vDash_L Q$.
$Q \wedge Q \vDash_L Q$.
$Q \wedge Q \nvDash_L P$.
This all looks so much more familiar!

Now go back to the question at the start. Consider the strings:
$\sigma_1 = (0,1,2)$
$\sigma_2 = (0,1,1)$.
Consider the question:
Is $\sigma_2$ a logical consequence in $L$ of $\sigma_1$?
Clearly the question does now make sense. It asks,
Is $P \wedge P$ a logical consequence in $L$ of $P \wedge Q$?
And the answer is: yes.

Logical Consequence

This is all a bit basic, but worth saying because it interacts with foundational questions (for example, "what are the relata of logical consequence?" or "what is more basic, semantics or inference?"). Suppose that $A = \{a_1, a_2, a_3\}$ is an alphabet. Suppose that $\sigma_1$ and $\sigma_2$ are non-empty strings from $A$. That is, they are finite sequences of the form
$\sigma_i : [0,n] \rightarrow A$.
Consider a claim of the form:
$\sigma_2$ is a logical consequence of $\sigma_1$.
I think this claim is seriously underspecified, because logical consequence can only make sense given a class of interpretations of the strings.

Suppose that $A$ above is the alphabet of an interpreted language $L$. Suppose that this also determines a special class $Sent(L)$ of strings, and a class $\Sigma(L)$ of interpretations for $L$ in such that way that, for any $I \in \Sigma(L)$, and any $\phi \in Sent(L)$,
$I \models \phi$
is defined, meaning,
the $L$-string $\phi$ is true in $I$.
Then consider the claim,
$\sigma_2$ is a logical consequence in $L$ of $\sigma_1$.
On the Bolzano-Tarski definition, this means,
$\sigma_1, \sigma_2 \in Sent(L)$ and, for any $L$-interpretation $I \in \Sigma(L)$,
if $I \models \sigma_1$ then $I \models \sigma_2$.

Monday, 20 August 2012

A Non-Conservation Theorem for Mereology

I'm a bit less sure of the content of this second paper, but popping it up here in case some reader notices an error. The previous post showed how to conservatively extend a theory $T$ with axioms $\mathsf{F}$ for mereology (more exactly, for fusions). The axioms of $\mathsf{F}$ are:
$\mathsf{D}_{\preceq}$: $\hspace{5mm} x \preceq y \leftrightarrow y = x \oplus y$.
$\mathsf{D}_{O}$: $\hspace{5mm} xOy \leftrightarrow \exists w(w \preceq x \wedge w \preceq y)$.
$\mathsf{D}_{At}$: $\hspace{3mm} At(x) \leftrightarrow \forall z(z \preceq x \rightarrow z = x)$.
$\mathsf{F}_1$: $\hspace{6mm} x \oplus x = x$.
$\mathsf{F}_2$: $\hspace{6mm} x \oplus y = y \oplus x$.
$\mathsf{F}_3$: $\hspace{6mm} x \oplus (y \oplus z) = (x \oplus y) \oplus z$.
$\mathsf{UF}$: $\hspace{5mm} \exists x \phi(x) \rightarrow \exists! z [\forall x(\phi(x) \rightarrow x \preceq z) \wedge \forall y \preceq z \exists x(\phi(x) \wedge xOy)]$.
Crucial for that result is the restriction of any axiom schemes of $T$ to the original language. At the end of the paper, "What Difference Does it Make?", I wondered if extending Peano arithmetic, $\mathsf{PA}$, with fusion theory and full induction yields a non-conservative extension.

This second paper, "Arithmetic with Fusions", is very much a preliminary draft, and it would be nice if any reader could find a snag. (Some earlier attempts did run into snags, but eventually, it now seems that the two main interpretability results are correct.) In the paper, we argue that the fusion extension of Peano arithmetic, denoted $\mathsf{PAF}$, interprets full second-order arithmetic, $Z_2$. If this is right, then $\mathsf{PAF}$ is a very strong theory indeed. One might also conclude that this sort of non-conservation undermines a fictionalist view of mereological fusions. (This is joint work with Thomas Schindler.)

Saturday, 18 August 2012

A Conservation Theorem for Mereology

Hartry Field, in his Science without Numbers (1980), introduced and developed a fictionalist programme for mathematical nominalism based on two major ideas:
1. An Elimination/Nominalization Result:
For certain kinds of mathematicized theories $T$ in science, the quantification over mathematical entities (numbers, real numbers, values of quantities, etc.) can be eliminated by "encoding" mixed relations (between concreta and mathematicalia, such as "$r$ is the mass-in-kg of $x$" or "$r$ is the value of scalar field $\phi$ at spacetime point $p$") and pure relations (amongst mathematicalia, such as "$r_1 = r_2 \times r_3$") into the purely concrete domain.

2. A Conservative Extension Result:
Given a theory $T$ entirely about concreta (a "nominalistic theory), the result of extending $T$ with axioms for sets yields a conservative extension when restricted to what one can prove about the concreta.
There is an important similarity between Field's programme and Hilbert's programme: for Hilbert, only finite mathematical entities (natural numbers, finite sets, finite graphs, etc.) are considered "real", while infinite sets are to be considered "useful fictions". Ideally, one would have shown that extending, say, $\mathsf{PA}$ with axioms for sets would give a conservative extension. However, of course, Gödel's results showed that this cannot hold in general. Gödel himself notes this in the famous foonote 48a of his 1931 paper. For example, extending $\mathsf{PA}$ with impredicative comprehension yields a theory that is arithmetically much stronger.

One might try and develop an analogous fictionalism for other kinds of entity that are sometimes considered metaphysically dubious (e.g., theoretical objects, possible worlds). One kind of fishy metaphysical entity is a mereological fusion. One might think that for any $a$ and $b$, there is always a fusion $a + b$. Or one might reject this in all cases. The analogous question then is: does extending a theory with axioms for mereological fusions make no difference? Does it yield a conservative extension? The answer to this is, "yes", under certain circumstances.

In a draft article, "What Difference Does it Make?", I sketch a conservation result for mereology. The basic idea is to begin with a theory $T$, which is then extended by a version of mereology I call Fusion theory, $\mathsf{F}$ (equivalent to Classical Mereology: the difference is that binary fusion, $x + y$, is treated as primitive). If the original entities are treated as atoms of the resulting combination (this is achieved by relativization of quantifiers to the predicate $At(x)$), $T^{At} \cup \mathsf{F}$, then anything provable in the extended theory about the original entities is provable in the original theory $T$.

The method is model-theoretic, and is analogous to Field's approach in his 1980. Take a structure $M$ for the original language $L$, with domain $X$, say. We try to construct a larger structure $M^{\ast}$ for the extended language, with the right properties. Let the larger domain be simply $X^{\ast} = \mathcal{P}(X) \setminus \{\varnothing\}$, so that we shall get a power set algebra (a complete lattice) with the least element removed. Then map the original elements $a \in X$ to their singletons $\{a\} \in X^{\ast}$. So, let $s(a) = \{a\}$. Finally, we define a structure $M^{\ast}$ for the full language by defining, each distinguished relation $(P_i)^M$ in $M$, a corresponding relation $s[(P_i)^M]$. In the larger structure we interpret $At$ as $s[X]$; we interpret the fusion symbol $+$ as $\cup$ and interpret the parthood symbol $\preceq$ as $\subseteq$ (we interpret the overlap predicate $xOy$ in the obvious way too).

The construction guarantees that
$M^{\ast} \models \mathsf{F}$.
Also, the singleton mapping $s$ embeds $M$ into $M^{\ast} | L$, with image $s[X]$, which is the denotation of $At$. So, for any $L$-sentence $\phi$,
$M \models \phi$ iff $M^{\ast} \models \phi^{At}$
(here $\phi^{At}$ is the relativization of $\phi$ to $At$).

It follows from this that
if $T^{At} \cup \mathsf{F}$ proves $\phi^{At}$, then $T$ proves $\phi$.
For if $T$ does not prove $\phi$, then there is a model $M \models T$ with $M \not \models \phi$. But then $M^{\ast} \models T^{At}$ and $M^{\ast} \not \models \phi^{At}$. And $M^{\ast} \models \mathsf{F}$. So, $M^{\ast} \models T^{At} \cup \mathsf{F}$. So, $T^{At} \cup \mathsf{F}$ does not prove $\phi^{At}$.

Friday, 17 August 2012

Language Relativity (or: Does a Cow Eat Without a Knife?)

I wonder if you know what
(1) Estas constipado?
means (in Spanish; no, sorry, I'm not doing the upside down question-mark thing). When I first heard this, with my pigeon Spanish, I took it to mean, "Are you constipated?", but it simply means, "Do you have a cold?"

Surely no one thinks that a string of phonemes (or letters, if you prefer orthography) has a meaning all of its own. A string can only mean something relative to some language. That is, if $\sigma$ is a string, then the meta-string
(2) $\sigma$ means $m$
does not, strictly speaking, express a complete thought. It should be,
(3) $\sigma$ means-in-$L$ $m$
where $L$ is an interpreted language.

Logicians and others have developed a very detailed theory over the last hundred years or so of uninterpreted languages. We sometimes use the notation "$\mathcal{L}$" to refer to uninterpreted languages. And, obviously, it doesn't make sense to ask of an string in $\mathcal{L}$ what it means. For it means nothing. Only when one has an interpreted language $L$ does the question, "what does $\sigma$ mean in $L$?" make sense. This is language relativity. Meaning/reference, and so on, has to be language relative (or intepretation relative). So, we sometimes represent an interpreted language as a pair $(\mathcal{L}, I)$.

Another example is:
(4) "sensible" is true of $x$ iff $x$ is sensible.
seems ok to a deflationist. But it is, strictly speaking, missing a parameter. And the parameter does make a difference, for:
(5) "sensible" is true in English of $x$ iff $x$ is sensible.
(6) "sensible" is true in Spanish of $x$ iff $x$ is sensitive.
The same orthographic string---i.e., finite string of Latin letters---has different meanings in English and Spanish (the somewhat different pronunciations, however, break the symmetry here).

Linda Wetzel gives the following example in her SEP article Types and Tokens.
Even being similar in appearance (say by sound or spelling) to a canonical exemplar token of the type is not enough to make a physical object/event a token of that type. The phonetic sequence
[Ah ‘key ess ‘oon ah ‘may sah]
is the same phonetic (type) spoken in Spanish or Yiddish. Yet if a Spanish speaker uses it to say a table goes here, she has not, in addition, said a cow eats without a knife in Yiddish. She has not said anything in Yiddish, however phonetically similar what she said might be to a sentence token of Yiddish. So her token is a token in Spanish, not Yiddish. Meaningful tokens are tokens in a language.
Wetzel makes two points here. The first concerns tokens: meaningful tokens are only meaningful in virtue of being tokens of some type in a language. The second point is to provide a neat example of language relativity, namely,
(7) [Ah ‘key ess ‘oon ah ‘may sah] means-in-Spanish that a table goes here.
(8) [Ah ‘key ess ‘oon ah ‘may sah] means-in-Yiddish that a cow eats without a knife.
(Unfortunately this example is only 99% neat, as my wife tells me that "acquí es una mesa" is not properly formed Spanish. Ho hum.)

As soon as one appreciates the importance of language relativity for linguistic notions (grammaticality, phonology, semantics, pragmatics), then one notices something that at first sight seems odd. Consider again,
(6) "sensible" is true in Spanish of $x$ iff $x$ is sensitive.
One might, naively, think this semantic fact is a contingent fact. After all, to find out this fact, one might jump onto an EasyJet flight, head off to Madrid, and check the speech behaviour of Madrileños.
But do a thought experiment. Suppose that $L$ is an interpreted language such that,
(9) "sensible" is true in $L$ of $x$ iff $x$ is from Belgium.
My intuition here is that $L$ is not Spanish. (This isn't a proof that $L$ isn't Spanish.) If that is right, then interpreted languages carry their interpretations essentially, and consequently semantic facts, such as (6), are necessities.

This conclusion (whether one accepts it or not) relates to the modal (and temporal) individuation of languages. I am inclined to say that even fluent competent English speakers in fact speak (or cognize) slightly different languages: idiolects. If the language I speak is $L_1$ and the language Robbie speaks is $L_2$, then there are certain differences---pronunciation, lexical, some semantic and pragmatic variation---between $L_1$ and $L_2$. One might say that $L_1$ and $L_2$ are "variants" of English but all this means is that $L_1$ and $L_2$ are both similar to English, a language that no one, strictly speaking, speaks!

As soon as one goes down this road, then all sorts of interesting phenomena happen. Semantic facts are facts about specific languages, and are necessities. There is no philosophical problem of reference. In principle, given an uninterpreted syntax $\mathcal{L}$, an interpretation can pair of any string with any meaning, generating countlessly many different languages. The famous questions about semantic indeterminacy become questions about what language one speaks/cognizes. Interesting questions about "reference magnetism" become questions about cognizable languages (or, at least, become closely connected with cognitive questions, rather than purely semantic ones).

For example, consider a non-standard model $M \models PA$. The Skolemite sceptic asks, or wonders, how it becomes determinate (or how we might know) that the interpretation of the language $L$ spoken by a number theorist when they do number theory is $\mathbb{N}$ or $M$. Well, if $\mathcal{L}$ is the underlying syntax, then perhaps the language $(\mathcal{L}, M)$ is, in some sense, not cognizable. One can (as I am doing) refer to this language, using singular terms or variables. But, perhaps, one can't speak this language.

So, to be as clear as possible, the rough idea is:
(i) The language $(\mathcal{L}, \mathbb{N})$ is cognizable.
(ii) The language $(\mathcal{L}, M)$ is not cognizable.
But even if (ii) is true, I'm still not sure why. Non-standard models are indeed strange (the most interesting instance of this for countable models of $PA$ is Tennenbaum's Theorem). On my view, cognizing a language $L$ consists in the matching up of the meanings of $L$ with the meanings one's mind assigns to strings. But why the human mind assigns the concept NUMBER to the string "number" in English (or variants) rather than the concept $M$-ELEMENT is not clear to me. Perhaps the concept $M$-ELEMENT is a concept one cannot "get" or "grasp" without first getting the concept NUMBER.

Thursday, 16 August 2012

CfP on Intentions: Philosophical and empirical issues

Call for papers: Topoi conference and special issue
INTENTIONS: PHILOSOPHICAL AND EMPIRICAL ISSUES
Rome, Italy, 29-30 November 2012


We are proud to announce that the first TOPOI CONFERENCE will be held in Rome in November 2012. This will be the first in a series of conferences, to be held every 2 years, sponsored by Topoi: An International Journal of Philosophy. In analogy with the journal format, each conference will focus on a specific theme (topos), and contributions presented to the conference will later appear in an issue of the journal dedicated to the same topic.


INVITED SPEAKERS
Marcel Brass (Ghent)
Cristiano Castelfranchi (Rome)
Elisabeth Pacherie (Paris)
Corrado Sinigaglia (Milan)
Bruno Verbeek (Leiden)


SUBMITTED PAPERS
Submissions of unpublished papers are welcome on any topic relevant to the conference theme (see below). Submitted articles should be in English, not exceed 5.000 words in length (including references), and be prepared for blind reviewing. Only original papers (i.e., not published or submitted for publication elsewhere) will be considered, since the authors of accepted contributions will be invited to submit a revised longer version of their papers for a special issue of Topoi, edited by Markus Schlosser (Leiden) and Fabio Paglieri (Rome).

Submission of papers by e-mail to: Fabio Paglieri (fabio.paglieri@istc.cnr.it)
Accepted file formats: .doc, .rtf., .odp, .pdf
Deadline for submission (full papers): 31 August 2012


RATIONALE AND THEME
Theorizing about human action has a long history in philosophy, and the nature of intention and intentional action has received a lot of attention in recent analytic philosophy. At the same time, intentional action has become an empirically studied phenomenon in psychology, cognitive neuroscience, artificial intelligence, and robotics. Many results obtained in these areas have been incorporated within the current philosophical debate, while at the same time scientists have often adopted in their experiments and models philosophical assumptions on the nature of intention and intentional action. As a result, the study of intentions is nowadays a thriving enterprise, where both conceptual and empirical issues are discussed in a dialogue across disciplines.
This conference aims to bring together empirically informed philosophers and philosophically savvy scientists to address a variety of central problems in the study of intention. Possible topics for contributions include:
What is the relevance of recent findings in neuroscience and experimental psychology for our philosophical understanding of intentions and intentional action?
What role can philosophical analysis play in the design of experiments and in the interpretation of empirical results?
What is the relation between intentions, choices, and other mental states and events (such as desires, goals, plans, beliefs, expectations, emotions, etc.)?
Are intentions necessarily conscious? Are they causally efficacious in virtue of being conscious?
What, exactly, are motor intentions, proximal intentions, and distal intentions?
What are the neural correlates of the different kinds of intentions?
What are the mechanisms responsible for intention recognition and action understanding?
What is the role of intentions in gaining self-knowledge and self-understanding?
What is the nature of group or shared intentions?
What role do intentions play in self-control and weakness of the will?
Do (or can) other animals or robots have intentions?
And so on.


PRACTICAL ARRANGEMENTS
The conference will be held at the Institute of Cognitive Sciences and Technologies of the National Research Council of Italy (ISTC-CNR), Via San Martino della Battaglia 44, Rome, Italy. The language of the conference is English. Attendance is free and no registration is required. All meals will be provided for the authors of accepted papers, but at the moment no reimbursement for accommodation and travel costs is expected to be available.


ORGANIZERS AND SPONSORSHIP
The conference is organized by Fabio Paglieri (Rome) and Markus Schlosser (Leiden), and sponsored by Topoi, Springer, the European Network for Social Intelligence (SINTELNET), and the Goal-Oriented Agents Lab (GOAL) of the ISTC-CNR.

For further information, please contact Fabio Paglieri, fabio.paglieri@istc.cnr.it

Saturday, 11 August 2012

Representational Impurities

Given an interpreted language $(L,I)$, of roughly the kind used in teaching predicate logic, there may be syntactically distinct strings which express the same proposition. For example,
If (John plays guitar and Paul drums) then Ringo whinges.
If [John plays guitar and Paul drums] then Ringo whinges.
These are syntactically distinct, because they are distinct strings of symbols. This has something to do with the use of brackets. In a sense, the brackets are a kind of "representational impurity". And, of course, brackets can be eliminated, at the cost of introducing Polish notation (for connectives and predicates). In a sense, Polish notation "quotients out" a certain kind of representational impurity.

Also, the following syntactically distinct string express the same proposition,
There is someone $x$ such that, for all $y$, $x$ does not like $y$.
There is someone $y$ such that, for all $x$, $y$ does not like $x$.
These are again syntactically distinct, because they are distinct strings of symbols. But they are equivalent in the sense that they are the result of applying a certain kind of substitution based on a permutation $\pi: Val(L) \rightarrow Var(L)$ of variables of the language $L$. This is sometimes called relabelling of variables, and explains why, for example,
$\int_0^1 x^2 dy = \int_0^1 y^2 dy$
The equivalences can be proved in predicate logic. So, the choice of variables is another kind of representational impurity. In fact variables can be eliminated, at the cost of introducing combinatory logic (what Quine calls predicate-functor logic). This is another example of "quotienting out" certain representational impurity.

So, brackets and choice of variables are examples of representational impurities.
And the "quotienting out" of these representational impurities in syntactic strings, by either going Polish, or going to predicate-functors, is not controversial. They are quite clear-cut cases of convention. (The "quotiented-out" languages are of course, very inconvenient! The resulting formulas are almost unintelligible to ordinary cognition.)

What is the underlying idea? First, one has an interpreted language $(L,I)$, and the interpretation function $I$ for $L$ generates certain semantic equivalence relations on the strings of $L$. In particular, a synonymy relation, which I'll write $\phi \equiv_{(L,I)} \theta$. To quotient out, one identifies:
i. a new interpreted language $(L^{\prime}, I^{\prime})$,
ii. a translation $^{\circ}: L \rightarrow L^{\prime}$, such that,
(a) if $\phi \equiv_{(L,I)} \theta$, then $\phi^{\circ} = \theta^{\circ}$
(b) $I \models \phi$ iff $I^{\prime} \models \phi^{\circ}$
So, equivalent but distinct $L$-strings are mapped to a single string in $L^{\prime}$.
For example, $\exists x \phi(x)$ and $\exists y \phi(y)$ in the ordinary quantificational language $L$ with variables will be mapped to the same syntactic string in the predicate-functor language $L^{\prime}$ without variables.

One might wonder if this process of eliminating representational impurities reaches a limit. This would mean that all synonymies have been eliminated. I suspect that there isn't a limit in the syntactic sense. And I also suspect that the standard criticisms of the correspondence theory as "mirroring" (a criticism one finds in the writings of many idealists and pragmatists) is really an expression of this conclusion: that syntactic representations are always "impure". These "impurities" are solely contributions of the representer, and are not there in unadorned reality.

Fortunately, such criticisms have little effect after Tarski (1935), "Der Wahrheitsbegriff". Tarski showed how truth is to be defined without assuming a mirroring relation. As we quotient out impurities, the translation and the (Tarskian) truth definition guarantee that required equivalences hold. That is, if $\phi$ and $\theta$ are syntactically distinct but equivalent in $(L,I)$, then each element of the equivalence class $[\phi]$ of equivalent $L$-strings has the same truth conditions as its image $\phi^{\circ}$ in $L^{\prime}$.

Saturday, 4 August 2012

So, what's your theory of truth?

I have written a little bit on theories of truth, so I'm sometimes asked what my own theory of truth is. I'm supposed to say something like,
Given $\phi$ one may infer $T\ulcorner \phi \urcorner$
or
$\forall x \in Sent(L_T) (T neg(x) \leftrightarrow \neg Tx)$
etc.
But I don't, because I think of them as being about the formal behaviour of a truth predicate, and not about truth.
Instead, I'd prefer this more representationalist view:
1. There are these things, called strings: they are sequences (usually finite) drawn from an alphabet.
2. There is syntax, which specifies that some of these syntactical strings are sentences (of some language $\mathcal{L}$).
3. And there are $\mathcal{L}$-interpretations, $\mathcal{I}$.
4. Then there is a semantic relation, written
$\mathcal{I} \models \phi$
which means "the sentence $\phi$ is true in the interpretation $\mathcal{I}$".
That's my theory of truth!
(And for my non-causal theory of reference, replace "sentence $\phi$" by "term $t$" and replace "$\mathcal{I} \models \phi$" by "$t^{\mathcal{I}} = a$").

But there are several objections:
1. (Non-classicist)
By an "interpretation" you mean a classical one? What if we want to discuss $n$-valued logic or truth in some Kripke frames or something exotic?
My response is: yes, entirely agree. The interpretations can be anything you find in a logic (or formal semantics) textbook: $n$-valued, $2^{\aleph_0}$-valued, supervaluational, Kripke frames, etc.
2. (Sentential deflationist)
Surely truth is a unary predicate, $Tx$, applicable to sentences. The function of $T$ is logical, to express schematic generalizations, which must satisfy disquotation/transparency?
My response is: I don't agree. Truth for linguistic entities is a binary relation, relating a string and an interpretation $\mathcal{I}$ (or an interpreted language $\mathbf{L} = (\mathcal{L}, \mathcal{I})$). The function of any unary truth predicate one studies is to denote some class of truths in an interpreted language, or to denotes some class of true propositions. For example, given $\mathbb{N}$, then define
$E = Th_{L}(\mathbb{N})$,
i.e., true arithmetic (well, codes of). Then, if the symbol $T$ is interpreted as $E$, we get,
$(\mathbb{N}, E) \models T\ulcorner \phi \urcorner \leftrightarrow \phi$,
for each $\phi \in L$. So, each T-sentence (so long as $\phi \in L$) comes out true in $(\mathbb{N}, E)$.
3. (Minimalist)
Yes, ok: truth for sentences is binary, not unary. Sentential deflationists are mistaken. But surely the primary notion of truth is for propositions; then it really is expressed by a unary predicate, $Tx$. The function of $T$ as applied to propositions is logical, to express generalizations, which must satisfy disquotation/transparency?
My response is: I'm not sure. I do tend to think that propositional truth is more basic somehow. But then no one has a good theory of propositions (for example, it's unclear how diagonalization and self-reference works for propositions).

[I am being a bit sneaky: my favourite formal theory of truth is KF, the Kripke-Feferman theory of truth.]

Friday, 3 August 2012

Identity, Indiscernibility and Individuation Criteria

This post is stimulated by some discussions with my research student Johannes Korbmacher about identity criteria (e.g., Horsten 2010, "Impredicative Identity Criteria"). I am interested here in how such identity criteria might be connected with other work on indiscernibility.

If $P$ is a unary predicate, then write,
$x \sim_P y$
for
$Px \leftrightarrow Py$
If $R$ is a binary predicate, then write,
$x \sim_{R,1} y$
for
$\forall z(Rzx \leftrightarrow Rzy)$
and
$x \sim_{R,2} y$
for
$\forall z(Rxz \leftrightarrow Ryz)$
Roughly, $x \sim_P y$ means that $x$ and $y$ are indiscernible by $P$; and $x \sim_{R,1} y$ means that $x$ and $y$ are indiscernible by $R$ on the first argument position; and $x \sim_{R,2} y$ means that $x$ and $y$ are indiscernible by $R$ on the second argument position.
It's easy to see how to generalize this to any primitive $k$-ary relation symbol.

The resulting formulas,
$x \sim_{R,i} y$,
are Hilbert-Bernays clauses.

The strongest (first-order) notion of indiscernibility expressible in a language $\mathcal{L}$ is given by the conjunction of all the Hilbert-Bernays clauses over the primitive predicate symbols. I.e., write:
$x \sim_{\mathcal{L}} y$
for
$\bigwedge \{x \sim_{R,i} y \mid R$ is an $n$-ary relation symbol in $\mathcal{L}$ and $1 \leq i \leq n\}$.
Some accessible technical results about such formulas are given in
Ketland 2006, "Structuralism and The Identity of Indiscernibles";
Ketland 2011, "Identity and Indiscernibility".
My favourite results are:
1. If $=$ is definable in $\mathcal{M}$, then it is defined by $x \sim_{\mathcal{L}} y$.
2. If $\mathcal{M} \models a \sim_{\mathcal{L}} b$, then $\pi_{ab} \in Aut(\mathcal{M})$
where $\pi_{ab}: M \rightarrow M$ is the transposition that swaps $a, b \in M$.

In the case of set theory, we have the signature $\sigma = \{\in, =\}$. Then the Axiom of Extensionality can be expressed as follows:
$x \sim_{\in,1} y \rightarrow x = y$.
If we prefix this with a relativization to sets, then we have,
$Set(x) \wedge Set(y) \rightarrow (x \sim_{\in,1} y \rightarrow x = y)$.
This kind of thing is sometimes called an individuation principle (I think this terminology is due to Quine). So, we sometimes say that sets are individuated by their members. But notice that we simply use a Hilbert-Bernays indiscernibility clause for the membership predicate.
Leibniz's Principle of Identity of Indiscernibles has the same form,
$Obj(x) \wedge Obj(y) \rightarrow (x \sim_{\iota,2} y \rightarrow x = y)$.
where $Obj(x)$ means "$x$ is an object", and we introduce $x \iota X$ to mean "$x$ instantiates $X$" and $x \sim_{\iota,2} y$ means
$\forall X(x \iota X \leftrightarrow y \iota X)$.
Similarly, for extensional properties, we have,
$Ext(X) \wedge Ext(Y) \rightarrow (X \sim_{\iota,1} Y \rightarrow X = Y)$.
(This generalizes to extensional relations too, when the instantiation relation is taken to be polyadic.)
So, going by analogy, one naturally expects all such individuation criteria to be formulated using Hilbert-Bernays indiscernibility clauses. So, an individuation criterion would then have the form,
$K(x) \wedge K(y) \rightarrow [\bigwedge \{x \sim_{R,i} y \mid (R,i) \in I_K \} \rightarrow x = y$].
where $I_K$ specifies a set of pairs of primitive symbols and argument positions.
So, for sets, $I_K$ is $\{(\in, 1)\}$.

Of course, there might be a kind $K$ of entities for which there is no such individuation criterion unless one assumes that the primitive identity predicate $x = y$ already is in $I_K$. In other words, $=$ is not definable in such structures from the other primitive notions/relations. It is easy to show that there are structures $\mathcal{M}$ in which $=$ is not definable. Examples are given in Ketland 2006 and Button 2006, "Realistic Structuralism's Identity Crisis: A Hybrid Solution". I call such structures "non-Quinian".

Because of such examples (and for other reasons), it is preferable to treat identity as a primitive notion.

Is There a Philosophical Problem of Reference?

Many philosophers worry about reference, and think we need to come up with a "theory of reference". This would explain how strings get "connected" to things: the string-thing relation. So, maybe this should be analysed in terms of causation. Causation glues cats to the word "cat". And so on. Or maybe one gets worried about how, e.g., "aleph_0" and $\aleph_0$ got glued together, when causation couldn't have done that---for $\aleph_0$ is a causally inert abstractum. In fact, the string "aleph_0" is also an abstractum. Worried by the unsurprising inability to reduce reference to causation, one might go deflationary and say that there is just a reference predicate, the binary predicate "$x$ refers to $y$", governed by disquotational axioms (e.g., ""red" refers to the set of red things") and one has this in one's language (which gives the illusion of genuine reference). Or perhaps one gives up, becomes a semantic nihilist and says outright: there is no reference relation at all.

Well, maybe there is no philosophical problem of reference at all: any relation of strings to things is a legitimate reference relation. So, the theory of reference is this:
$\mathcal{I}(\sigma) = x$
where $\mathcal{I}$ is some function: any function you like. Because $\mathcal{I}$ is a parameter here, reference is really a ternary relation,
$\sigma$ refers to $x$ relative to $\mathcal{I}$.
Many authors believe, however, that reference is a binary relation that strings bear to referents. That is,
$\sigma$ refers to $x$.
With no language or interpretation relativity! It seems to me that this violates the single most basic principle of language---that anything can mean anything in some language. The language relativity is essential. And yet the view that reference is a binary relation without language relativity seems fairly standard.

Chris Gauker gave a talk on a related topic at MCMP several months back. And Chris's view is an example of the conventional view of what is required from a theory of reference:
The problem
What is reference? Offhand, it appears to be a relation, just as being heavier than is a relation. Moreover, it is a relation that holds between words and other things. It holds between "chair" and the chair I'm sitting in, between "chair" and the chair next door, and between "chair" and the chairs of the past and future. It holds between "Socrates" and Socrates, between "meson" and mesons, and between "beautiful" and beautiful things. Of course, the reason it holds between these things is not just that it holds between every word and everything. For instance, it doesn't hold between "basketball" and daffodils. The fact that the relation being heavier than holds between an atom of oxygen and an atom of hydrogen, as well as between Pavarotti and Diana Ross, but not between everything and everything, ought to make us wonder what being heavier than amounts to, if we don't already know. Likewise, the combination of diversity and specificity exhibited by the reference relation ought to make us wonder what reference is. We mustn't just call it an "unanalyzable primitive".
Accounts of reference that might satisfy us fall into two classes. On the one hand, we might seek what I'll call an analysis of reference. An analysis takes the form:
$t$ refers to $a$ if and only if $\dots$.
(Gauker 1990, "Semantics without reference", Notre Dame Journal of Formal Logic, 438.)
This seems to me to be a bit like a grammatical mistake, because it omits the crucial language parameter,
$t$ refers to $a$ in $(\mathcal{L}, \mathcal{I})$ if and only if $\dots$.
And then the answer is:
$t$ refers to $a$ in $(\mathcal{L}, \mathcal{I})$ if and only if $\mathcal{I}(t) = a$.
There are no constraints at all on what the interpretation function $\mathcal{I}$ might be. It can be any old string-thing function. On this proposal, it isn't that the famous semantic indeterminacy puzzles disappear (e.g., Quinian inscrutability of reference, Kripkenstein, Putnam's "model-theoretic argument", ec.). Rather, the puzzles turn out, on reflection, to be puzzles about what language one speaks. The question isn't how languages get referentially glued to things---they already are, in every mathematically possible way (generating uncountably many different languages). The question is how the mind gets glued to some particular language: a problem of language cognition.