Arabic numeral arithmetic: semantics

What we would like to do is figure out a way of associating elements of $L_{ANA}$ with meanings. In order to do this, we have to make use of a model—that is, one of the things that we introduced here.

Which model, exactly, should we be using to interpret $L_{ANA}$? We’ll follow the standard way of doing things and say that the model of $L_{ANA}$ is the one defined in (1), which has as its domain the set $ℕ$ of natural numbers.

$\mathcal{M}_{ANA} = ⟨ℕ, \{\True, \False\}, I_{ANA}⟩$

That is, $D$ is defined to be $ℕ$—i.e., the set $\{0, 1, 2, ...\}$.

What can we say about the model’s interpretation function, $I_{ANA}$? Well, because we defined $atom$ (here) to be the set $\{⌜0⌝, ..., ⌜9⌝\}$, we know that this set is the domain of $I_{ANA}$: it is the job of the model’s interpretation function to map things from this set onto interpretations of some kind.

What about the codomain of $I_{ANA}$? Since we’re having $I_{ANA}$ map elements from the set $\{⌜0⌝, ..., ⌜9⌝\}$, we should choose the codomain of $I_{ANA}$—the set it maps these symbols to—to be $ℕ$: the interpretation of any of the symbols $⌜0⌝$ through $⌜9⌝$ is just a natural number!

Specifically, we’ll define $I_{ANA}$ as in (2).

$I_{ANA} : atom → ℕ$

$I_{ANA}(⌜0⌝) = 0$
$I_{ANA}(⌜1⌝) = 1$
$I_{ANA}(⌜2⌝) = 2$
$I_{ANA}(⌜3⌝) = 3$
$I_{ANA}(⌜4⌝) = 4$
$I_{ANA}(⌜5⌝) = 5$
$I_{ANA}(⌜6⌝) = 6$
$I_{ANA}(⌜7⌝) = 7$
$I_{ANA}(⌜8⌝) = 8$
$I_{ANA}(⌜9⌝) = 9$

Hopefully, none of these choices is particularly surprising! Still, it’s important to point out that—while it looks a bit like we are using the interpretation function to simply “remove the corner brackets ($⌜, ⌝$)”—that’s just due to the fact that we are employing the very language that we are interpreting as part of our metalanguage.¹ Said another way: there is nothing special about the symbol $⌜1⌝$ that makes it a particularly good candidate for being mapped onto the number 1. The choice reflects how we use the language $L_{ANA}$, but it is arbitrary nonetheless—a genuine Lewis-style convention.

Alright. The time has come to define our interpretation function (or “interpretation brackets”, “denotation function”, what have you) $⟦·⟧_{\mathcal{M}_{ANA}}$ over the entirety of $L_{ANA}$. We will show how to do this using two different presentations.² One presentation uses English and provides the rules mostly as conditional if then statements. The other presentation is a bit more formally systematic: it provides formal rules for constructing proofs. Importantly, both presentations say exactly the same thing.

Rules in English

On the first presentation, we’ll append additional clauses to the syntactic rules that were stated in Arabic numeral arithmetic: syntax. This presentation is given in (3).

Rule 1. $atom = \{⌜0⌝, ..., ⌜9⌝\}$, and for any $x ∈ atom$, $⟦x⟧_{\mathcal{M}_{ANA}} = I_{ANA}(x)$.

Rule 2. $atom ⊆ n$.

Rule 3. If $x, y ∈ n$, then $x^{⌢}y ∈ n$, and $⟦x^{⌢}y⟧_{\mathcal{M}_{ANA}} = ⟦x⟧_{\mathcal{M}_{ANA}} * 10^{length(y)} + ⟦y⟧_{\mathcal{M}_{ANA}}$.

Rule 4. $n ⊆ np$.

Rule 5. If $x, y ∈ np$, then $⌜(⌝^{⌢}x^{⌢}⌜+⌝^{⌢}y^{⌢}⌜)⌝, ⌜(⌝^{⌢}x^{⌢}⌜×⌝^{⌢}y^{⌢}⌜)⌝ ∈ np$, and also:
- $⟦⌜(⌝^{⌢}x^{⌢}⌜+⌝^{⌢}y^{⌢}⌜)⌝⟧_{\mathcal{M}_{ANA}} =⟦x⟧_{\mathcal{M}_{ANA}} + ⟦y⟧_{\mathcal{M}_{ANA}}$
- $⟦⌜(⌝^{⌢}x^{⌢}⌜×⌝^{⌢}y^{⌢}⌜)⌝⟧_{\mathcal{M}_{ANA}} =⟦x⟧_{\mathcal{M}_{ANA}} * ⟦y⟧_{\mathcal{M}_{ANA}}$
Rule 6. If $x, y ∈ np$, then $x^{⌢}⌜{=}⌝^{⌢}y ∈ s$, and $⟦x^{⌢}⌜{=}⌝^{⌢}y⟧_{\mathcal{M}_{ANA}} = \True$ if $⟦x⟧_{\mathcal{M}_{ANA}}$ and $⟦y⟧_{\mathcal{M}_{ANA}}$ are the same number; otherwise, $⟦x^{⌢}⌜{=}⌝^{⌢}y⟧_{\mathcal{M}_{ANA}} = \False$.

Rule 7. Closure clause: nothing else is in $n$, $np$, or $s$.

Note that we are using quite a lot of mathematical notation in the metalanguage! That’s fine—there is no point in pretending that we don’t know how to add and multiply, etc. One thing to note is the use of the function called ‘$length$’ in Rule 3. This is a function that maps strings onto natural numbers—kind of like the interpretation function does! That is:

$length : Σ_{ANA}^* → ℕ$

That is, its domain is entire the set of strings over the alphabet (here I’ve used the “Kleene star” notation), and its codomain is the set of natural numbers.

Of course, $length$ does not behave like the interpretation function at all. For instance, $length(⌜12⌝)$ is not $12$—it is $2$!

Rules as proof constructors

On the second presentation, we’ll present the exact same collection of rules, but using a slightly different notation. Rather than writing out each rule in English, we’ll encode it as a kind of rule for constructing proofs. Here, each proof, or derivation, is just a proof that some string in $L_{ANA}$ has a particular category, as well as a particular interpretation.

Lexicon. \[\begin{prooftree}\AxiomC{$⟨⌜0⌝, I_{ANA}(⌜0⌝)⟩ ⊢ atom$}\end{prooftree}\]
\[\begin{prooftree}\AxiomC{$⟨⌜1⌝, I_{ANA}(⌜1⌝)⟩ ⊢ atom$}\end{prooftree}\]
\[\begin{prooftree}\AxiomC{⋮}\end{prooftree}\]
\[\begin{prooftree}\AxiomC{$⟨⌜9⌝, I_{ANA}(⌜9⌝)⟩ ⊢ atom$}\end{prooftree}\] Rules. \[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ atom$} \RightLabel{Rule 2}\UnaryInfC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \RightLabel{Rule 3}\BinaryInfC{$⟨x^{⌢}y, ⟦x⟧_{\mathcal{M}_{ANA}} * 10^{length(y)} + ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \RightLabel{Rule 4}\UnaryInfC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \RightLabel{Rule 5}\BinaryInfC{$⟨⌜(⌝^{⌢}x^{⌢}⌜+⌝^{⌢}y^{⌢}⌜)⌝, ⟦x⟧_{\mathcal{M}_{ANA}} + ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \RightLabel{Rule 5}\BinaryInfC{$⟨⌜(⌝^{⌢}x^{⌢}⌜×⌝^{⌢}y^{⌢}⌜)⌝, ⟦x⟧_{\mathcal{M}_{ANA}} * ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \end{prooftree}\]
\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \RightLabel{Rule 6}\BinaryInfC{$⟨x^{⌢}⌜{=}⌝^{⌢}y, ⟦x⟧_{\mathcal{M}_{ANA}} = ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ s$} \end{prooftree}\]

It is worth—that is, I highly recommend—reading each of the rules in (5) carefully and comparing it to its corresponding rule in (3), in order to convince yourself that its two versions say exactly the same thing. Note that the closure clause is only implied on the second presentation.

References

Keenan, Edward L., and Lawrence S. Moss. 2016. Mathematical Structures in Language. CSLI Lecture Notes. CSLI Publications.

Footnotes

Recall that our metalanguage is just the language that we’re using to conduct Introduction to Semantics (LIN4803/LIN6804). It consists of our natural language idiolects (as spoken and written) and a little bit of set-theoretic notation, as well as certain other mathematical notation. I’m assuming our idiolects (as written) make Arabic numeral arithmetic available.↩︎
After all, according to Keenan and Moss (2016), “if you can’t say something two ways, you can’t say it at all”. Not sure if this is what they had in mind.↩︎

--- title: "Arabic numeral arithmetic: semantics" bibliography: ../../semantics.bib format: html: css: ../styles.css html-math-method: mathjax mathjax-config: loader: {load: ['[tex]/bussproofs','[tex]/bbox','[tex]/colorbox']} tex: packages: {'[+]': ['bussproofs','bbox','colorbox']} --- ::: {.hidden} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \def\True{\ct{T}} \def\False{\ct{F}} $$ ::: What we would like to do is figure out a way of associating elements of $L_{ANA}$ with *meanings*. In order to do this, we have to make use of a *model*---that is, one of the things that we introduced [here](overview.html). Which model, exactly, should we be using to *interpret* $L_{ANA}$? We'll follow the standard way of doing things and say that the model of $L_{ANA}$ is the one defined in (@ex-ana-model), which has as its domain the set $ℕ$ of natural numbers. (@ex-ana-model) $\mathcal{M}_{ANA} = ⟨ℕ, \{\True, \False\}, I_{ANA}⟩$ That is, $D$ is defined to be $ℕ$---i.e., the set $\{0, 1, 2, ...\}$. What can we say about the model's interpretation function, $I_{ANA}$? Well, because we defined $atom$ ([here](ana_syntax.html)) to be the set $\{⌜0⌝, ..., ⌜9⌝\}$, we know that this set is the *domain* of $I_{ANA}$: it is the job of the model's interpretation function to map things from this set onto interpretations of some kind. What about the codomain of $I_{ANA}$? Since we're having $I_{ANA}$ map elements *from* the set $\{⌜0⌝, ..., ⌜9⌝\}$, we should choose the *codomain* of $I_{ANA}$---the set it maps these symbols *to*---to be $ℕ$: the interpretation of any of the symbols $⌜0⌝$ through $⌜9⌝$ is just a natural number! Specifically, we'll define $I_{ANA}$ as in (@ex-ana_atom_interp). (@ex-ana_atom_interp) $I_{ANA} : atom → ℕ$ $I_{ANA}(⌜0⌝) = 0$ $I_{ANA}(⌜1⌝) = 1$ $I_{ANA}(⌜2⌝) = 2$ $I_{ANA}(⌜3⌝) = 3$ $I_{ANA}(⌜4⌝) = 4$ $I_{ANA}(⌜5⌝) = 5$ $I_{ANA}(⌜6⌝) = 6$ $I_{ANA}(⌜7⌝) = 7$ $I_{ANA}(⌜8⌝) = 8$ $I_{ANA}(⌜9⌝) = 9$ Hopefully, none of these choices is particularly surprising! Still, it's important to point out that---while it looks a bit like we are using the interpretation function to simply "remove the corner brackets ($⌜, ⌝$)"---that's just due to the fact that we are employing the very language that we are interpreting as part of our *metalanguage*.^[ Recall that our metalanguage is just the language that we're using to conduct Introduction to Semantics (LIN4803/LIN6804). It consists of our natural language idiolects (as spoken and written) and a little bit of set-theoretic notation, as well as certain other mathematical notation. I'm assuming our idiolects (as written) make Arabic numeral arithmetic available. ] Said another way: there is nothing special about the symbol $⌜1⌝$ that makes it a particularly good candidate for being mapped onto the number 1. The choice reflects how we *use* the language $L_{ANA}$, but it is arbitrary nonetheless---a genuine Lewis-style convention. Alright. The time has come to define our interpretation function (or "interpretation brackets", "denotation function", what have you) $⟦·⟧_{\mathcal{M}_{ANA}}$ over the entirety of $L_{ANA}$. We will show how to do this using two different presentations.^[ After all, according to @keenan_mathematical_2016, "if you can't say something two ways, you can't say it at all". Not sure if this is what they had in mind. ] One presentation uses English and provides the rules mostly as conditional *if then* statements. The other presentation is a bit more formally systematic: it provides formal rules for constructing *proofs*. Importantly, both presentations *say* exactly the same thing. ### Rules in English On the first presentation, we'll append additional clauses to the syntactic rules that were stated in [Arabic numeral arithmetic: syntax](ana_syntax.html). This presentation is given in (@ex-ana-semantics). (@ex-ana-semantics) Rule 1. $atom = \{⌜0⌝, ..., ⌜9⌝\}$, and for any $x ∈ atom$, $⟦x⟧_{\mathcal{M}_{ANA}} = I_{ANA}(x)$. Rule 2. $atom ⊆ n$. Rule 3. If $x, y ∈ n$, then $x^{⌢}y ∈ n$, and $⟦x^{⌢}y⟧_{\mathcal{M}_{ANA}} = ⟦x⟧_{\mathcal{M}_{ANA}} * 10^{length(y)} + ⟦y⟧_{\mathcal{M}_{ANA}}$. Rule 4. $n ⊆ np$. Rule 5. If $x, y ∈ np$, then $⌜(⌝^{⌢}x^{⌢}⌜+⌝^{⌢}y^{⌢}⌜)⌝, ⌜(⌝^{⌢}x^{⌢}⌜×⌝^{⌢}y^{⌢}⌜)⌝ ∈ np$, and also: - $⟦⌜(⌝^{⌢}x^{⌢}⌜+⌝^{⌢}y^{⌢}⌜)⌝⟧_{\mathcal{M}_{ANA}} =⟦x⟧_{\mathcal{M}_{ANA}} + ⟦y⟧_{\mathcal{M}_{ANA}}$ - $⟦⌜(⌝^{⌢}x^{⌢}⌜×⌝^{⌢}y^{⌢}⌜)⌝⟧_{\mathcal{M}_{ANA}} =⟦x⟧_{\mathcal{M}_{ANA}} * ⟦y⟧_{\mathcal{M}_{ANA}}$ Rule 6. If $x, y ∈ np$, then $x^{⌢}⌜{=}⌝^{⌢}y ∈ s$, and $⟦x^{⌢}⌜{=}⌝^{⌢}y⟧_{\mathcal{M}_{ANA}} = \True$ if $⟦x⟧_{\mathcal{M}_{ANA}}$ and $⟦y⟧_{\mathcal{M}_{ANA}}$ are the same number; otherwise, $⟦x^{⌢}⌜{=}⌝^{⌢}y⟧_{\mathcal{M}_{ANA}} = \False$. Rule 7. Closure clause: nothing else is in $n$, $np$, or $s$. Note that we are using quite a lot of mathematical notation in the metalanguage! That's fine---there is no point in pretending that we don't know how to add and multiply, etc. One thing to note is the use of the function called '$length$' in Rule 3. This is a function that maps *strings* onto *natural numbers*---kind of like the interpretation function does! That is: (@ex-length-function) $length : Σ_{ANA}^* → ℕ$ That is, its domain is entire the set of strings over the alphabet (here I've used the ["Kleene star" notation](https://en.wikipedia.org/wiki/Kleene_star)), and its codomain is the set of natural numbers. Of course, $length$ does not *behave* like the interpretation function at all. For instance, $length(⌜12⌝)$ is not $12$---it is $2$! ### Rules as proof constructors On the second presentation, we'll present the *exact same collection of rules*, but using a slightly different notation. Rather than writing out each rule in English, we'll encode it as a kind of rule for constructing *proofs*. Here, each proof, or *derivation*, is just a proof that some string in $L_{ANA}$ has a particular *category*, as well as a particular *interpretation*. (@ex-ana-proofs) Lexicon. \begin{prooftree}\AxiomC{$⟨⌜0⌝, I_{ANA}(⌜0⌝)⟩ ⊢ atom$}\end{prooftree} \begin{prooftree}\AxiomC{$⟨⌜1⌝, I_{ANA}(⌜1⌝)⟩ ⊢ atom$}\end{prooftree} \begin{prooftree}\AxiomC{⋮}\end{prooftree} \begin{prooftree}\AxiomC{$⟨⌜9⌝, I_{ANA}(⌜9⌝)⟩ ⊢ atom$}\end{prooftree} Rules. \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ atom$} \RightLabel{Rule 2}\UnaryInfC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \RightLabel{Rule 3}\BinaryInfC{$⟨x^{⌢}y, ⟦x⟧_{\mathcal{M}_{ANA}} * 10^{length(y)} + ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \RightLabel{Rule 4}\UnaryInfC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \RightLabel{Rule 5}\BinaryInfC{$⟨⌜(⌝^{⌢}x^{⌢}⌜+⌝^{⌢}y^{⌢}⌜)⌝, ⟦x⟧_{\mathcal{M}_{ANA}} + ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \RightLabel{Rule 5}\BinaryInfC{$⟨⌜(⌝^{⌢}x^{⌢}⌜×⌝^{⌢}y^{⌢}⌜)⌝, ⟦x⟧_{\mathcal{M}_{ANA}} * ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \end{prooftree} \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ np$} \RightLabel{Rule 6}\BinaryInfC{$⟨x^{⌢}⌜{=}⌝^{⌢}y, ⟦x⟧_{\mathcal{M}_{ANA}} = ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ s$} \end{prooftree} It is worth---that is, I highly recommend---reading each of the rules in (@ex-ana-proofs) carefully and comparing it to its corresponding rule in (@ex-ana-semantics), in order to convince yourself that its two versions say exactly the same thing. Note that the closure clause is only implied on the second presentation.