Assignment

Due on Monday, October 13th.

Part 1

There is a planet orbiting Proxima Centauri that has a population of organisms which are very human-like. As it happens, a group within the larger population uses a system a lot like Arabic numeral arithmetic to perform arithmetic calculations, except that the syntax and semantics are slightly different. This group uses a base-7 number system, and the atomic numerals have different meanings than they do in Arabic numeral arithmetic. We can call the language they use for arithmetic calculations $L_{NANA}$ (not Arabic numeral arithmetic).

To be more precise, for this group within the population which lives on the planet orbiting Proxima Centauri, the set of atomic numerals is the following one:

\[atom = \{⌜1⌝, ⌜2⌝, ⌜3⌝, ⌜4⌝, ⌜5⌝, ⌜6⌝, ⌜7⌝\}\]

Meanwhile, the interpretation function of the model employed by this group, $I_{NANA}$, is the following one:

\[I_{NANA}(⌜1⌝) = 6\] \[I_{NANA}(⌜2⌝) = 5\] \[I_{NANA}(⌜3⌝) = 4\] \[I_{NANA}(⌜4⌝) = 3\] \[I_{NANA}(⌜5⌝) = 2\] \[I_{NANA}(⌜6⌝) = 1\] \[I_{NANA}(⌜7⌝) = 0\]

The full model of $L_{NANA}$ employed by this group is thus:

\[\mathcal{M}_{NANA} = ⟨ℕ, \{\True, \False\}, I_{NANA}⟩\]

One more thing. Instead of the Rule 3 which we defined in Arabic numeral arithmetic: semantics, these guys use the following alternative Rule 3:

\[\begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \RightLabel{Rule 3}\BinaryInfC{$⟨x^{⌢}y, ⟦x⟧_{\mathcal{M}_{ANA}} * 7^{length(y)} + ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \end{prooftree}\]

The rules of for generating strings of $L_{NANA}$ and their interpretations are otherwise identical to those for generating strings of $L_{ANA}$ and their interpretations.

Show that $⌜((53 + 6) × 5) = 24⌝$ is a sentence (i.e., an $s$) of $L_{NANA}$, and show what its interpretation in $\mathcal{M}_{NANA}$ is.

Part 2

Using the English grammar fragment defined in A first approximation of English in conjunction with the model defined in An English model and example, show that the string in (1) is a sentence (i.e., an $s$), and show what its interpretation is.

Bella doesn’t jump and run.

Part 3

As you know, English does not only have sentences like (1). It also has sentences like (2).

Bella doesn’t jump and Bella doesn’t run.

Let us propose that in addition to the word and, English has another word, and2, which is pronounced exactly the same way as and is pronounced (i.e., [æːnd]), but which coordinates sentences instead of verb phrases. Thus and is the variant of the coordinator that appears in (1), while and2 is the variant of the coordinator that appears in (2).

Provide a lexical entry for the variant of the coordinator which appears in (2)—something that can be added to the lexicon given in A first approximation of English. Give a derivation showing that (2) is a sentence with a meaning, using this lexical entry. Does your lexical entry ensure that (1) and (2) entail each other? If so, how does it ensure this?

--- title: "Assignment" 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} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \def\True{\ct{T}} \def\False{\ct{F}} $$ ::: Due on Monday, October 13th.        ### Part 1 There is a planet orbiting [Proxima Centauri](https://en.wikipedia.org/wiki/Proxima_Centauri) that has a population of organisms which are very human-like. As it happens, a group within the larger population uses a system a lot like Arabic numeral arithmetic to perform arithmetic calculations, except that the syntax and semantics are slightly different. This group uses a base-7 number system, and the atomic numerals have different meanings than they do in Arabic numeral arithmetic. We can call the language they use for arithmetic calculations $L_{NANA}$ (not Arabic numeral arithmetic). To be more precise, for this group within the population which lives on the planet orbiting Proxima Centauri, the set of *atomic* numerals is the following one: $$atom = \{⌜1⌝, ⌜2⌝, ⌜3⌝, ⌜4⌝, ⌜5⌝, ⌜6⌝, ⌜7⌝\}$$ Meanwhile, the interpretation function of the model employed by this group, $I_{NANA}$, is the following one: $$I_{NANA}(⌜1⌝) = 6$$ $$I_{NANA}(⌜2⌝) = 5$$ $$I_{NANA}(⌜3⌝) = 4$$ $$I_{NANA}(⌜4⌝) = 3$$ $$I_{NANA}(⌜5⌝) = 2$$ $$I_{NANA}(⌜6⌝) = 1$$ $$I_{NANA}(⌜7⌝) = 0$$ The full model of $L_{NANA}$ employed by this group is thus: $$\mathcal{M}_{NANA} = ⟨ℕ, \{\True, \False\}, I_{NANA}⟩$$ One more thing. Instead of the Rule 3 which we defined in [Arabic numeral arithmetic: semantics](ana_semantics.html), these guys use the following *alternative* Rule 3: \begin{prooftree} \AxiomC{$⟨x, ⟦x⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \AxiomC{$⟨y, ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \RightLabel{Rule 3}\BinaryInfC{$⟨x^{⌢}y, ⟦x⟧_{\mathcal{M}_{ANA}} * 7^{length(y)} + ⟦y⟧_{\mathcal{M}_{ANA}}⟩ ⊢ n$} \end{prooftree} The rules of for generating strings of $L_{NANA}$ and their interpretations are otherwise identical to those for generating strings of $L_{ANA}$ and their interpretations. Show that $⌜((53 + 6) × 5) = 24⌝$ is a sentence (i.e., an $s$) of $L_{NANA}$, and show what its interpretation in $\mathcal{M}_{NANA}$ is. ### Part 2 Using the English grammar fragment defined in [A first approximation of English](english.html) in conjunction with the model defined in [An English model and example](english_examples.html), show that the string in (@ex-bella) is a sentence (i.e., an $s$), and show what its interpretation is. (@ex-bella) Bella doesn't jump and run. ### Part 3 As you know, English does not only have sentences like (@ex-bella). It *also* has sentences like (@ex-bella-two-clauses). (@ex-bella-two-clauses) Bella doesn't jump and Bella doesn't run. Let us propose that in addition to the word *and*, English has another word, *and2*, which is *pronounced* exactly the same way as *and* is pronounced (i.e., [æːnd]), but which coordinates *sentences* instead of verb phrases. Thus *and* is the variant of the coordinator that appears in (@ex-bella), while *and2* is the variant of the coordinator that appears in (@ex-bella-two-clauses). Provide a lexical entry for the variant of the coordinator which appears in (@ex-bella-two-clauses)---something that can be added to the lexicon given in [A first approximation of English](english.html). Give a derivation showing that (@ex-bella-two-clauses) is a sentence with a meaning, using this lexical entry. Does your lexical entry ensure that (@ex-bella) and (@ex-bella-two-clauses) entail each other? If so, how does it ensure this?