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In On Function and Concept, Gottlob Frege makes a distinction between concepts and objects. Specifically, he claims that objects are *saturated object-expressions* while concepts are *unsaturated function-expressions*. My goal is to illuminate what the *object-concept* distinction entails in the context of simply-typed lambda calculus $(\lambda^{\rightarrow})$ and whether or not this entailment is susceptible to problems like the $\mathtt{\text{concept horse paradox}}$.

Frege’s On Function and Concept (1892) claims that there is a substantive difference between concepts and objects. This distinction is described on pg. 147:

Statements in general, just like equations or inequalities or expressions in Analysis, can be imagined to be split up into two parts; one complete in itself, and the other in need of supplementation, or unsaturated. Thus, e.g., we split up the sentence Caesar conquered Gaul into Caesar and conquered Gaul.

**Definition 1.** Complete parts of statements are saturated’ while incomplete ones are unsaturated’.

**Example 1.** Consider the following statement:

$$\text{Fido jumped over the fence’}$$

By **Definition 1**, we can see that (1) is split up into the saturated Fido’ object-expression and jumped over the fence’ unsaturated function-expression.

*Remark 1.* These two categories, as it turns out, have simple analogues in $\lambda^{\rightarrow}$. Unsaturated function-expressions are terms with functional types, e.g. $N^{\sigma\rightarrow\tau}$ or $N:\sigma\rightarrow\tau$, while saturated object-expressions are terms with non-functional types, e.g. $M^{\sigma}$ or $M:\sigma$.

**Definition 2.** We say that if $M$ gets type $\sigma$ and $N$ gets type $\sigma\rightarrow\tau$ then the application $NM$ is legal (as $N$ is considered a function from terms of type $\sigma$ to terms of type $\tau$) and gets type $\tau$.

**Definition 3.** The set of types of $\lambda^{\rightarrow}$, denoted by Type($\lambda^{\rightarrow}$) is inductively defined as follows. We write $\mathbb{T}=\text{Type}(\lambda^{\rightarrow})$ where

$\begin{array}{ll}\alpha,\alpha^{\prime},\alpha^{\prime\prime},\ldots\in\mathbb{T} & \text{(type variables);}\\\sigma,\tau\in\mathbb{T}\Rightarrow(\sigma\rightarrow\tau)\in\mathbb{T} &\text{(function space variables).}\end{array}$

**Definition 4.** We will now define a new variant of $\mathbb{T}$ that more closely models Frege’s natural language notions of concept and object. This new variant, $\mathbb{T_{\mathcal{F}}}$, has the following restrictions:

- There are three and only three types: $\begin{cases}\alpha\in\mathbb{T_{\mathcal{F}}} & \text{(}objects\text{);}\\\mathbf{\text{H}}\in\mathbb{T_{\mathcal{F}}} & \text{(truth-values);}\\(\alpha\rightarrow\mathbf{\text{H}})\in\mathbb{T_{\mathcal{F}}} & \text{(}concepts\text{).}\end{cases}$
- Objects have an unrestricted domain.
- A truth-value is a boolean type where $\mathbf{\text{H}}=\{\text{true’},\text{false’}\}$.
- A
*concept*is a functional type that takes an object as an input and returns a truth value. This mapping is done by some valuation function $v$ where $v$ maps to $\text{true’}$ if the function-expression applied to the argument(s) is true, e.g., in (1), if Fido*did*jump over a fence then $v(\text{Fido’})=\text{true’}$; otherwise, $v$ maps to $\text{false’}$. - Note that $(\mathbf{\text{H}}\rightarrow\alpha)\notin\mathbb{T_{\mathcal{F}}}$.

*Remark 2.* Until otherwise noted, we will work only with types in $\mathbb{T_{\mathcal{F}}}$.

**Definition 5.** Let $M$ be an untyped $\lambda^{\rightarrow}$-term. Given the non-functional type $\alpha$, we say that $M:\alpha$ is a saturated *object-expression*, or simply an *object*.

**Definition 6.** Let $N$ be an untyped $\lambda^{\rightarrow}$-term. Given the functional type $\alpha\rightarrow\mathbf{\text{H}}$, we say that $N:\alpha\rightarrow\mathbf{\text{H}}$ is an *unsaturated function-expression*, or simply a *concept*.

**Definition 7.** We will now make a semantic distinction between two kinds of uses of the token is’ in natural language. To illustrate the distinction, consider:

- Mark Twain is Samuel Clemens’: here, we have the
*is’ of identity*, or $x=x$. The equality operator $=$ is used as a mathematical formalism and may not necessarily be computable in $\lambda^{\rightarrow}$. So, $v(\text{Mark Twain’})=\text{true’}$ iff $\text{Mark Twain’}=\text{Samuel Clemens’}$. - Mark Twain is dead’: here, we have the
*is’ of predication*which, if combined with some property $P$ will yield a functional type of form $\alpha\rightarrow\mathbf{\text{H}}$ for some object $\alpha$. As in Definition 4, if $\alpha$ has property $P$, $v(\alpha)=\text{true’}$; otherwise, $v$ maps to $\text{false’}$.

**Lemma 1.** Consider the same sentence from **Example 1**:

$$\text{Fido jumped over the fence’}$$

This sentence can be formulated in $\lambda^{\rightarrow}$ like so:

$$\Gamma\vdash J:\alpha\rightarrow\mathbf{\text{H}},\,\Gamma\vdash F:\alpha\Rightarrow\Gamma\vdash(JF):\mathbf{\text{H}}$$

*Proof.* We will set the object-expression Fido’ $=F$ and the (one-place) function-expression $\left(x\right)$ jumped over the fence’ $=J\left(x\right)$. Note that we will drop the $\left(x\right)$ in practice for $\lambda$-calculus wff satisfiability. By **Definition 5**, $F$ must be of type $\alpha$. And to make $(JF)$ legal, by **Definition 2**, $J$ must be of type $\alpha\rightarrow\mathbf{\text{H}}$. Therefore, a correctly typed $(JF)$ from $J:\alpha\rightarrow\mathbf{\text{H}}$ and $F:\alpha$ looks like
$$\left(JF\right):\mathbf{\text{H}}$$
which is precisely what we wanted to show in (3). $\square$

*Proof.* We can also derive a more formally-rigorous proof. Consider:
$$\dfrac{\Gamma\vdash J:\alpha\rightarrow\mathbf{\text{H}}\,\,\,\,\,\,\,\Gamma\vdash F:\alpha}{\Gamma\vdash JF:\mathbf{\text{H}}}{\scriptscriptstyle (\rightarrow e)}$$

where we get the same result: $JF$ is of type $\mathbf{\text{H}}$. $\square$

*Remark 3.* Natural language predicates (e.g. is blue’, jumped over the fence’, is a concept’, conquered Gaul’, may have stolen my wallet’, etc.) are *concepts* in $\mathbb{T_{\mathcal{F}}}$ while subjects, be they proper nouns, improper nouns, pronouns, etc. (e.g. the car’, Fido’, he’, the tall man’, the Brooklyn Bridge’, etc.) are *objects*. We will now look at more nontrivial examples of natural language propositions and attempt to model them in $\lambda^{\rightarrow}$ under $\mathbb{T_{\mathcal{F}}}$.

**Example 2.** Consider the statement

$$\text{The concept horse is a concept.’}$$

*Claim 1.* We will set the *object-expression* The concept horse’ $=T_{ch}$ and the (one-place) *function-expression* $\left(x\right)$ is a concept’ $=C$. **Example 2** is witness to a semantic ambiguity between natural language and our type system ($\mathbb{T_{\mathcal{F}}}$). More specifically, (6) evaluates to true’ in natural language, but false’ under $\mathbb{T_{\mathcal{F}}}$, i.e. $CT_{ch}:\mathbf{\text{H}}$ has an ambiguous truth-value. So, if $T_{ch}:\alpha$ is the *saturated object-expression* the concept horse’ and $C:\alpha\rightarrow\mathbf{\text{H}}$ is the *unsaturated function-expression* is a concept’, then the mapping $v:\alpha\rightarrow\mathbf{\text{H}}$ is ambiguous.

*Proof.* Consider the following natural language statement

$$\text{The house is a house.’}$$

Trivially, (7) is true. Per **Definition 7**, the is’ here seems to be an *is’ of identity*, so $v(\text{house’})=\text{true’}$ since $\text{house’}=\text{house’}$.

Now, let’s define a new set of types $\mathbb{T_{\mathcal{F}}^{\mathrm{*}}}$ where $\mathbb{T_{\mathcal{F}}^{\mathrm{*}}}$ is just like $\mathbb{T_{\mathcal{F}}}$ except that $\mathbb{T_{\mathcal{F}}^{\mathrm{*}}}$ has one and only one additional type: $\beta\in\mathbb{T_{\mathcal{F}}^{\mathrm{*}}}$. We will call all terms of type $\beta$ *houses*. Now, we have an ambiguity in (7); it is unclear whether we’re asking if the identity $\text{house’}=\text{house’}$ holds or whether we’re asking if the *saturated object-expression* the house’ is of type $\beta$. The former is true per the first half of this proof. The latter is false, as any object-expression is of type $\alpha$ in $\mathbb{T_{\mathcal{F}}}$ and, by extension, also in $\mathbb{T_{\mathcal{F}}^{\mathrm{*}}}$ and $\alpha\neq\beta$. This same kind of ambiguity happens in (7). $\square$

**Definition 8.** Given the proof of *Claim 1*, we will now denote two variations of $v$, essentially defining two valuations for $\alpha\rightarrow\mathbf{\text{H}}$:

- $v_{0}$ for meta-language predicates;
- $v_{\mathfrak{L}}$ for natural language predicates.

In light of this, we now have two interpretations for statements that involve *objects*, *concepts*, and truth-values. So, in some cases, $v_{0}(CT_{ch}:\mathbf{\text{H}})$ will evaluate to false’, but $v_{\mathfrak{L}}(CT_{ch}:\mathbf{\text{H}})$ evaluates to true’ or vice-versa. This is known as the $\mathtt{\text{concept horse paradox}}$.

**Theorem 1.** Given any object-expression, $\mathbb{O}:\alpha$, and if we let the function-expression $\left(x\right)$ *is a concept*‘ $=C$, $v_{0}(C\mathbb{O}:\mathbf{\text{H}})$ will always evaluate to false’.

*Proof.* From the proof of *Claim 1* and **Definition 8**. $\square$

**Example 3.** Now consider the statement:

$$\text{Seabiscuit is the concept horse.’}$$

*Claim 2.* We will set the *object-expression* Seabiscuit’ $=S$ and the (one-place) function-expression $\left(x\right)$ is the concept horse’ $=C_{h}$. **Example 3** avoids the ambiguity in (6) because its truth value happens to be false under both valuations, so $v_{0}(C_{h}S:\mathbf{\text{H}})=v_{\mathfrak{L}}(C_{h}S:\mathbf{\text{H}})$.

*Proof.* By **Theorem 1**, $v_{0}(C_{h}S:\mathbf{\text{H}})$, will always evaluate to false’. In natural language, Seabiscuit’ is not the concept horse’ (at best, it is an instance of the concept horse’), so $v_{\mathfrak{L}}(C_{h}S:\mathbf{\text{H}})$ will evaluate to false’ here as well. Since both valuations return false’, we avoid the paradox. $\square$

**Example 4.** Consider the statement:

$$\text{Anna is something Bill is not — namely, a student.’}$$

*Remark 4.* **Example 4** is not ambiguous as it only uses terms found in natural language. This is trivial to prove (since the only valid valuation is $v_{\mathfrak{L}}$) However, (9) introduces another problem with $\mathbb{T_{\mathcal{F}}}$ (more specifically, with $\lambda^{\rightarrow}$): subtyping. It seems intuitive that being a student’ should be a member of things Bill is not’. As it turns out, $\lambda^{\rightarrow}$ has no way of representing this.

The most straightforward solution to the semantic problem of the $\mathtt{\text{concept horse paradox}}$ is a simple one. Instead of using terms like *concept* and *object*, that not only have a meaning in the meta-language, but also in natural languages, we will use terms that have no meaning in natural language. Consider a new type system, $T_{\mathcal{F}}^{\mathrm{**}}$.

The types of $\mathbb{T_{\mathcal{F}}^{\mathrm{*}}}$: $\begin{cases}\alpha\in\mathbb{T_{\mathcal{F}}} & \text{(}foo\text{);}\\\mathbf{\text{H}}\in\mathbb{T_{\mathcal{F}}} & \text{(baz);}\\(\alpha\rightarrow\mathbf{\text{H}})\in\mathbb{T_{\mathcal{F}}} & \text{(}foobaz\text{).}\end{cases}$

Under such a schema, statements like The concept horse is [a] foobaz’ are meaningless in natural language, but are semantically well-formed in the meta-language. Thus, we can infer that we need to valuate the statement with $v_{0}$ as opposed to $v_{\mathfrak{L}}$ and avoid any ambiguity.

*Remark 5.* The caveat here is that we lose some expressiveness in natural languages. In **Example 3**, we were able to refer both to the semantics of natural language and the meta-language without ambiguity. This would no longer be possible.

Another proposed solution is syntactic in nature. We previously showed that Fido jumped over the fence’ (**Example 1**) can be formulated as follows:

$$\dfrac{\Gamma\vdash J:\alpha\rightarrow\mathbf{\text{H}}\,\,\,\,\,\,\,\Gamma\vdash F:\alpha}{\Gamma\vdash JF:\mathbf{\text{H}}}{\scriptscriptstyle (\rightarrow e)}$$

We will now introduce a new type of $\lambda^{\rightarrow}$-term that will differentiate between what valuation $v$ one should use to determine the truth-value of some $M:\mathbf{\text{H}}$. To do this, consider a new type system $\mathbb{T_{\mathcal{F}}^{\mathrm{\omega}}}$.

The types of $\mathbb{T_{\mathcal{F}}^{\mathrm{\omega}}}$: $\begin{cases} \alpha\in\mathbb{\mathbb{T_{\mathcal{F}}^{\mathrm{\omega}}}} & \text{(}objects\text{);}\\\mathbf{\text{H}}\in\mathbb{T_{\mathcal{F}}^{\mathrm{\omega}}} & \text{(truth-values);}\\(\alpha\rightarrow\omega\rightarrow\mathbf{\text{H}})\in\mathbb{T_{\mathcal{F}}^{\mathrm{\omega}}} & \text{(unrestricted }concepts\text{);}\\(\omega\rightarrow\mathbf{\text{H}})\in\mathbb{T_{\mathcal{F}}^{\mathrm{\omega}}} & \text{(restricted }concepts\text{);} \\\omega\in\mathbb{T_{\mathcal{F}}^{\mathrm{\omega}}} & \text{(}\textbf{worlds}\text{).} \end{cases}$

A derivation of (10) would now look like the following:

$$\dfrac{\dfrac{\Gamma\vdash J:\alpha\rightarrow\omega\rightarrow\mathbf{\text{H}}\,\,\,\,\,\,\,\Gamma\vdash F:\alpha\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}{\Gamma\vdash JF:\omega\rightarrow\mathbf{\text{H}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma\vdash W:\omega}{\scriptscriptstyle (\rightarrow e)}}{\Gamma\vdash JFW:\mathbf{\text{H}}}{\scriptscriptstyle (\rightarrow e)}$$

The purpose of $W$ is to pick out one (or several, if there are no ambiguities) valuating function(s). So, in **Example 1**, $W$ picks out $v_{\mathfrak{L}}$, in **Example 3**, $W$ picks out $\{v_{\mathfrak{L}},v_{0}\}$, and in **Example 2**, $W$ either picks out $v_{\mathfrak{L}}$ or $v_{0}$ but not both. A benefit of $W:\omega$ is that it can pick out the meta-language ($v_{0}$), meta-meta-language ($v_{1}$), meta-meta-meta-langauge ($v_{2}$), etc. ($v_{n}$). Furthermore, $W$ can also pick out combinations of languages ($\{v_{\mathfrak{L}},v_{0},v_{2},v_{n}\}$), provided the truth value stays consistent.

*Remark 6.* A caveat of this syntactic addition is that $W$ tends to be implicit and thus, the $\mathtt{\text{concept horse paradox}}$ merely shifts (now we have an ambiguity of $W$) and does not completely disappear. A rather elegant fix is letting $W$ pick out $v_{\mathfrak{L}}$ by default. In such a case, the paradox would dissolve and we would still preserve the truth-value in sentences like **Example 3**.