Sense, reference, and computation

Frege’s theory of sense and reference is systematically introduced for both singular terms and complete sentences in his seminal essay Sinn und Bedeutung. The theory seems to provide a compelling answer to the paradox of identity with respect to singular terms, for we are able to say that the cognitive value of $a=a$ and $a=b$ is generally different because $a$ and $b$ may differ in their sense, despite being coreferential. However, the theory becomes highly controversial when we consider the sense and reference of sentences, which are taken to express a thought (in modern terminology, a proposition) and refer to a truth value, which, since referents are always objects, are seen as the simplest representatives of logical objects. Frege is well aware of this counterintuitive aspect of his doctrine, as he immediately attempts to justify it by noting that what he calls an object can only be properly understood in connection to concepts and relations and then attempts to substantiate his claim that the reference of a sentence is its truth value by remarking that the latter remain unchanged when a part of the sentence is replaced by a coreferential expression, that is, that sentences with the same truth value are interchangeable salva veritate. Later on, in the preface of \gga, Frege remarks that the introduction of truth-values may seem strange at first, but adds that the fact that everything becomes much simpler and sharper with those objects puts a great weight in the balance in favor of his own conception.⁶⁶6 Cf. \gga I, x. In the ideography, concepts and relations are treated as particular cases of functions that output truth values and Frege explicitly states that an object is anything that is not a function.

Besides the poor justifications for Frege’s adoption of his strange doctrine of truth values, it is disappointing that almost no considerations about sense or thoughts can be found in \gga, at least if we assume that the real purpose of the distinction between sense and reference is to offer a solution to the paradox of identity. Although Frege still seems to have a strong conviction that a criterion of identity for senses is of fundamental importance, as we can judge from, for instance, a December 1906 letter to Husserl, where Frege writes:

It seems to me that an objective criterion is necessary for recognizing a thought again as the same, for without it logical analysis is impossible. (Frege, 1980, p. 70)

It is possible that Frege had come to recognize that the most important notion for the vindication of his logicism is that of reference, as pointed out by Simons (1992), and that for those purposes all that we need to know about sense is that a sentence needs to express a thought in order to refer at all. In fact, now the horizontal stroke symbolism

$$\F[1]A$$

$$\Fn[1]A$$

that has the opposite effect of yielding the false if $A$ does not refer to the true and the true otherwise. The turnstile judgment has kept its role, being the only judgment form of the revised ideography. It is now fully explained in terms of reference as well, now asserting that the expression $A$ refers to the true. There is no way to formally assert the fact that a sentence expresses a thought in \gga, which is to say to fulfill the role previously played by the content stroke.

In general, it would seem that Frege contradicts himself by downplaying the value of sense and treating his doctrine of truth values as a mere technical device for the successful accomplishment of his logicism, as argued in Ruffino (1997) and Duarte (2009). This can be seen more clearly in light of the discarded manuscript that was written before the introduction of the distinction between sense and reference to the ideography in \gga. In §69 of \gla, immediately after the explicit definition of the concept of number is proposed by Frege, he attempts to confirm the fruitfulness of his definition by sketching a proof of a crucial theorem that says that the number that falls under the concept $F$ is equal to the number that falls under the concept $G$ iff $F$ and $G$ are in one-to-one correspondence (Hume’s Principle). The proof consists in showing that both sides of the biconditional imply each other. But in the ideography, where a biconditional is always represented by an equality, the formalization of the proof cannot be completed without propositional extensionality. In \gga, what makes the derivation of propositional extensionality as a theorem possible is the introduction of Basic Law IV, which roughly states that the truth values denoted by the sentences $A$ and $B$ either coincide or not:

$$\neg(\F[1]A=\Fn[1]B)\varsupset(\F[1]A=\F[1]B).$$

But this is only allowed as an axiom in the ideography because of the distinction between sense and reference. If those observations are correct, Frege conceived his theory of sense and reference primarily as a logical apparatus necessary to overcome technical obstacles rather than as a solution to the paradox of identity as it is commonly thought.⁷⁷7 See e.g. Ruffino (1997, §3) and Duarte (2009, p.167). That would explain why Frege does not do justice to the notion of sense in \gga.

What comes closer to the proposal of a criterion of identity for senses in Frege’s writings is the equipollence principle that two sentences $A$ and $B$ express the same thought provided that anyone who accepts $A$ as true must also immediately accept $B$ as true and vice versa. Sundholm (1994) has called attention to the fact that this idea has been informally suggested on many occasions by Frege, most notably in the opening passage of his Kurze Übersicht meiner logischen Lehren:

Now two propositions $A$ and $B$ can stand in such a relation that anyone who recognizes the content of $A$ as true must thereby also recognize the content of $B$ as true and, conversely […] So one has to separate off from the content of a proposition the part that alone can be accepted as true or rejected as false. I call this part the thought expressed by the proposition. (Frege, 1906, pp.197–98)

If we now replace one word of the sentence by another having the same reference, but a different sense, this can have no bearing upon the reference of the sentence. Yet we can see that in such a case the thought changes; since, e.g., the thought in the sentence ‘The morning star is a body illuminated by the Sun’ differs from that in the sentence ‘The evening star is a body illuminated by the Sun.’ Anybody who did not know that the evening star is the morning star might hold the one thought to be true, the other false. (Frege, 1892, p.62)

Yet, much remains to be done before both equipollence principles for singular terms and sentences can be adopted as satisfactory identity criteria for senses, for we do not have a rigorous account of what it means to “immediately” accept a proposition as true. In Funktion und Begriff, Frege has expressly stated that the sense of two expressions is equal up to renaming of bound variables, that is, that we could write $x^{2}-4x$ directly as $y^{2}-4y$ without altering its sense, however, it can be very difficult to determine the extent of strictness in his account of sameness of sense, considering that Frege also happens to claim that the two halves of Basic Law V express the same sense, but in a different way.⁸⁸8 Both claims are found in Frege (1891, p.27). Frege’s views on sameness of sense have been dealt with in more detail in Sundholm (1994, p.304–307), Klement (2016, §4), and Bentzen (2020c, §2.1).

In type theory, a construction is a humanly computable procedure or program that is generally accepted as a method for obtaining a mathematical object. Following the so-called propositions-as-types correspondence (Howard, 1980), both propositions and sets are specifications of constructions that are structured through the unifying concept of a type. The elements of a type are commonly called terms, and, as expected, terms are interpreted as constructions.

For a more concise semantic explanation, it is convenient to base our type theory on two equality forms of judgment that state that two expressions are equal types and also that two expressions are equal terms of a type. I shall write those equalities with a triple bar notation to emphasize the fact that they are judgmental relations, meaning that they only occur in assertions and therefore are not subject to operations such as those determined by logical connectives:

$$A\equiv B\type\qquad\qquad a\equiv b:A.$$

The whole process of meaning explanation starts with the specification of a native computation system that takes the form of a programming language, typically the untyped lambda-calculus extended with constants for the type formers, constructors, and eliminators of the type theory. In this paper, I will consider dependent function types $\Pi_{x:A}{B}$, equality types $a=_{A}b$, and the natural numbers ${\rm Nature}$. First, we specify the syntax of the programming language:

	$\displaystyle\mathsf{var}\,\,\,:=\>$	$\displaystyle x\>|\>y\>|\>z\>|\>...\>|\>x^{\prime}\>|\>y^{\prime}\>|\>z^{\prime}\>|\>...$
	$\displaystyle\mathsf{expr}:=\>$	$\displaystyle\mathsf{var}$
		$\displaystyle\pitype{\mathsf{var}:\mathsf{expr}}\mathsf{expr}\>|\>\fun{\mathsf{var}}\mathsf{expr}\>|\>\mathsf{expr}(\mathsf{expr}$
		$\displaystyle\mathsf{expr}=_{\mathsf{expr}}\mathsf{expr}\>|\>\refl(\mathsf{expr})\>|\>\eqrec(\mathsf{expr},\mathsf{expr},\mathsf{expr})$
		$\displaystyle{\rm Nature}\>|\>\zero\>|\>\succ(\mathsf{expr})\>|\>\natrec(\mathsf{expr},\mathsf{expr},\mathsf{expr}).$

This transition relation is fully captured by the computation rules that I now symbolically describe, where $\shouldsf{a}\longmapsto\shouldsf{a^{\prime}}$ indicates the fact that $\shouldsf{a}$ transitions to $\shouldsf{a^{\prime}}$ and $\shouldsf{a}\val$ means that $a$ is a value, which is to say that $\shouldsf{a}\longmapsto\shouldsf{a}$ is the case. I will start considering the rules for the dependent function type, which generalizes the notion of universal quantification and indexed product of sets:

$$\frac{}{\prod_{x:\shouldsf{A}}\shouldsf{B}\val}\quad\frac{}{\fun{x}\shouldsf{a}\val}$$

$$\frac{\shouldsf{a}\longmapsto\shouldsf{a^{\prime}}}{\shouldsf{a(b)}\longmapsto\shouldsf{a^{\prime}(b)}}$$

$$\frac{}{(\fun{x}\shouldsf{a})(\shouldsf{b})\longmapsto\shouldsf{a}[\shouldsf{b}/x]}\quad\frac{}{\fun{x}\shouldsf{a}(x)\longmapsto\shouldsf{a}.}$$

The following rules provide a full computational account of the equality type $a=_{A}b$, the type-theoretic counterpart of the usual notion of an equality proposition. Unlike the judgmental equality $a\equiv b:A$, $a=_{A}b$ does not have any assertive force. The constructor $\refl(a)$ represents the reflexivity of equality and the eliminator $\eqrec(a,b)$ generalizes the principle of salva veritate that Frege borrows from Leibniz:

$$\frac{}{\shouldsf{a=_{A}b}\val}\quad\frac{}{\refl(\shouldsf{a})\val}$$

$$\frac{\shouldsf{a\longmapsto a^{\prime}}}{\eqrec(\shouldsf{a,b})\longmapsto\eqrec(\shouldsf{a^{\prime},b})}$$

$$\frac{}{\eqrec(\refl(\shouldsf{a}),\shouldsf{b})\longmapsto\shouldsf{b}.}$$

$$\frac{}{{\rm Nature}\val}\quad\frac{}{\zero\val}\quad\frac{}{\succ(\shouldsf{a})\val}$$

$$\frac{\shouldsf{a\longmapsto a^{\prime}}}{\natrec(\shouldsf{a,b,c})\longmapsto\natrec(\shouldsf{a^{\prime},b,c})}$$

$$\frac{}{\natrec(\zero,\shouldsf{a,b})\longmapsto\shouldsf{a}}$$

$$\frac{}{\natrec(\succ\shouldsf{(a),b,c})\longmapsto\shouldsf{(c(a))}(\natrec(\shouldsf{a,b,c})).}$$

The transition relation divides terms into two classes: those that are themselves values are called canonical, and those that are not themselves values but may eventually reach a value if we keep iterating the transition process are called non-canonical. This iteration is made precise with the notion of evaluation, a computation that halts when it finds the value that a term transitions to after an indefinite number of transition steps. I shall write $\shouldsf{a}\downarrow\shouldsf{a^{\prime}}$ to mean that $\shouldsf{a}$ evaluates to $\shouldsf{a^{\prime}}$. Since we are dealing with untyped computations, the evaluation of a term will not always terminate, but it suffices to characterize closed expressions as programs that when executed output the value they evaluate to. Notice that natural numbers are considered as canonical terms whereas, taking into account Frege’s conception of numbers as extensions of concepts, one might expect such number terms to evaluate to lambda terms, the canonical terms representing value-ranges. This can be done with a technique known as Church encoding, with which we can represent the natural numbers using lambda notation. But the idea goes against the spirit of type theory in that it conflates two very distinct data types, numbers and functions.

Roughly, we explain what a type is by specifying the terms of that type following an idea that can be traced back to the constructive conception of set advocated by Bishop (1967) and the interpretation of the intuitionistic logical constants proposed by Heyting (1934). In fact, the first thing we do to explain what types are is distinguishing between canonical and non-canonical types. Any expression that evaluates to a canonical type is a type, and a canonical type is explained by prescribing what their canonical terms are.

The meanings of the type and term equality judgments are mutually explained in a similar way. To account for the former, we assume that $A$ and $A^{\prime}$ are closed expressions, and assert that

$A$ and $A^{\prime}$ are equal types provided that they evaluate to the same canonical type up to type equality.

$$\frac{A\downarrow\prod_{x:B}C\quad A^{\prime}\downarrow\prod_{x:B^{\prime}}C^{\prime}\quad B\equiv B^{\prime}\type\quad x:B\vdash C\equiv C^{\prime}}{A\equiv A^{\prime}\type}$$

$$\frac{A\downarrow a=_{B}b\quad A^{\prime}\downarrow a^{\prime}=_{B^{\prime}}b^{\prime}\quad B\equiv B^{\prime}\type\quad a\equiv a^{\prime}:B\quad b\equiv b^{\prime}:B}{A\equiv A^{\prime}\type}$$

$$\frac{A\downarrow{\rm Nature}\quad A^{\prime}\downarrow{\rm Nature}}{A\equiv A^{\prime}\type.}$$

$$\frac{a:A\vdash a:A^{\prime}\quad a:A^{\prime}\vdash a:A}{A\equiv A^{\prime}\type.}$$

Now, in order to give a full account of term equality we assume that we are given closed terms $a$ and $b$ as well as a type $A$, and evaluate them all, declaring that

$a$ and $b$ are equal terms of type $A$ provided that $a$ and $b$ evaluate to equal canonical terms of the canonical type which $A$ evaluates to.

$$\frac{A\downarrow\prod_{x:B}C\quad a\downarrow\fun xM\quad a^{\prime}\downarrow\fun xM^{\prime}\quad x:B\vdash M\equiv M^{\prime}:C}{a\equiv a^{\prime}:A}$$

$$\frac{A\downarrow b=_{B}b^{\prime}\quad a\downarrow\refl(b)\quad a^{\prime}\downarrow\refl(b^{\prime})\quad b\equiv b^{\prime}:B}{a\equiv a^{\prime}:A}$$

$$\frac{A\downarrow{\rm Nature}\quad a\downarrow\zero\quad a^{\prime}\downarrow\zero}{a\equiv a^{\prime}:A}$$

$$\frac{A\downarrow{\rm Nature}\quad a\downarrow\succ(M)\quad a^{\prime}\downarrow\succ(M^{\prime})\quad M\equiv M^{\prime}:{\rm Nature}}{a\equiv a^{\prime}:A.}$$

$$x:A\vdash f:B$$

Because only closed terms are taken into account in the validation of a hypothetical judgment, it can be shown that the following principle, which can be regarded as a stronger version of the so-called Axiom K (Streicher, 1993), obtains

$$x:A,p:x=_{A}x\vdash p\equiv\refl(x):x=_{A}x$$

It only remains to be asked how this account of sameness of sense can be linked to the equipollence principles that we have previously discussed. For the sake of argument, let us first assume that $A$ and $B$ are sentences, which is to say that they evaluate to canonical types. We have to show that $A$ and $B$ have the same sense when anyone who accepts $A$ will immediately accept $B$ and vice versa. One interpretation of the immediacy in the latter condition is that proposed in Sundholm (1994)

$$\frac{a:A}{a:B}\quad\text{and}\quad\frac{b:B}{b:A}$$

If we choose to interpret immediacy in a purely computational way instead, then we have that anyone who accepts $A$ will immediately accept $B$ if, without any reference to the notions of transition or evaluation, one is justified in an making assertion of the truth of $B$ from one’s assertion of the truth of $A$, or, equivalently, if from one’s construction of a term $a:A$ one can construct a (possibly new) term $b:B$ by means that do not involve the computation. It is plain that sameness of sense is given by equipollence under this view, for this condition describes precisely what it means for two types to have the same computational behavior. Finally, if we are given an open type

$$x:A\vdash P(x)\type$$

If we were to interpret equality of reference for types in a strictly Fregean sense, that is, following the idea that two sentences are coreferential if they have the same truth value, then obviously the relation that we would be looking for would be that of logical equivalence, which, type theoretically, translates to the existence of two functions $f:A\to B$ and $g:B\to A$ between the types $A$ and $B$.¹¹¹¹11 We typically write $A\to B$ as an abbreviation for $\Pi_{x:A}B$ when the type $B$ does not depend on $x:A$. But it is worth stressing that actual equality of reference does not follow from logical equivalence, for two types may imply each other without computing to the same canonical type. Put differently, if we had a type universe $\mathcal{U}$, whose terms are smaller types, then the following principle of propositional extensionality

$$(A\to B)\to(B\to A)\to(A=_{\mathcal{U}}B)$$

would be false in the meaning explanations, just as it was in \bs because $A$ and $B$ may have different contents. In general, we do not even have that $g(f(a))=_{A}a$ and $f(g(b))=_{B}b$ obtain, for every $a:A$ and $b:B$. The only thing logical equivalence tells us is that if we constructed a term of $A$ we know how to find another term of $B$ and vice versa. It comes with no other guarantees, as Sundholm (1994) notes.

Over the last few years, homotopy type theory (UFP, 2013) has emerged as a new foundation for mathematics that unites type theory and homotopy theory via a homotopical interpretation of type theory where a type $A$ is viewed as a space, a term $a:A$ as a point of the space $A$, an equality term $p:a=_{A}b$ as a path from point $a$ to point $b$ in the space $A$ and so on.¹²¹²12 For a philosophical introduction to homotopy type theory, see Ladyman and Presnell (2016) and Bentzen (2020a).

One of the main ingredients that reinforces this interpretation is the univalence axiom, which offers a formal justification for the common view among mathematicians that two mathematical objects are equal when they are isomorphic. Roughly, the univalence axiom implies that two types are identical just in case they are equivalent in a technical sense:

$$\mathsf{ua}:A\simeq B\to A=_{\mathcal{U}}B.$$

And this new sort of identification imposes a weaker understanding of propositional equality where in general a closed term $p:a=_{A}b$ does not entail $a\equiv b:A$, because univalence introduces a new canonical term $\mathsf{ua}(e)$ to the equality type that is not judgmentally equal to a reflexivity term, assuming that $e$ is an equivalence. Considering that $\refl(a)$ is interpreted as the constant path at the point $a$, univalence ensures the existence of non-trivial paths.

Yet, this homotopical view of equality goes against the constructive theory of sense and reference that I have expounded, precisely because if propositional equality is to express coreferentiality, then the reference of a term cannot be its value in homotopy type theory. In the presence of non-trivial paths, that is, if we are allowed to construct a closed term $p:a=_{A}b$ such that $p\not\equiv\refl(a)$, then we have that $a\downarrow a^{\prime}$ and $b\downarrow b^{\prime}$ but $a^{\prime}\not\equiv b^{\prime}$, meaning that the interpretation of propositional equality as equality of values is lost.