Introduction

It is somewhat an item of common sense that sentences of natural language may at times be false rather than true. Yet, despite the obviousness of this observation, the role played by falsity in the context of modern logic is at best very modest. Indeed, the received view on the matter, dating back to Frege ((Frege 1919),(Frege 1979) "Logik"), defends that falsity is a psychological notion, but not a logical one, because all instances of false propositions can be circumvented by treating them as instances of true propositions prefixed with a negation operator. In this article, we argue that falsity holds his spot as a substantial logical notion in the context of intuitionistic logic. We do so by first by presenting the role of false proposition in Franz Brentano’s reform of traditional logic, which stands in strong opposition to the received view. We then move to the $BHK$ semantics for intuitionistic logic (christened after Brouwer, Heyting, and Kolmogorov). This, we argue, inherits some fundamental traits of Brentano’s treatment of false propositions, and thus constitutes a witness for a system where the notion of falsity is conceptually prior to the one of negation.

Falsity in Brentano’s syllogistic

Brentano was active in Vienna at the end of the 19th century, initially a scholar of Aristotle, he then devoted himself to the systematic investigation of the notion of intentionality, which he recovered from the scholastic tradition and made the cornerstone of his very own doctrine of descriptive psychology. According to Brentano, intentionality constitutes the fundamental feature of all acts of consciousness, to the extent that:

Every mental phenomenon is characterized by what the Scholastics of the Middle Ages called the intentional (or mental) in-existence of an object, and what we might call […] reference to a content, direction toward an object. […] In presentation something is presented, in judgement something is affirmed or denied, in love loved, in hate hated, in desire desired and so on. (Brentano 2012) .

Accordingly, descriptive psychology aimed at providing a systematic description of the structure of all acts of consciousness, under the fundamental tenet that there could be no conscious act without a correlative intentional object, and all objects are such just because they could be correlative objects of some conscious act.

Brentano’s interest in logic is a direct offspring of his philosophical ideas. Naturally, if all acts are determined by their correlative object, then the same should apply to the acts of judgments which were the specific domain of Aristotelian syllogistic.

While traditional syllogistic held judgments to be of the form “S is P”, Brentano held that the fundamental form of judgment is one of affirmation or denial of the existence of some representation, namely:

In addition to those basic judgments of thetic form, Brentano also admitted qualifying judgments, that ascribed or denied some property of some object which has previously been affirmed. In (Brentano 1956), Brentano then showed how much of the traditional syllogistic was translatable into his systems, and how several of its shortcomings were there naturally amended. A simple and yet illustrative example, is Brentano’s translation of Aristotle’s square of opposition, which we report below in both versions, using $\vartriangleleft$ for qualifying acts.

Following the cartesian tradition, Brentano defended that a representation $\varphi$ could be affirmed whenever $\varphi$ is evident to the subject, where evidence was taken as a primitive notion expressing the grasp of the undoubtable truth of the represented content.

What about $\varphi \; false$? Brentano held denial to stand for, disappointment¹ a primitive, undoubtable evidence of things not being in a certain way. However an act of denial is an act of consciousness, and, as such, it must have some correlative intentional object. If one now says that the correlative object of $\varphi\; false$ $is$ the situation that $\varphi$ does not obtain, then they run into a contradiction, to the extent that $\varphi\; false$ is in the end not an act of consciousness at all.

Brentano’s solution was to say that to deny $\varphi$ is to pose a certain modal qualification on acts representing $\varphi$. In particular, to deny $\varphi$ is to say that any individual affirming $\varphi$, and that is, having evidence for the representation $\varphi$, must be doing so incorrectly.

Anyone who says, “No S is P” is thinking of someone judging that “An S is P,” and declaring that in thinking of him in this way he is thinking of someone who judges incorrectly, someone who maintains something contradictory to his own judgement. (Brentano 2012).

Thus, an act of denial of some representation $\varphi$ is grounded in the undoubtable awareness that things are not the way expressed by the content $\varphi$, but does not have $\varphi$ or its negation as a correlative object. Instead, denial of $\varphi$ is glossed as an affirmation about any other possible individual, stating that, insofar as they judge $\varphi$ with evidence, they must be incorrect in doing so. As far as other individuals and their judgments are existing objects, contradiction is avoided.

Brentano would then go on to generalize this approach to all representations of negative form, so that negation itself would be understood as a linguistic fiction, whose logical structure is to be glossed in term of denial, and thus, of incorrectness of other individuals asserting the same content.

Now let us see how the logician can simplify these operations […] All he has to do is to create the fiction that there are negative objects, too.[…]The fact that such fictions are useful in logic has led many to believe that logic has non-things as well as things as its object and, accordingly, that the concept of its object is more general than the concept of a thing. This is, however, thoroughly incorrect. (Brentano 2012)

The topic of falsity can hardly serve as a proper presentation of Brentano’s brilliant contributions to logic and philosophy, and thus, we shall only notice here how the Brentanian treatment of false propositions fundamentally differs from the received view.² In particular, the latter reduces all instances of falsity to instances of negation, and thus rejects denial as logically substantial speech act. On the contrary, Brentano’s position on negation endorsed the following two claims:

1. Negation as a propositional operator is dependent upon the speech act of denial, and;

2. denial of a representation $\varphi$ can be glossed as a modal qualification on judgments of $\varphi$: if someone judges $\varphi$ affirmatively, they are in that incorrect.

While the machinery of modern classical logic was mostly modeled after the Fregean approach, some of Brentano’s ideas indirectly found their way into systems of non-classical logic. Two influential names among the disciples of Brentano were K. Twardowski and E. Husserl. While the former played a major role in the birth of the polish school of logic in Lvov-Warsaw, Husserl’s ideas also had some influence in the development of certain strand of modern logic. It is well known, for example, that Arend Heyting came to know of Husserl’s “Logical investigations” through Oskar Becker, and that it played some role in his formalization of intuitionistic logic. We now inspect Heyting’s own take on negation, and argue it falls much closer to the Brentanian approach than to the received view.

Negation in $BHK$ semantics

Intuitionistic mathematics was initiated by the work of L.E.J. Brouwer, after the idea that mathematical objects are primarily mental construction of the working mathematician, and thus that mathematics should admit no non-constructive method of proof. Brouwer’s ideas remained mostly informal, but the first formalization of the brouwerian principles was given by his disciple A. Heyting, resulting in the system nowadays known as intuitionistic logic.

The so-called $BHK$ semantics for intuitionistic logic specifies the meaning of logical constant in terms of their contribution to the construction of mathematical objects. Heyting, in particular, introduces the semantics as follows:

A mathematical proposition expresses a certain expectation […] Perhaps the word intention’, coined by the phenomenologists, expresses even better what is meant here. We also use the word ‘proposition’ for the intention which is linguistically expressed by the proposition […] The affirmation of a proposition means the fulfillment of an intention. (Benacerraf and Putnam 1983) (A. Heyting, The intuitionistic foundation of mathematics)

Thus, truth of a mathematical proposition means availability of evidence (fulfillment) for it, which, in the case of mathematics simply means availability of a proof. Since to have evidence of a proposition is to know it holds, and thus to be able to affirm it, Heyting’s explanation implies the validity of the following identities:

According to this principle, Heyting lays down the explanation of logical constants as follows:³

$\bot$ There is no proof of $\bot$

$\phi \ \& \ \psi$ A proof of $\phi$ and a proof of $\psi$

$\phi \ \lor \ \psi$ A proof of $\phi$ or a proof of $\psi$

$\phi \supset \psi$ A method yielding a proof of $\psi$ from a proof of $\phi$

Prima facie, the semantics is silent about both falsity and negation. However, since we are given that truth is the same as affirmability, which is the same as provability, we must also endorse the following characterization of falsity in intuitionistic terms .

On the other hand, Intuitionistic logic does not typically take negation as a primitive constant, rather, it must be explained in term of something else. In particular, we have

which means that $\neg \varphi \; true$ is the same as $\varphi\supset \bot\; true$, but what does the latter mean? Given the tables above, $\varphi \supset \bot$ means that we have an effective procedure that given $\varphi\;true$ yields $\bot\; true$. Plainly, because $\varphi \; true$ means $\varphi$ is provable, $\varphi\supset \bot$ yields from a proof of $\varphi$, a proof of $\bot$. Notice that, by definition, there is no evidence of $\bot$ (in fact, the meaning of the constant $\bot$ is defined by stipulating that it cannot be proven) so we have just shown with $\varphi \supset \bot$ that there can be no proof of $\varphi$.

If we recall that a proof of $\varphi$ is nothing but what makes $\varphi$ evident, then it is easy to see how close we have landed to the Brentanian explanation. In fact, we have explained $\neg \varphi$ as a method that yields evidence of $\bot$ given evidence of $\varphi$, and that, given the meaning of $\bot$, is in itself evidence of the fact that there can be no evidence of $\varphi$, which is the same as saying that, if anyone affirms $\varphi$ evidently, they must be incorrect in doing so. In the case of Brentano, this impossibility statement was justified by appealing to disappointment. Because it is plain to the speaker that things are in a certain way, it is immediately given with this that they could never not be the way in which they evidently are, and so those who judge otherwise must be mistaken. In the case of $BHK$, instead, the impossibility statement is justified by the grasp of the meaning stipulated for $\bot$. It was decided that $\bot$ is a proposition without proof, so anybody who judges in such a way that allows to obtain a proof of $\bot$ must therefore be mistaken. Whether and how far these two justifications are related, is not something that can be decided here, but it is important to notice that, according to $\star$ above, $\bot$ is a proposition which is stipulated unknowable, and thus cannot be affirmed, but only denied, precisely because it has, by stipulation, no admissible proofs.Therefore, the justification for the construction of negative expression rests, both in the case of Brentano and of $BHK$, onto their respective explanation of the notion of falsity and denial.

In this sense, $BHK$ semantics vindicates Brentano’s claim 2: the negation of $\varphi$ in $BHK$ semantics is a modal statement about any other judgment of $\varphi$, qualifying it to the extent that they can never be evident.

What about Brentano’s claim 1?

From $\star$ and the explanation of $\neg \varphi$ it is easy to see that denial and negation are intuitionistically interderivable, i.e, we have:

Does this entail the notions are identical? We contend no. Indeed, it is clear from the explanation of $\neg \varphi$ just above, that understanding the meaning of $\neg \varphi$ depends on the understanding of there being one proposition, $\bot$, which can’t be true but only ever false. Therefore, the understanding of affirmation of negative proposition presupposes the understanding of falsity of that same proposition, that is, its denial. On the contrary, the explanation of $\bot$ can never, on pain of circularity, make reference to affirmation of sentences of negative form. In this sense, we conclude, the notions of falsity and denial are conceptually prior to the notion of affirmative negative proposition. This vindicates Brentano’s claim 1 in the context of $BHK$ semantics, and suggests it as a witness to the fact that falsity and denial deserve their own place in the architecture of modern logic.

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