Four pillars of bivalent logic

The history of logic can easily fill entire shelves and the following discussion cannot even aspire to provide something like an overview. As a first cut, and strangely quite akin to what we actually can still find in the idea of an ‘empirical’ approach to social science, much of the debate in logic focuses on four pillars: the principle of identity,Footnote ¹³ the principle of non-contradiction,Footnote ¹⁴tertium non datur Footnote ¹⁵ and the sentence of the ‘ground’.Footnote ¹⁶ The identity principle postulates an identity between being and thought and therewith defines the theme of logic itself. Without it, it would not be possible to speak about truth at all. With the principle, it is possible to separate being and non-being and to define the object of reflection as ‘true being’ or the ‘being of being’. Of course, the philosophical debate around what identity means and how thought and the being of being relate to each other ranges from Plato's cave, to Leibniz, who possibly provided the most elaborate discussion on the identity principle,Footnote ¹⁷ and finally to quantum mechanics.Footnote ¹⁸ Thus, without going down that road too far and complicate things unnecessarily, one can say the principle establishes that ‘being’ is not just a subjective motive within reason, but transcends the individual ‘belief’ and manifests itself as an objective ‘thing’ before the thinking mind.Footnote ¹⁹ This unity between being and thought is followed by a cosmology that divides the world into things and their relations (the two logical variables), where either something is, or something is not and that frames ‘epistemology’ as the problem of how to align the categories and definitions of our language with the properties of those things and relations, for example via verification or falsification.Footnote ²⁰

Having the theme of logic established, the sentence of non-contradiction shows that true being has to be thought of without contradictions.Footnote ²¹ While the identity principle provides the theme, the principle of non-contradiction secures that the process is not characterised by a simultaneous existence of positive and negative values. Logical values like positive/negative or being/non-being cannot be true at the same time. Something cannot be and not-be, a sentence cannot be right and wrong at the same time. If we would allow for contradictions, the basic convictions about true being would be shattered: every contradiction breaks the unity of the world. The principle of tertium non datur shows that always either something is or something is not. There is no other ‘third’ possibility: a coin will show head or tail – and nothing in between. The ‘sentence of the ground’Footnote ²² states that there is a ground on which we can found our reasoning. Of course, for most of Western tradition, this foundation was found in God. Without God, Descartes would have lived in despair and Kant unable to postulate his things-in-themselves. As cursory and necessarily incomplete as this discussion certainly is, we can nevertheless see that: (a) identity, non-contradiction, tertium non datur, and sentence of the ground provide the pillars of bivalent logic; that (b) logic operates with the form being/non-being; that (c) these four axioms determine the (hierarchical) relationship between human existence, nature, and God (with God being on top); and (d) they negates the existence of undecidable questions and put them into the category of non-being.

Exactly at this point, the re-emergence of paradoxes at the turn of the twentieth century is relevant. Bertrand Russell showed that paradoxes can be proven mathematically when mathematical concepts are deferred onto themselves.Footnote ²³ To take a less technical example than Russell (who focused on the question whether a set can be a member of itself), let us consider Epidemides the Crete who says ‘All Cretans are liars’. Now, can we believe him? If he is a liar he actually speaks the truth and if he speaks the truth he is a liar. Paradoxes are true and false at the same time. If it is taken as true, then it is false. If it is taken as false, its truth conditions are fulfilled and therefore it is by definition true. These kind of sentences are false if they are true (but then they are false) and they are true if they are false. As soon as we enter one side, we are immediately thrown to the other and back again. One finds oneself ‘oscillating’ between these two options because the condition of possibility is at the same time the conditions of its impossibility: A because non-A. Or as Wittgenstein once remarked: paradoxes don't lead astray, they don't lead anywhere.

Within the framework of bivalent logic, these kinds of questions would have to be treated as non-existent, as a failure of our thought, or the result of the limits of our knowledge. However, the way Russell derived the paradox made it impossible to negate its existence or to claim it's a problem of limited knowledge. Now if we agree that paradoxes do exist, their solution is neither to be found in an empirical test, nor is it a priori decidable as bivalent logic would need to argue. If nature is unable to determine the ‘answer to’ paradoxes, the truth-conditional theory of meaning is inadequate for their analysis. Paradoxes are neither solved by looking harder at the facts, nor by any other means that could establish some reference between the sentence and the ‘fact’.

The discussion around paradoxes resulted in nothing less than the question about the very foundations of logic. Does the self-reference of logic, its own condition of possibility, point to itself – or to something else? If logic cannot provide for its own foundations and cannot fulfil its own criteria of rigour and coherence, its assumed objectivity cannot be sustained. In the subsequent debate it became clear that paradoxes emerge because self-referential questions like ‘the proof of proof’, the ‘falsification of falsification’ or the ‘logic of logic’, cannot be answered within the classic system itself. There is no ultimate proof that mathematical proofs are necessary; criteria like verification or falsification cannot be justified on their own grounds. The reason is that the third pillar – the tertium non datur – cannot be fully established in complex systems. Due to incompleteness, self-referential questions will always either produce contradictions, antinomies, and paradoxes despite the ‘necessity’ of logical operations.Footnote ²⁴ If paradoxes cannot be solved by negation, then the question arises what to do with them. This question structured the subsequent debate and gave rise to two different solutions: the possible worlds semantic of logical positivism and Wittgenstein's language games.

Possible worlds: logical positivism, conventialism, and semantic conception of truth

Alfred Tarski realised that two assumptions are necessary for the construction of paradoxes: semantic closure of language and validity of basic logical rules.Footnote ²⁵ Without this assumed validity, it would be impossible to grant logic – and science for that matter – any objectivity. Semantic closure is based on the idea that there exists one and only one ‘natural’ or ‘genuine’ language in which all other languages can be translated. Hence, if we find that language (formal logic for example) and can prove that a sentence is true in that language, then it's also true in all other languages. With these two assumptions, logic provides the means whereby we can analyse both language of nature and the laws of thought and thereby analyse the structure of true being. Unfortunately, paradoxes now stand at the entrance door.

For Tarski, the ‘mistake’ was to conflate the semantics of a language with the language itself. The basic idea is to have semantic terms like ‘denotation’ or ‘truth’ not to be found within the same language as the objects to which they refer: one cannot establish the truth of sentences of one language within that language. Consequently, Tarski along with his colleague Rudolf Carnap proposed a solution by abandoning semantic closure and allowing for the existence of a plurality of non-reducible languages,Footnote ²⁶ or in their vocabulary: possible worlds. The problem for them is not the consistency of logic itself, but the fusion of different languages: the language within we speak (object-language) and the language about we speak (meta-language). While the first one is ‘subjective’, the second one tries to take an outside position to describe this language objectively. Meta-language is assumed to be constitutive for how the object-language is to be interpreted; it sets the boundaries in which an object-language is applicable. Carnap and Tarski formulate the problem of paradoxes as a problem of levels, types, or part-whole relationships to break the circle within ‘how does logic explain logic’ in order to find a solution to paradoxes without having to sacrifice logical necessity a priori.

It eventually should be noted that logical positivists realised that the assumption of a general and objective truth needs to be abandoned once and for all: whether or not some sentence is true or false depends on the language in which one speaks (in which world we are in). Questions of inferences and validity of truth conditions depend on which language a particular sentence belongs to. Without a necessary connection between signs and objects, without a ‘natural’ meaning, mathematical theorems cannot themselves determine their meaning. The choice of which language is to be preferred is a matter of a choice, that is, a matter of convention. If we wish to observe the world in terms of classical logic with strong rules of inference, we are free to do so. But there is no reason why logical systems with weaker rules might or might not be used. That is to say categories of true and false can only be examined after conventional decisions: logical positivism is a form of conventionalism as it claims that everything that is true is true by convention.Footnote ²⁷ This ‘possible world’ led Rudolf Carnap to formulate his famous tolerance principle. As Section 17 of The Logical Syntax of Language explains:

The principle is for Carnap deeply connected with the question of ‘intersubjectivity’.Footnote ²⁸ But the conventionalism of the Vienna Circle does not imply a conventionalist epistemology. Quite to the contrary. For logical positivist, if we want to make inferences we need to observe the world. If we want to know whether swans are white, and that the name ‘white’ is a convention, we nevertheless have to observe swans.

This belief is, in my understanding, deeply entrenched in the positivist distinction between meta-language and object-language. For Carnap, meta-language is essentially logical syntax: each system is viewed simply as a set of symbols and rules for their manipulation and consequently, the meta-language is independent of any question concerning the meaning of symbols which is rooted in the object-language.Footnote ²⁹ All formal models can therefore be analysed from a neutral standpoint. Consequently, for logical positivists, this ‘meta-language’, and thus the ‘self’ and its conventions are objectively given. Conventions were just observable entities and ultimately an empirical question. But that means that science and the logical operations themselves are excluded from ‘conventionalism’. Thus, despite this overall acknowledgement of the heterogeneity of languages, for logical positivist, this does not include an abandonment of the ‘scientific’ method.

Wittgenstein's language games

Wittgenstein showed that the avenue proposed by Tarski and Carnap fails to deliver what it promises. The distinction between meta-language and object-language requires a clear boundary that does not exist: or where does this distinction itself belong to? Wittgenstein realised that he had to opt for semantic closure (which then set off ordinary language philosophy) and give up logical necessity a priori by introducing the ‘use theory of meaning’ and the concepts of rule following and language games. Of course, I cannot do justice to the complexity of these concepts and only hope that a short remark on the general thrust is sufficient for the purpose of this discussion.

Wittgenstein's ‘use theory of meaning’ attempts to step beyond the solution in terms of ‘levels’ as termini of both languages obtain their meaning in their use.Footnote ³⁰ According to Wittgenstein, the meaning of words derives not from some essence or property, but from its use. Crucially, the use theory of meaning argues that meaning cannot be ‘shown’, ‘tested’, or ‘derived’, but needs to be reconstructed. It needs to be shown how some notion works within a language game. He thereby breaks the connection between logic, language, and reality and shows that the self-reference of logic does not point to logic itself, but to praxis, rules, and life-worlds.

As the associated concept of rule following has been subject of a broader debate already, I think there is no need to discuss it at length at this point.Footnote ³¹ What is more important at this point is the self-referential logic of ‘social’ institutions or language games: each game consists of constitutive rules which define the game and thereby serve as conditions of possibility to ascribe meaning to moves within the game.

Analogously, without those constitutive rules, mathematics would be just ink on a piece of paper, a chess figure just a piece of wood or marble – without any meaning. Without those rules, we would not be in the position to say that ‘this is right’ or ‘wrong’ and we would not know how to go on within the game or even able to ascribe expertise or wit.Footnote ³³ With these constitutive rules, the ink becomes a mathematical proof and the piece of marble a pawn. The same holds true for a variable in logic.Footnote ³⁴ Especially in his later work, logical operations become part of language games:Footnote ³⁵ each language game has its own constitutive rules and operations where the very idea of what, for example, it means to ‘add’ something differs. Sometimes one plus one makes two; sometimes it makes zero.

The crucial aspect is that Wittgenstein redefines the entire problematique of epistemology from reference to the drawing of boundaries. Wittgenstein's conventionalism implies an intersubjective epistemology. Tarski excluded science from his conventionalism and believed a rational or ideal language could be built upon logical atoms where words have fixed meaning and new sentences could be constructed rationally from prior ones according to the laws of logic. For Wittgenstein, language games overlap, show similarities and differences, change continuously, and refer to other language games. This means that when language games change, concepts and notions change and through that the meaning of words.Footnote ³⁶ The primary question is not one of reference, but how the constitutive rules of a language game are drawn. In contrast to the Vienna Circle, Wittgenstein's conventionalism implies an intersubjective epistemology.

With the turn to language games, Wittgenstein thus redefines the way complexity is described: while Tarski and Carnap structured complexity in levels to define epistemology as the problem of reference, Wittgenstein puts emphasis on the drawing of boundaries and selections. Consequently, language cannot be defined as the ‘sum’ of all language games. The concept of language games cannot simply replace the logical atom: logic does not provide a comprehensive account of how language changes. Rather, semantic changes indicate a change of perspective, a new conceptualisation of a fact referring to some social or societal change. I think this is the answer of Wittgenstein to Russell's paradox: paradoxes are produced by the conceptual apparatus we use. To solve or unfold them requires the introduction of new distinctions or the change of the conceptual framework.

As a summary then, this ‘linguistic turn’ and Wittgenstein's critique on logical positivism stands behind the constructivist's rejection of bivalent logic. This is the reason why for constructivism, the distance and opposition to logical positivism and to early analytic philosophy is important – and why the ‘abandon’ epistemology sounds really strange: for constructivists, the entire epistemology defined in terms of reference has come to an end and has been replaced by one of boundaries. Most of the critique of epistemology seems to be based on an understanding that associates epistemology with reference, not boundaries.

Intersubjectivity and the (ir)relevance of ‘the world’?

The previous section has argued that the notions of rule-following and language games break with the identity principle as a possible basis of logic. This implies that the entire constructivist project can, and I personally think it actually has to, be seen as the attempt to break free from the shackles of bivalent logic and develop an entirely new logic. A logic that replaces identity by difference, that puts paradoxes and contradictions right at the centre of inquiry and finally leaves behind the either or logic of the tertium non datur to realise that reality is polycontextual. This also implies that for constructivists, references to ontology and epistemology ‘in themselves’ don't say anything, especially when they are defined in terms of bivalent logic: With the advent of a new, multivariate logic, the very meaning of ontology and epistemology changes.

What does this new logic look like? The vantage point for constructivists is not some prefixed identity like a tree or car – but difference, the ‘making’ of an identity resulting from qualitative judgements that allow the pen to count as one object – and not as its five different parts. To observe these qualitative judgements, the constitutive boundaries, the logical ‘bases’ of constructivist thought are extended by (at least) a third logical value: ‘undecideness’. That means – the vantage point is an unstructured ‘reality’ where only in time and through qualitative ‘judgements’ (inscribed not in reasoning but discourses) is it possible for observers to ascribe the values being/non-being. The world is structured by discourses, communication, or language ‘in use’. The sentence ‘it is p’ refers not to an a priori given reality, but results from social processes. Hence, being and non-being are not universals, but result from social processes whereby the world is structured by bringing and inscribing semantic distinctions onto the world. There is no a priori separation of subject and object that might serve as some kind of ground or hook. All that is can only be through previous setting of boundaries: truth can only exist in contrast to error, the whole in contrast to the parts, the system in contrast to its environment. These differences are primary before any identity through differentiation and naming can be established.

What distinctions are used, how boundaries are drawn, determines what will be seen and what will be left unnoticed. This includes categorisation processes as Wittgenstein emphasised: ‘how we group words into kinds will depend on the aim of the classification – and on our own inclination.’Footnote ³⁷ From this perspective, categories do not represent material characteristics or intrinsic properties, and the ‘materiality’ of objects certainly does not determine processes of categorisation, but, as Mary Douglas once summarised a discussion on the materiality of risks:

Apart from the cultural dependence of perception and classification, this quote nicely shows how shared conventions, expectations, and cultural categories are the very ‘stuff’ of social structures. But this ‘stuff’ is not framed in terms of subjects facing an object, but in terms of how genuine intersubjective processes constitute objects and subjects in the first place.

Here is a difference between constructivists and logical positivists. As the discussion of the Vienna Circle has shown, a simple reference to ‘intersubjectivity’ does not do the trick. As much as intersubjectivity is understood as some space between pre-existing subjects, the subject-object distinction is maintained as ‘the other mind’ is framed similarly to an ‘object’ that an individual has to face, observe, and infer meaning from. In its attempt to leave this subject-object distinction behind, the ‘constructivist’ turn from the subject to language thus also needs to (re)define intersubjectivity and disconnect meaning and the mind. This is not the place to provide an overview over available alternatives. Rather, I would like to simply point out that a hook for ‘ontology’ in human existence – as the idea of a layered ontology in general – appears to be problematic. To start from existing subjects and derive social categories as some higher form of ‘personality’ already failed with Husserl's phenomenology. What is at stake seems to be a different concept of what the world is: For constructivists, the world is not crowded by substances or simply the sum of all things, but only process: The world is the unmarked space of distinction through which we look at her. This has important repercussions for ‘necessity’ and consequently ‘causality’ – points that cannot be dealt with at this occasion. Although it is tempting to show why the accusation that constructivists buy into a Humean concept of causality is problematic, as some critical realists repeatedly do, I simply would like to emphasise some repercussions of intersubjectivity as self-referential processes have for an understanding of the relevance of materiality for social processes: not the materiality of things produces meaning, but self-referential processes give meaning to the materiality of things.

That implies that the primary question is not how ideas derive from its material bases and then impact on it, but rather that there is a distinction between the physical and the social world. What is needed first of all is some concept of self-referential processes and how this is linked with notions of society or sociality. From this perspective, the kind of materiality we find in social processes is different from matter we find ‘in the world’. The usual examples testifying the importance of the world – like ‘a plane can fly, a tree is outside the window, or “if I wish to know how many cars are parked in my drive then the easiest way is to probably go and look”’Footnote ³⁹ seems simply beside the point.

What is of interest is not the materiality of things per se, but the ascribed meaning or the ‘function’ of some ‘thing’ in some discourse, system, or society. John R. Searle has reminded us that the material properties of things do not determine their function: money is money not because of the paper used to print the symbols on it. The meaning of money is not reducible to its physics, particular with rise of cyber money. The semantics of ‘money’ has no reality beyond its talk. As Searle confirms:

In the same way, institutions like money, marriage, and property which structure expectations, inhabit cultural categories and are based on shared conventions. Institutions are linked to questions of recognition and status, facts which are constituted intersubjectively and do not have a material counterpart. I guess my wife would not be very pleased if I would try to explain the function or the meaning of the ring on my finger by pointing to its physical properties. And as John Austin reminded us, such a reference to ‘the world’ would even be potentially misleading. It would distort the view to the self-referential processes those institutions are based upon.

Acknowledging self-reference has important repercussions for the kind of materiality relevant for the observation or reconstruction of social processes. Without wanting to critique Colin Wight, I think an example he provided might serve as an illustration to highlight the different notions of materiality at play. In his rejoinder to Friedrich Kratochwil, he writes:

Apart from the rather obvious epistemological problems Colin Wight raises, it is rather obvious that there is no need for constructivist thought for ‘solving’ this problem. Any game theorist and positivist would be able to do that with his tool kit just fine. They might even run a regression and identify some correlations. What shines through in this passage, however, is the subject – object distinction and a rather unproblematic treatment of logical necessity which stands in contrast to his rejection of bivalent logic. Ultimately I think the conclusions Wight wants to draw are a classic case of a non-sequitur. He jumps from the problem of ‘counting’ to establishing the ‘materiality’ of cars and thus a world beyond the ‘productive power of discourses’. A problem, for which the counting of cars is irrelevant. His example is parallel to the example ‘a bullet in the head kills you’ for establishing the materiality of the body. That of course leaves the question open why such acts are classified differently.