What we need to do in order to separate empirical and normative science in the Analytic Metaphysics of Quality is to identify romantic clusters – some kind of intuitions – that can result in the formation of abstract conceptual systems. It could be said that subjective quality includes concepts whose meaning is taken for granted, objective quality includes empirically justified conceptual systems and normative quality includes abstract conceptual systems such as predicate logic.
Normative quality is not decisively empirical or non-empirical in the sense that it would involve manipulation of patterns which belong to the inorganic, biological, social or intellectual level. Addressing the issue of whether normative quality is subjective or objective is not even required for understanding the nature of normative quality. It is only required if we are to explain normative quality in terms of objective quality. If normative quality is seen to emerge from objective quality, then according to the Analytic Metaphysics of Quality it does so by being the category of intellectualizations about intellectualizations.
I call the first level of normative quality the syntax level. Syntax is the study of constructing valid statements of some language. The language may be a formal one, like propositional calculus, or a natural language, such as English. Syntax rules are used to determine what kind of thoughts about thoughts are deemed normatively intelligible. Syntax rules, like everything else in the Analytic Metaphysics of Quality, are derived from romantic quality – that is, from some form of intuitive understanding that is direct enough to resemble perception. In order to study what this intuition is like – what it feels like – let us examine the following axiomatization of propositional logic by Jan Łukasiewicz. The reader is not required to understand nearly anything of it. The only important thing to understand is that all theorems of sentential logic can be derived from these axioms. In other words, any proof within sentential logic can be ultimately grounded on some or all of these axioms.
- A1, Simplification A → (B → A)
- A2, Frege’s axiom A → (B → C) → ((A → B) → (B → C))
- A3, Transposition (¬A → ¬B) → (B → A)
By examining the axioms I can intuitively guess their syntax to adhere to the following rules:
- The axioms consist of terms (A, B, C), operators (→, ¬) and brackets.
- Anything enclosed in brackets can be treated as if it were a term.
- Implication (→) must be preceded and followed by a term.
- Negation (¬) must be followed by a term and must not be preceded by a term.
Let us suppose propositional calculus would be presented to me for the first time, but with an additional bogus axiom:
- A1, Simplification A → (B → A)
- A2, Frege’s axiom A → (B → C) → ((A → B) → (B → C))
- A3, Transposition (¬A → ¬B) → (B → A)
- A4, Foobar )¬→
Being familiar with basic mathematics I would make some sense of all of these axioms except the fourth. The fourth would remain unintelligible, because it would appear to be syntactically unrelated to the rest. I could not definitely rule out the possibility that it has a purpose. Perhaps there are exceptional circumstances, in which it makes sense to interpret ) and → as terms, like A and B, and ¬ as a binary relation, like →, so that ¬A would be invalid but A¬B would be valid. But to assume so would seem much more far-fetched than what is required to perceive the other axioms as meaningful. This uneven distribution of “far-fetchedness” among the axioms could probably even be measured by using some criteria, but my initial perception of it is not based on measurement but intuition. The fourth axiom feels disharmonious when presented in conjunction with the others, and I attribute that feeling of disharmony, or cognitive dissonance, to a form of normative intuition I call harmony intuition.
Harmony intuition is used to assess whether abstract expressions, such as strings of symbols, could effortlessly be understood to adhere to certain syntax rules. The above example is quite obvious, but a less obvious one can be exemplified by the strings 11 and 01. The former string is harmonious within two relatively familiar mathematical syntaxes, the binary and the decimal syntax, whereas the latter is intelligible only as a binary number. If this is observed intuitively rather than by employing some dialectic method – if this is considered obvious – the obvious observation arises from harmony intuition.
- Also known as PURPOSE
The next level in the Analytic Metaphysics of Quality is the semantics level. The semantics level emerges from the syntax level and is another form of normative quality. Semantics is the study of meaning. In the context of normative quality, “meaning” arises from rules governing how terms may be replaced with propositions or other terms. The semantics level emerges from the syntax level and not the other way around, because the syntax level is used to introduce the notion of a term in the first place. Without syntactic ontological entities semantic ontological entities would have nothing to work with.
Normative meaning is theoretical: it is not related to the issue of how the expression “an actual feeling of joy” manages to refer to said emotion as romantic quality. Instead, it is concerned with manipulating terms correctly with regards to their abstract relations to each other. Despite this, normative semantics does include romantic quality if it is possible to have intuitions about normative semantic meaning, as such intuitions are romantic quality.
To provide an example of a semantic pattern, one semantic convention is that q in Łukasiewicz’s first axiom (p → (q → p)) may be replaced with p when making a proof, thus obtaining the proposition (p → (p → p)). But this is a bit hardcore for those who don't like formalisms.
A more intuitive example of a semantic pattern can be found in the notion that “the opposite of black” is usually thought of as white. From a physical point of view it could also be argued that a mirror surface is the opposite of black, as perfect black would reflect no light and a perfect mirror surface would reflect all light. Hence the notion, that white is the opposite of black, is true within a language game that presupposes we are talking about colors and black is a color but “mirror” is not, but it might not be true in another kind of language game.
This shows, how “oppositeness” is a semantic pattern that does not necessarily tell anything about colours or light but of our way of conceptualizing them. In a normative language game “black” and “white” could be denoted by terms and “oppositeness” by a binary relation. These formal constructs could be manipulated correctly, that is, so that the oppositeness of black and white is preserved, by someone who does not know these terms refer to colors the eye can see. The romantic cluster that underlies semantic patterns may be called significance intuition.
The difference between syntactic and sematic truth is that given our previous language game about black and white, “black is the opposite of white” is both syntactically and semantically correct, but “black is not the opposite of white” is syntactically correct but semantically incorrect. “Black white opposite not is not” is not even syntactically correct.
Phenomenal–synesthetic primality checks are probably semantic patterns. The intuition, that synesthetic sense-data corresponds with the normative properties of numbers, seems to be a form of significance intuition.
- Also known as UNDERSTAND
The third normative level of static quality is called the metatheory level. The possibility to have a normative metatheory emerges once there first is an object-level theory, with syntax and semantics, of which a metatheory can be made. The metatheory, too, has syntax and semantics, but it differs from the object-level theory by being a theory about another theory. That other theory is the object-level theory. If theories are thought to consist of a normative environment (rules of the system) and normative agents (entities to which the rules apply), the environment of a metatheory consists of agents of the object-level theory. The agents of the metatheory are separate from the agents of the object-level theory in the sense that different rules can be found to apply to them.
An example of a theory and a metatheory can be found in the rules of chess and chess strategy. The rules include statements like "A bishop only moves diagonally", but the strategy includes statements like "Don’t sacrifice a rook for a pawn". Even if you do sacrifice a rook for a pawn you are still playing chess – you are just more likely to lose. But if you move a bishop non-diagonally you are no longer playing chess at all. Chess books describe strategic metatheories of chess that are built on the object-level rules of chess.
The metatheory level connects to a romantic cluster I call awareness intuition. Among other things, this form of intuition allows one to use a normative system to achieve a goal. In the case of chess the goal is usually to win. And from a normative point of view the rules of chess are not only about moving tangible pieces – they also form a system with syntax and semantics.
- Also known as WISDOM
Next, let us study the fourth and final level of normative quality. Sometimes syntactically and semantically different theories exhibit similar properties. I call such similarities analogies. Analogies exist normatively in the form of metatheories that can express the similarities between two theories by having both of them as object-level theories. Therefore the analogy level emerges from the metatheory level. For an example of an analogy let us consider the following three cases. The first one is the Barber paradox, which is an informal presentation of Russell’s paradox:
Suppose that in a village there is a barber, who is a man, and shaves all men who don’t shave themselves. Does the barber shave himself?
The second case has sometimes been used to argue that God doesn’t exist:
Suppose god is omnipotent. Can he create such a heavy rock that he himself cannot lift it?
The third case is the Grelling–Nelson paradox:
Let us call an adjective ”autological” if it describes itself. ”Short” and ”unhyphenated” are autological adjectives, because the former is short by virtue of having only five letters, and the latter is not hyphenated. Let us call an adjective ”heterological” if it doesn’t describe itself. ”Long” and ”Armenian” are heterological adjectives, because the former is short and the latter isn’t a word of the Armenian language. Is ”heterological” a heterological adjective?
In all paradoxes there is a pattern – either a barber, god or an adjective, that supposedly has some attribute. In the Barber paradox that attribute is the shaving of all men who don’t shave themselves. Suppose the barber does that. In that case it is not true that the barber shaves all men who do not shave themselves. But if he doesn’t do that, he doesn’t shave himself, although he should.
In the god paradox the attribute is the capacity to create a rock so heavy the creator can’t lift it. Suppose an omnipotent god can do that. In that case god would not be omnipotent, because he could not lift the rock. But if he can’t do that, he can’t create something, although he should be omnipotent.
In the Grelling–Nelson paradox the attribute is to describe all adjectives that don’t describe themselves. Suppose the adjective ”heterological” does that. Then it describes itself, although it shouldn’t. Suppose it doesn’t describe itself. Then it should describe itself.
Although a barber, god and an adjective are very different patterns, the normative structure of the statements is similar: all are paradoxes by virtue of stating something to have properties that contradict each other. Our ability to conceptualize the general notions of paradox, tautology and such are analogy patterns. A tautology in sentential logic is as much a tautology as a tautology in lambda calculus, despite those two systems being quite different. Even the notions of syntax, semantics and metatheory – the contents of the lower normative levels – can only be generalized in the form of an analogy. Sentential logic alone cannot provide an argument for its axioms to be similar to the Peano axioms by virtue being axioms. That is to say, our ability to speak of the axioms of sentential logic and the axioms of Peano arithmetic, as if both belonged to the category of ”axioms”, is a consequence of us understanding analogies.
For those with more mathematical background information a more sophisticated example of an analogy pattern can be found in the similarities between Gödel’s incompleteness theories in logic and the halting problem in computer science. But I will not go into the details of that here.
Intuitive perception of analogies is done with what I call unity intuition. The name is hopefully self-explanatory: two normative entities are perceived as similar despite them originating from different theories, and this is seen as a manifestation of – excuse me – “cosmic oneness”, that is, unity.