From Semantics to Reality

What does Model Theory tell us about Apophaticism?

Table of Contents

0. Preamble

1. An Introduction to Model Theory

2. Fundamental Results: Syntax and Semantics

   1. Soundness and Completeness

   2. Compactness

   3. Löwenheim-Skolem’s Theorem

   4. Non-standard Models

   5. Tarski’s Theorem

   6. Gödel’s Incompleteness Theorem

3. The Shades of Verticality

4. The Universality of Unsayability

If you are interested in mathematics, this essay is for you. If you are not, you may still want to consider Sections 3 and 4 below.

The thesis is, as before, that formal reasoning is horizontally necessary and powerful but vertically insufficient.

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Preamble

You can skip this section if you want to dive straight in.

In Philosophy is the Poor Man’s Poetry, I demonstrated that:

1. Lower objectivity and its absolutisation of symbols insulates the ego from the transformation that Higher Objectivity demands

2. Both Early and Late Wittgenstein clarify that all philosophical confusion results from the improper use of language. This improper use is, in fact, the fallacy of misplaced concreteness, the absolutisation of symbols. Proper use, however, inevitably leads us to a form of apophaticism; the deepest truths, fundamental value remains ultimately transcendental

3. An acknowledgement of the limits of the proper use of language necessitates verticality in the apprehension of Reality

4. History can either be factual or narrative, and it is meaningful only to the extent that it is narrative (emplotted). Myth, therefore, is the purest form of history

5. Poetry and philosophy are both ultimately symbolic, but philosophy in its unwillingness to sit with contradictions, in its insistence on the “encompassment” of Truth in syntax is therefore bounded as an act of horizontality, whereas Poetry may well inspire one to intuitive leap upwards, albeit at the risk of confusion

6. Philosophy is not of no value. It is critical to recognise that in the absence of clarity, our intuitions may well be leading ourselves astray

In this essay, I return to this question more formally, mathematically, in order to harmonise the various attempts to demarcate the limits of reason.¹

The thesis is: formal reasoning is horizontally necessary and powerful but vertically insufficient.

The point is: this isn’t “woo-mysticism”. It is a structural fact about formal, conceptual systems, built into the very nature of Reality itself.

Once we clarify the purpose of verticality and horizontality, and recognise the need for personal transformation on collision with Reality, we will then answer the decisive question: what kind of interpretant must one become for Reality to disclose itself correctly?

Some vocabulary:

1. Verticality: the mode of intelligibility by which the necessary principle governing a symbol is intuited. We further refine this notion in the Shades of Verticality section below.

2. Horizontality: the mode of intelligibility by which the particulars, and distinctions of a symbol are clarified

3. World-Anchoring: semiotic and syntactic fidelity, the non-arbitrary constraint that the extant world imposes on admissible interpretations. When an interpretation is wrong, Reality itself resists it: prediction fails, action misfires, coordination breaks, and the interpretation can only be saved by endless ad hoc patches: the opposite of a stable, eternal principle which survives variation and multiplicity.

An Introduction to Model Theory

“If proof theory is about the sacred, then model theory is about the profane." — Dirk van Dalen

Every formal system of symbols and sources has two layers: a syntactic, and a semantic. Let’s consider the raw components of each.

Definitions

1. Syntax
   The formal, syntactic side: symbols, well-formed (“grammatically-correct”) expressions, and rules for deriving one expression from others (proof rules).

2. Semantics
   The interpretive, semantic side: how expressions are assigned meanings in a given structure, and therefore evaluated as true or false.

3. Language (L)
   A specified vocabulary: logical symbols (and/or/not/”for all”/”there exists”) and non-logical symbols (constant symbols, function symbols, relation/predicate symbols).

4. Domain (D)
   A set of objects that the language is interpreted over. This is the set of possible sources we’re concerned with.

5. Term
   A name-like expression meant to denote an element of D: variables, constants, and function-applications built from them.

6. Formula
   A well-formed statement built from relations applied to terms, combined using logical symbols (and/or/not) and quantifiers (for all/there exists).

7. Sentence
   A formula with no free variables (i.e., something that can be simply true/false in a structure without needing a variable to be externally specified). An example of a sentence is: “For all bottles, bottles are made of plastic.” Here, the variable is “bottles”, and it is not free within the sentence, since we’re applying it to all the bottles. “Bottles are made of plastic.” is not a formal sentence, since we still have to specify which bottles we are referring to, i.e. the variable is free.

8. Theory (T)
   A set of sentences in the language, L (Think: axioms, or constraints stated syntactically.)

9. Model (M)
   A set, D, with an interpretation of L on a domain, D: it specifies what each constant denotes in D, what each function does on D, and which subsets of objects in the set D satisfy each relation. (So: a “model” is a set of objects with interpretations.)

10. Satisfaction / Truthhood-in-a-model
    M models S means: the sentence S is true in the model, M. We say M satisfies S. Similarly, M models T means that M satisfies every sentence in a Theory, T.

11. Derivability / Provability
    T derives S means that there exists a formal proof of S from T using the proof rules of L. That is, it is possible to manipulate the sentences of T correctly (according to the rules of the Language) to derive S.

Basically, a formula is syntactic (a symbol), whereas a proposition is semantic (a source). A theory is syntactic, whereas a model is semantic. Formulae are derived syntactically within a Theory (by manipulating symbols), but are satisfied semantically within a Model by their reference to a proposition in the model.

A theory and its proofs are what can be moved around “on the page” (horizontality), while a model and satisfaction are what determine if those sentences are true or false (verticality).

Intuitively speaking, Interpretation is therefore the act of mapping formulae to propositions, i.e. the choice of a Model for a Theory in a given Language.

Note that here the word Model is used technically and is actually the set of sources for a set of potential symbols, rather than a “model” of some other system. In this formal context, the Theory is the map, and the Model is the territory. (This just goes further to show how critical context is to interpretation.)

Fundamental Results: Syntax and Semantics

We will not prove these here: they’re left as an exercise to the reader.²

2.1 Soundness and completeness

1. We introduce two further ideas:

   1. Soundness: if a sentence can be proved from a theory, it is true in every model of that theory.

   2. Completeness: if a sentence is true in every model of a theory, it can be proved from that theory.

2. The idea: Within first-order logic, proof and semantic consequence line up perfectly: all derivable sentences are true in every model of a theory, and further, if a sentence is true in every model of a theory, it must also be provable from that theory

3. A simple example: If the theory defines the axioms “all humans are mortal” and “Socrates is human,” then the sentence “Socrates is mortal” is both provable and true in every possible interpretation of the original axioms. Provability and truth track each other.

4. What does this mean? This is good news: it demonstrates that formal reasoning is not arbitrary. Horizontality has real weight and meaning. However, here’s the caveat: formal reasoning does not identify an intended interpretation (i.e. model). It can only demonstrate what holds true in every possible interpretation of the original theory.

But how can we know which interpretation is correct? Does this even make sense?

2.2 Compactness

1. The idea: If every finite subset of a theory has a model, then the whole theory has a model.

2. The intuition: If we have a theory (which could be infinitely large), and if for every finite set of sentences in that theory, there exists an interpretation (i.e. a sub-model), then the whole theory (which could have infinitely many sentences) will also have an interpretation as a whole which “locally” looks like each of the sub-models.
   
   In a way, this is quite nice, it indicates that there must exist interpretations that “stitch together” or “harmonise” all the various submodels into a global model that makes sense of the theory as a whole.
   
   The problem is: these global interpretations can often be quite weird, surprising, or have unintentional consequences.
   

3. A simple example: Imagine a theory that simply says: “There exists a number greater than 1,” “greater than 2,” “greater than 3,” and so on for every finite number. Any finite subset of this theory is easy to satisfy (just pick a number bigger than the largest number mentioned). Compactness says however that the whole set is then satisfiable i.e. there exists a model of the world that admits a “number” that behaves like “Infinity” which satisfies each of those sentences. This number is not “infinity” (which is not a number) but a kind of “ghost number” added “at the end” of the number line. This new, alien model satisfies all the rules above but may not be what we intended to describe.
   

4. The caveat: Horizontality can provide you with interpretations that are locally coherent to “pieces” of Reality, but this does not guarantee an interpretation of Reality that is globally faithful. The lesson here is important, related to the problem of induction in a deep sense: the Truth is not necessarily incremental. It may well have to be apprehended all at once.

You cannot count to Infinity. You have to leap there in one go.

2.3 Löwenheim–Skolem Theorem

Pre-requisite: This might not make sense if you are unfamiliar with Cantor, but the key idea is that there exist different sizes of infinite sets. Some infinitys really are larger than others: countable infinities are just the smallest size.

1. The idea: If a first-order theory has an infinite model, then it must have a countably infinite model.

2. The intuition: Formally written language often cannot even pin down how big the interpreted universe is. Descriptions of Reality underdetermines how large it is.

3. A simple example: You may well think you’ve written down a theory that describes a large, elaborate universe of objects. However, if your description has any interpretation with infinitely many objects, then there must be a model (interpretation) which satisfies that theory but which is only countably infinite, even if this interpretation behaves as if it were much larger.

4. The caveat: A formal description of an infinite universe will yield a coherent interpretation of a universe that may be smaller than the universe you intended to describe. The lesson is similar to the above: our symbols can constrain reality without uniquely determining it.

   Verticality is what prevents us from mistaking “adequate constraint” for “complete capture.”

2.4 Non-categoricity and non-standard models

1. The idea: Many important theories have multiple, distinct (non-isomorphic) models.

2. The intuition: Outside of some very basic cases, it is often impossible to set up a theory (which could be infinitely large) that uniquely describes a universe of objects. You may write axioms to describe some familiar universe, but the same axioms can just as well admit many distinct, weird, and alien universes, all of which are just as describable with that theory.

3. A simple example: It is desirable to write down a description of “the natural numbers” to strictly and uniquely mean the set {0, 1, 2, 3, …}. However, there exist many other sets with “extra” number-like objects, for instance, ghostly integers that come after all the standard ones, while still obeying the formal rules you wrote down. This is similar to the weird “numberising” of “Infinity” mentioned in the section on compactness above.

4. The caveat: The point here is simply that syntax does not uniquely determine semantics.

   
   Adding axioms can constrain the set of universes that satisfy the axioms, but language itself cannot force a unique interpretation. If you insist on apprehending a unique, intended universe, you simply cannot do so syntactically: something more is needed: higher-order constraint, context, and ultimately the formation of the interpretant, what we have earlier referred to as verticality.

2.5 Tarski’s Theorem

1. The idea: For sufficiently rich (i.e. expressive) languages, a notion of “truth-hood” cannot be defined within the language. You need a meta-language within which to evaluate the truth-hood of the “sub-language”

2. The intuition: To speak of “truth” for a system, you must step outside that system. No formal system of symbols can fully contain the measures of its own truthhood.

3. A simple example: If a language was sufficiently expressive, it will admit self-referential statements whose truth value are indeterminable. The classic metaphor here is “The Liar’s Paradox”: “this sentence is false”. Tarski’s result formally determines the strict requirement for a higher vantage-point from which to evaluate the truthhood of symbols.

4. The caveat: This result itself is an insistence on apophatic structure: the whole cannot be captured from within the same plane without paradox or incoherence. Verticality is the disciplined willingness to ascend a level: to find the higher vantage-point, from which to evaluate a symbol’s truthhood.

2.6 Gödel’s Incompleteness Theorem

1. The idea: In any sufficiently strong, consistent (non-contradictory) formal system, there exist statements that are true (in the sense we “mean”) but which are not provable within the system.

2. The intuition: Even when a system is sound, i.e. does not prove contradictions, then syntactic closure is not semantic completion. Proof fundamentally cannot exhaust truth.

3. A simple example: Imagine a rich, non-contradictory legal code. No matter how carefully drafted, there must be propositions about the “world” the code describes that are true but cannot be derived from the code’s own statutes. This is always true, and there is no amount of additional enrichment of this legal code will eliminate that problem.

4. The caveat: Gödel’s result states there will always be more truth in a system than it can possibly prove. Horizontality (syntax, symbolic manipulation, and proof) remain critical, but we should never forget that there will always be true statements that the system cannot demonstrate. Reality always exceeds any possible description of it.

The Shades of Verticality

So, what does this all mean?

1. All in all, we’re illustrating that horizontality and symbolic manipulation is powerful, internally well-behaved, and yet structurally underdetermines Reality. True sentences will lead via horizontality to more true descriptions, but there simply does not exist a total description of Reality.

2. Horizontality gives rigor, coherence, clarity, and transmission, but it’s simply not enough, by its own admission. Verticality, on the other hand, the need for a “higher” vantage point, and the need for apprehension of the universe as a whole, as a gestalt, is the only possible way by which the Truth is apprehensible. Without world-anchoring, one can have immaculate derivations inside an unintended universe.

3. Symbolism and logic admits its own humility: logic necessarily leads to apophaticism. In fact, it proves it itself. Syntax is never enough.

The underdetermination of interpretations of Reality by language and logic is a warning. It is incredibly easy to be led astray.

Reality always exceeds description.

“That is your Lord’s Path—perfectly straight. We have already made the signs clear to those who are mindful.” — Quran 6:126

The Problem: So, if our syntax and logic can be immaculate, and still underdetermine Reality, the decisive question remains: what constitutes a true and faithful interpretation? Let us return to the modes of intelligibility:

1. Horizontality is clarity by articulation, deduction, and reason. It is the work of distinction: definitions, inferential relations, proof, refutation, disambiguation. It instrumentalises language to ensure that symbols (which can themselves be interpretations or models of Reality) are well-formed, stable, logically sound, and admit no contradictions.

2. Verticality is clarity by ascent. It is the work of elevation: the act by which symbols cease to be mere marks and become answerable to what they mean. There are therefore, two kinds of verticality worth highlighting:
   

   1. Semantic verticality: from syntax to truth-in-a-model, i.e. it is the discovery or formation of an interpretation or model that satisfies the theory we have written down. A sentence admits no notion of truthhood in the absence of interpretation. It can, at best, be grammatically correct/well-defined.
      It can only be true or false in the light of an interpretation: when a universe is discovered or defined, the language’s symbols are afforded meanings (proposition here is the technical term) and the sentence is then evaluated in the light of whether or not the interpretation satisfies the sentence. This is verticality in the technical, narrow sense: the ascent from a set of symbols, to their meanings in the world. Their truth depends on the world you’re considering.
      
      This is the leap from syntax to semantics.
      

   2. Noetic verticality: from truth-in-a-model to Reality. Interpretations are arrived at constantly. One might always be always able to imagine a world where the sentences of a theory are true. The deeper ascent, then, is not just the formation of an interpretation or semantics with which to evaluate the syntax, but even further, the apprehension of Reality with which to evaluate the semantics at all (themselves necessarily conceptual, stated in Tarski’s “meta-language”): the formation of the interpretant into the kind of subject for whom the admissible interpretations are not arbitrary. Here verticality refers to a disciplined attunement: increasingly submitting interpretation to the constraints of Reality until meaning is not simply imagined, but disclosed.
      
      This is the leap from semantics to Reality.

3. Importantly, even semantic verticality is insufficient, and in fact a “bit horizontal” since it remains ultimately imaginal and conceptual. It is necessary to apprehend and understand the truthhood of lower-level horizontality, but this must itself be ultimately evaluated from a standpoint beyond all conceptualisation. From the View from Everywhere.
   
   Semantic verticality can simply “manufacture” coherence. It is the fabrication of fresh symbols (new coherent interpretations). These must then be trimmed by collision with Reality. (Doesn’t this remind you of natural selection?)
   

4. We introduce the notion of world-anchoring as the fundamental resistance that Reality imposes on interpretation.

   
   “Either make the tree good, and his fruit good; or else make the tree corrupt, and his fruit corrupt: for the tree is known by his fruit.” — The Bible, Matthew 12:33
   
   Practical success and failure, predictiveness, correctability, coordination with others, its moral and aesthetic costs, and perhaps most of all the invariability of explanations.³

Of course, this entire discourse itself is ultimately itself symbolic and representational. I am here metaphorising (interpreting) Reality and further metaphorising (interpreting) the modes of intelligiblity by which it is apprehensible.

This is a map of the maps between symbols and the Real, and importantly, a demonstration that even this map clarifies the insufficiency of the maps it represents.

Syntax cannot substitute for collision: higher objectivity can never simply be conceptual, and must involve collision with Reality. It must involve Reality’s self-disclosure.

The Universality of Unsayability

The framework stipulated above is not exclusive, even though it is clothed in the language of formal systems, and mathematical logic. The same shape appears wherever thought has attempted to describe Reality fully. We always run into the same obstacles.

Horizontality: Method, process, calculation, and critique clarify, stabilise, transmit, and organise. They remain, however, fundamentally insufficient. Horizontality refines concept and language, but Verticality concerns an openness to Reality’s disclosure.

Let’s consider some of the forms this idea has taken:

1. We have already discussed Wittgenstein and the proper use of Language or McGilchrist and his left-right brained distinctions in Philosophy is the Poor Man’s Poetry.
   

2. Simone Weil’s attention is a form of disciplined receptivity to Reality. She believes it is a mode of truth. The Truth unveils itself to a subject who has cultivated stillness, patience, and honesty. It is received by the subject’s transformation rather than his grit. This, of course, is simply noetic verticality. The world is not exhausted by the will of mankind: it discloses itself to the pure, instead.
   
   You do not act on Reality. It acts on you.
   

3. Thomas Kuhn distinguishes between “normal” science and scientific revolution (i.e. paradigm shifts). Normal science simply accumulates, refines, extends, locally interprets results within the broader context of an existing theory and interpretation of Reality. This is horizontality.
   
   Paradigm-shifts are qualitatively and fundamentally distinct. They change what counts as problems, puzzles, evidence, processes, and even solutions. A paradigm is not merely a set of propositions, but a brand-new interpretation of Reality. It is a change in what constitutes the meaning of scientific endeavour. This is verticality.

4. In Gadamer’s Truth and Method, understanding is not exhausted by method. The “understanding” of a symbol or text is simply an event. Texts are not simply dead, static symbols with a pre-defined meaning, but rather symbols that continually yield fresh interpretations within the context of changing contexts, traditions, and interpreters. Again, we must be careful here not to devolve into pure relativism. No, instead, meaning appears via the merging of the broader paradigm of the interpreter with the symbols being interpreted, and meanings are resisted by Reality, must be world-anchored in order to be admissible. There is an act of verticality implicit in all interpretive activity, an act of re-integration critical for the interpretation to matter. Interpretation is dialogic between the symbol and the interpreter. The reader is interpreted just as much as he interprets.
   

5. In his Discourse on Thinking, Heidegger distinguishes between “calculative thinking” from “meditative openness”. Calculation orders and controls, granting power, prediction, technique. It delimits Reality, narrows what can manifest, because it ultimately seeks to instrumentalise, rather than interpret, the symbols of Reality. They are simply resources to be arranged, organised, and exploited. This is horizontality. Meditative thought aims at Reality’s disclosure itself: the void in which a symbol’s meaningful-ness arises. This is verticality in its most ontological register.
   

   This is obviously not the disavowal of rigour, but a recognition that rigour occurs within a prior openness that rigour cannot fully generate. We must have a meta-language first, an intended model, an ambient universe, a series of fresh interpretations upon which to apply rigour. The question is one of the quality of our openness, the posture by which we let the Real approach us. The point is not the dissolution of reason, but its just and proper use (horizontal) in the approach of Reality, and the subsequent need for verticality, intuition to determine, choose, interpret Reality’s disclosures. Reason necessarily reaches a point in its depiction of Reality where any further progress necessitates a change in the one who reasons.

You can probably think up hundreds of further examples. The symbolism, language, register may vary, but the ultimate structure underpinning all intelligibility is persistent. Reality itself resists the absolutisation of the horizontal, and this is demonstrable horizontally.

I will say that this structure⁴ is world-anchored. This is the universality of unsayability.

It is not that nothing can be said, but that Reality is fundamentally inexhaustible by what is sayable.

You simply cannot count to Infinity. You have to get there in one fell swoop.

1. This essay, I suppose, is my Critique of Pure Reason.

2. I believe in you.

3. Consider Deutsch, The Beginning of Infinity on the invariability of good explanations/interpretations.

4. The structure beingthe limits of horizontality, the need for verticality tempered by world-anchoring)