Hermeneutics: Applying Mathematical Logic To The Quran (Part 2)

1. Introduction to Higher-Order Logic

This conjunction, 'adjunction' and 'mixing' of what should be independent— shown in part 1— is not only inevitable but is the necessary next step (The rules of inference indicate this). Because here we make a direct and immediate inference to how we first came to define our terms; and this is what a 'higher order' means in logic: it's that which explains how we arrive at the lower order. Applied to our case, this duality, in application and in definition, is where the binary system of zeros and ones comes from (0:1). Because it's given that the emergence of higher order operations corresponds to 'greater degrees' of 'freedom' as far as manipulation of statements and the ways in which a statement can be interpreted. And as far as the conjunction of the two orders of truth and falsehood, of tautology and absorption is concerned, we obtain the law we call 'non-contradiction'.

2. The Principle of Non-Contradiction

This law is the foundation of all reasoning, a principle that has fueled remarkable advances in philosophy, mathematics, and computing. It forms the basis of the binary arithmetic system of zeros and ones, which in turn underlies the digital circuitry and logic gates that make modern computers possible. At its core, this law emerges from the interplay between higher and lower orders of truth and falsehood. The binary nature of truth and falsehood, of opposites, is the embodiment of the law of non-contradiction, but again, we obtain this law only as the conjunction of the rules of tautology and absorption which explain the full extent of this law, going past the bare statement of 'a proposition cannot be both true and false simultaneously in the same context', and actually shows how this dynamic unfolds; it tells the story right from the beginning, so to speak, and explains how we can have the interchange of truth and falsehood in the same context, by positing the structure of 'what is' and 'what could be'.

This diagram is a logic gate I constructed to show how the law of non-contradiction is composed of the rules of replacement 'Tautology' and Absorption. The laws of thought are reducible to conjunctions of rules of replacement, and PNC specifically, is composed of the ones just mentioned. Here's a breakdown:

1) This here is an 'operator' —< (that serves in these gates the function of distribution, it distributes —in a 'predicative' sort of way—the function of the law to both sides of the dyad.

2) This 'distribution' takes into account the 'trans-ordinal' relation discussed in part 1. Rules of inference establish these orders'. Rules of replacement 'govern' relations between the them, whereas laws of thought are the fundamental axioms from which we infer that there are different orders.

3) We can extend this by saying that the rules of logic represent the 'deconstruction' and exposition of the laws of thought, which in this context are presuppositons of all rules of logic.

4) It's from the laws of thought that we derive the notion of 'order', or 'inference' and of 'replacement'. a 'presupposition' is a non-contradictory notion, the same goes for 'unity', or 'exclusion'. The more complex operations of logic arise from these suppositions as central axioms

Deconstructing the Formula

The absorption formula states: p ^ (p v q) = p and p v (p ^ q) = p. If we hold that conjunction and disjunction exist between the two orders of truth p,q, then it intuitively follows what function p serves in both cases. If an assertion in general is equivalent to an immediate representation (a thought, an idea, or 'appearance' in the sense of Kant), we can see how this is a condition/state 'distinct' from the subject that's its correlate. The subject preserves his/her independence from the 'fact', and this independence (or distance) allows the subject to continuously 're-present' or 'act' or 'operate' upon the fact, to reject or add, modify, or defer. These presuppose an 'independent' subject. This defines p for us. Now the operators (^ and v) when added to it, produce the following:

In a binary relation:

• If x is 'stable', then y is in 'flux'

• If x is 'in-flux', then y is stable

  
  

The same basic idea translates when applied to the absorption formula.

p ^ (p v q) = p: 'Subject' 'stable', analogically, 'there's change' in the system (any system), only because of the underlying 'unity' of the subject. But this unity is not of the 'substantive' sort, not a relation between the concept and its limitation, but the necessary 'indeterminacy' of any element in the system by contrast to the determinacy of any definite element (which is its natural limit). These two, as noted, operate as a dyad.

You can begin to make out what a 'stable' subject is in relation to the validity of a 'falsehood' which constitutes the absorption rule. A 'true falsehood' or an 'indeterminate' subject is not a unity in the 'constructive sense' (this would be 'associative' -> see part 4), but a 'presupposition' in the broadest sense of that word, a variable, insofar as 'variability' is functionally 'discrete'.

Thus the conjunction p ^ denotes the 'stability' with which a 'change' in the 'configuration' of the system is expressed (p v q), whereas a disjunction p v denotes the 'flux' itself, which we've just described as 'stable' (p ^ q). We find in the exception (in the conventional sense), what allows us to hold it as a rule, and in the rule, see what can be held 'exceptional'. In absorption, what's primarily asserted is the 'validity' of the 'exception', and the exception in the logical sense is the natural 'correlate' of the 'valid' (or 'real').

Think of it as a 'precondition'; its place in this order isn't that of a 'discrete' proposition, but it's nonetheless what becomes 'propositional' (becomes 'infected') by mere inference from true proposition. You can say it's 'propositionalized' in its denotation as the 'precondition' of a proposition though it is not, in itself, a proposition.

3. Practical Application: A Mechanical Analogy

A car (or any mechanical system, your body included, operates this way too) serves as a great example of this in everyday life. Pressing the gas pedal accelerates the car; it's here where the 'action' is, the point of 'mechanical emphasis', the center of force. In a second, the emphasis will shift over to some other part of the vehicle, this is despite the fact that every part of the system is instrumental to the effect/output of any given part.

This 'shifting' of emphasis from and to different parts of the car according to the situation is non-contradictory, because it involves a duality between some immediate function and the possibility of another within the one system; and in each case, all peripheral parts, the parts that are not emphasized, are absorbed, and this absorption is necessary for the maintenance of this change. In a mathematical context, we'd use a variable to denote the 'rest of the system' as opposed to the true, given part; like (x + 7) = 15, it may be obvious what x stands for there (8), but the point of the variable is to 'denote' the possibility of 'any' number; and 'number' covers' the entire spectrum of natural numbers. It is thus a propositional function (an existential quantifier, but more on this below).

Notice that when we shift our focus over to the variable, the absorbed part, we also shift the truth-value over to it in real time, absorbing in the process what used to be the true, or real part. This is tautology and absorption functioning together.

Application to Quranic Analysis

1. From Logic to Textual Analysis

Having ventured deeply into the domain of logic, let us return to the original topic: how this relates to the Quran and our understanding of its terms. To start, how is this law of non-contradiction useful in this case?

The short answer is that the Quran is composed of words as the simplest units of meaning, and to understand a verse, a chapter, a narrative, one must understand the words, and to understand the words in their true sense and import, we must analyze them. There it is: the inevitability of analysis leads back to the problem of method, and as far as method is concerned, we could look no further than logic.

2. The Analytical Framework

If we actually apply the principle of non-contradiction to the analysis of a Quranic term, what first happens is that we relinquish, or abdicate, whatever 'meaning' we suppose the word to have. Non-contradiction, when applied, has the effect of negating anything already in existence, because, since it's the simplest, most fundamental law, it doesn't add anything to anything; it is, in itself, the beginning of logical analysis, of thought itself.

This difference is a good sign, as it means that we're in sync with these rules, that we're beginning to make the transition from the domain of convention to that of pure reason. There's an isomorphic relation between the mind of the thinker and the rules according to which the mind naturally experiences objects, and the result is that the object is arranged in precisely the manner in which it should.

3. Case Study: The Word 'Ard' الارض

The term is stripped down to that state of general validity, or truth. Think of it this way: there's no 'meaningful term' as such here; there's what Cassirer would call an 'analytic' Identity (Substance & Function) , a general object that we are able to analyze. The concept of validity is applicable here insofar as we have a concept to which the rules of logic can be applied in the widest/broadest sense. We're essentially running a 'logical' module in parallel to the word whose meaning we are only vaguely acquainted with. The aim here is to have the logical module act as a course correction procedure as we proceed to analyze the term. 'Ard' here, then, is understood in terms of its general 'validity' and its possibility. This way of conceiving the term is actually the same as that which modern science first conceptualized the concept of energy.

The word 'Ard' recurs 461 times in the Quran, in various morphological and grammatical forms; these represent the boundary of the word, the limit within which it is understood as an individual concept. This defines the universe of discourse for the word and how we're going to understand it. This universe of discourse is essential in that it's what decides how much factors into the overall meaning and sense of the word. The law of Non-contradiction fits this whole domain within its binary structure, saying that, at the most fundamental level, there's only the assertion of the word and the possibility of asserting it.

The Mechanics of Meaning

1. Absorption and Tautology in Practice

It's precisely here, in this new domain of analysis, where the rules of absorption and tautology which make up the law of non-contradiction will apply. They will provide us with the 'restriction' necessary for continued exposition of our terms. Absorption will maintain for us every morphological or contextual form of the word in the background as we transition (as we did in the car analogy) from one modification to another; and tautology will keep these possible meanings —absorbed in the background—valid in addition to any particular determination of meaning.

2. Contextual Analysis

Thus if in one context the word Ard expresses a domestic sense, like verse 22 of chapter 2, where the current verse, say, 6, of chapter 78 in which it also appears expresses a different sense of the term, the first meaning (the domestic meaning) is kept as relevant as the new meaning, and equally accounted for, functioning as a possible mediating or analytical factor into every new expression or rendition of the term. It's not the particular form or meaning of the word that these laws pertain to → so much as the general validity and possibility of a word at all.

3. Synthesis: The Dynamic Structure

Thus we must interpret this variance as to what the term could mean in different contexts as essentially and mainly a reflection of the dynamic of validity and possibility (or variability). Thus tautology, jointly with absorption, gives us continuous access to immediate as well as possible terms, and allows us to rotate around these two dynamic poles, restricting us to the word as such, and the possible conditions of the word.

Thus we can say that the law of non-contradiction exclusively provides the structure of validity and possibility: any immediately given expression → say Ard الارض, and any possible, non-immediate, or derivable meaning of the word (e.g 'Yarda' يرضى).

Non-contradiction only provides this outline; the rules of tautology and absorption are what really operationalizes this law. It gives us the duality and makes possible that to-and-fro movement from what's immediately the case, and what's not immediately the case, but is relevant to it. All in all, it positively restricts us to this dyad.

The Law Of Excluded Middle

1. Expanding Traditional Understanding

It's important to consider that Non-contradiction only gives us the binary structure of true and false, valid and possible; but it does not determine the elements of its own structure; the either-or function of determination is carried out by the law of excluded middle. This is an example of how we circumvent Russell's paradox using the same essential rules of logic. But non-contradiction isn't usually thought of in terms of the two orders of truth and falsehood, a crucial oversight, that in turn affects how we understand LEM. In the traditional sense, the principle only states that 'every proposition must be either true or false; there is no third option or "middle ground"' but this is incomplete. The reality is, we must show how it is possible to assert that something is either true or false. This is attained when we consider that there's a higher level from which these determinations are made. Hence, the law of excluded middle is fully represented through considerations of the ground of each determination.

2. Universal Quantification

This is indeed the case; in fact, the law of excluded middle is either the same as, or relies on universal quantification—the idea that a given statement or property applies to all elements within a specified domain of discourse.

For all x, P(x) is true. A universally quantified statement ∀x P(x) for all x is true if P(x) holds for every possible x in the domain. It is false if there exists at least one x in the domain for which P(x) does not hold.

3. The Role of Predication

This is a clearer exposition of the law of excluded middle, because to state either truth or false involves 'predication', and since there are two orders of truth and falsehood, it can only mean that we can have truth and falsehood as 'assertions'; by having lower order truth or falsehood as the genus p(x), and their assertions as limiting cases, x. Thus the universal quantifier captures, more accurately than the classical law of the excluded middle, the assertability of either truth or falsehood, by showing how each assertion has a basis in the self-same system. Moreover, universal quantification fully accounts for the absorptional and tautological rules operative within the same structure: using a variable to maintain the relativity of assertions.

Understanding the Formula

1. Scope and Domain

The p in the formula denotes the generally valid at any given instance, and x as the placeholder for either truth or falsehood. But the scope of p(x) is important, quantifying truth for example, involves all assertable functions and values of truth within the domain, and likewise of falsehood; this is what 'for-all' x means; we don't only assert truth from the standpoint of higher order truth, but we take into account the various possible expressions of truth which non-contradiction through absorption and tautology establish for us. This link is absolutely essential.

2. Real-World Applications

I can think of many examples of this rule, in the quantifier sense, in real life: what we conceive, we ourselves produce, or in the Stoic maxim, we 'reap what we sow'. Two objects perceived in space is, this law tells us, the result of implementing the concept of 'two-ness'; that we decided to focus on the numerical aspect of the object results from an application of the concept of number to the object, and that we specifically see two, results from a similar though more 'differentiated' application → namely, a numerical concept. This bridges that gap which Kant and others have tried to close between representation and the faculty of representing; truth and falsehood, this framework tells us, results purely from our capacity to apply them. But what we apply, we apply in the fullest sense. In a nutshell: that we recognize 'true' arises from a higher order application of truth; that we recognize false arises from a higher order application of falsehood. Predication arises at precisely this level, hence 'predicate logic'.

Rules and Applications

1. The Need for Additional Rules

More than anything, this shows that non-contradiction alone is not enough; it relies through and through upon the ability to determine the poles of non-contradiction, but the determinability of said poles involves their direct application, a new rule is thus factored that accounts for this 'granular' aspect.

2. Contraposition and Exportation

Just as non-contradiction is specified by the two rules of tautology and absorption, so is the law of excluded middle; for the latter, we have the rules of contraposition and exportation. Think of the function of an instance of legal violation or infringement and how it ties to, or 'implicates', the whole legal system; we say 'it breaks the law', or, 'it's punishable by x penalty'. There's what seems to be a seamless and implicational transition from the instance to the precedence, an immediate 'incitement', as it were which says that this instance of infringement, or 'enforcement' is representative of the system at large (the system at large being the precedent); now the context is not important, you can substitute it for any other, you can talk about any whole from the standpoint of any instances of that whole.

It's what gives us p(x) as the premise of x, or allows us to infer the general from the instance; but inference in the sense that I'm always and continuously able to infer the precedent given the instance; so it's not an immediate inference so much as the 'possibility' of inference, as a 'power' or capacity within the analysis; in a word, 'predicability'. This is contraposition.

'Object', 'System', 'Whole', 'Predicand', are you using these terms 'interchangeably'?

Let's me clarify what i mean by 'predicability', 'object', 'system', 'whole' in this context of contraposition. I admittedly, I used these terms synonymously, but what they indicate is one and the same reality. By 'precedent' or 'object', we're not referring to a fixed, concrete entity, but rather to what we might call an 'inductive object' - a guiding principle that operates more like a compass direction than a destination. An objective is but an 'orienting principle', inasmuch as an 'object', so long as it's concieved as something whose 'reality' is 'to be derived'.

Think of how a compass works: 'North' (N) isn't a specific place you can reach; it's a directional principle that guides movement and orientation. Similarly, when we talk about a 'system' in contraposition, we're referring to an indexical framework that guides understanding rather than a concrete, bounded entity. It's not a particular 'point/place' that we can isolate or reach; instead, it's an objective in the process-sense - a continuous guiding principle that helps organize and direct our logical operations.

Let's carry this discussion over to part 3