1. Contraposition and Variable Perspective

Contraposition does not appear to tell us from which standpoint a totality may be viewed, but almost as a matter of course, seems to leave this undecided. When we say 'all languages of the world', we're applying contraposition to the concept 'language', but the standpoint is not a particular language, but 'any language'. Here, 'any' is a variable that captures a higher level falsehood (-q). We call -q a higher order falsehood because, by contrast to a lower order falsehood, or falsehood as such, because it's predicable of some term, making it a hypothetic syllogism. Lower order falsehood proceeds a priori; it does not reveal anything—it is the state that precedes any determination period, precedes it in the same sense in which Deleuze holds 'difference' to precede identity. The same goes for lower order truth; it's the already valid, note the predicable sense of validity.

Deleuze's 'difference' and why it's logically important

‘Difference’ here is precisely the sense in which -p is used in propositional logic. The ‘starting point’ of any determination is ‘undefinable’, and ‘necessarily so’. This is embodied in the disjunctive syllogism -p ∴ q. Being ‘undefinable’, we can only be made aware of it ‘as defined’. The expression ‘undefinable’ is itself a definition; and the ‘definition’ of the ‘undefined’ is the hypothetical syllogism proper (-q ∴ p). This ties the two rules together. Kant uses this concept in the transcendental asesthetic in his description of space

"The true correlate of sensibility (assertoric truth q), that is, the thing in itself, is not known, nor can it be known at all through these representations, nor do we ever ask any questions about it in experience" The Critique Of Pure Reason (B45, A30, A29)

To understand by words such as 'undefinable' or 'unknowable' some actual 'inaccessible' something, would be wrong, 'undefinable' is a logically valid assertion; a logical function. To miss this is to confuse logical validity for a judgement. Again Tautology validates everything, it has nothing to do with a judgement per se, but with its 'validity' as something asserted.

2. Beyond Simple Mereology

The condition of 'any', and of contraposition specifically, is not that which we see in mereology, of 'yellow' being an instance of the genus 'color', or 'silver' being an instance of 'metal'. No—when you perceive the system at large, your wholistic disposition sets up that condition which allows you to view the whole through whatever quality or property the part possesses. You can choose, for example, to study the whole substance 'metal' from the standpoint of density or elasticity, and actually choose to glean in other aspects of metal properties which you may call elastic or dense; you're not restricted to any part. What's important here is knowing that contraposition is simply the program that makes this possible, because it uses a variable, 'any' as its standpoint, not a definite part.

For instance, when we study a specific expression of 'Ard' 'contrapositively', we're not merely looking at a subordinate case of a general concept. Instead, we're examining a unique configuration that reveals something essential about all other instances of 'Ard'. Each manifestation carries within it patterns that resonate throughout the entire system, much like how a single genetic mutation can illuminate our understanding of the entire genome.

Application to Textual Analysis

1. The Case of 'Ard' ارض

Thus, we have 'Ard' on the one hand, and 'every possible modification' and 'expression' of Ard'. Our analysis of the term does not exit this circle, and cannot determine anything outside it. Should it do so, like citing a dictionary, then, despite pertaining to the word Ard, and could be considered as falling under the rubric of 'possible meanings of Ard', it would violate the law, because it contains judgements inconsistent with the structural sense which the logical scheme is building up. Again, the logic is concerned only with how we arrange and condition our relation to the word, before any meaning such as a dictionary would throw at us.

Note the Restriction: This is important. This shows how the analysis is restricted to the sphere of the given term.

What we mean by 'every possible modification' are not just modifications to the meaning of the word 'Ard' according to the different contexts in which it appears, this is given, but modifications to the ways in which the word embodies any given rule of logic. If we understand the word 'Ard' in terms of the rule of absorption, then this would count as a 'meaning' (or, a truth value, if you will); if we understand it in terms of the rule of exportation, then this too would be another meaning; if we understand it in terms of the rule of tautology, then this would be yet another meaning. Some may object, saying that these are only structural aspects of the interpretation of the word whose sole purpose is to give us 'meaning' (the original sense of meaning; of 'Ard') rather than be objectively meaningful themselves, but this would be the same objection which many had leveled at Einstein's theories of special and general relativity—for this is precisely the methodological principle through which Einstein arrived at his revolutionary theories.

The Interplay of Logical Rules

1. The Role of Excluded Middle

The law of excluded middle is essential in this respect: it says that we can only determine either of the structural conditions of non-contradiction. But since truth and falsehood become diversified into different structural expressions and conditions of truth and falsehood as necessary implications of the existence of two orders of truth and falsehood and of the mediating role of the rules of replacement, the differentiation of the function of exclusion into contrapositive and exportive relations is revealed as necessary.

2. Contraposition and Exportation

Contraposition affirmingly states the valid in possibility; it thus uses a higher order falsehood (the truly false) -q to state a lower order falsehood -p, thereby pointing to all that which can appear or manifest as true. It's a 'validating' relation; it doesn't actually 'state' a truth, though it's anticipation of truth is a statement, if we approach it assertorically. Every valid proposition falls within the range of contrapositive judgement; through contraposition we attain a permanent posture towards the valid but this is symbolically achieved by 'pointing' rather to the 'ground' of validity, a falsehood; by so doing, it maintains a positive relation, or an 'openness' towards what 'can be validated', but contrary to a hypothetical syllogism, not from the standpoint of some truth, for this would be 'restrictive', but from no definite standpoint -p.

To be Clear, you never perform or produce a contrapositive judgement using contraposition alone; you can only do this by another rule, a rule that allows you to mediate between the 'whole' and the undefined, possible instance—this is exportation:

3. The Function of Exportation

This rule says that we never determine anything unless through another thing. For example, if you're going to work, exportation in the broadest sense will focus on something like the relation of 'going' to the object 'work'. There's a mediation here: 'work', the objective, is 'mediated' or 'indicated' by a relation of 'going'. It focuses on the expedient 'relation' between the objective and something that determines the objective.

The 'expedient' is what expedites the objective in the proposition. This can be any meaningful (the general sense) unit that's not the objective:

I'm, Going , To (expedients) ==> Work (objective)

We understand these non-objective units 'expediently', as determining the objective. But we can see the contrapositive 'import' in this, in how the determinations 'shape' the objectivity of the 'objective'; that is to say, they 'determine' from their own standpoint, the manner in which the objectivity of the 'object' is understood!. This is the 'Contrapositive' side of the equation whose exportive side is the objective determinations themselves.

Synthesis and Conclusion

From a certain perspective, exportation is how we determine anything in the first place, because our knowledge is always discursive; it uses 'means' to reach an end. But at the same time, contraposition is how we have objectivity in the first place—it gives both the object and all possible means of determining it at the same time insofar as the object is, in the widest sense, true or false. To be more precise, for contraposition, the whole structure viewed as a unit is our objective, but to the universal quantifier, or the law of excluded middle, it's the positivity of the elements of the structure in relation to the structure at large. You can say that contraposition is cosmological, but universal quantification is genealogical. Thus on analyzing the law of excluded middle, you will find that it's primarily defined by the rules of contraposition and exportation.

Symbolic analysis

Understanding Truth Orders and Reference

1. Basic Symbols and Their Meanings

p stands for 'true', a placeholder for generally known ideas before reference
q stands for higher order truth, representing instances or limiting cases of 'true'
The relationship between p and q shows how reference works:
- p is like 'yellow' as it exists before mention
- q is the act of referring to or citing p
- Thus "p therefore q" (modus ponens) shows their interdependence

2. Negation Forms

-p refers to possibility of original truth (before reference)
-q is a limiting case of false, an instance of reference to the possible
The contraposition formula (-q → -p) demonstrates:
- You can only refer to possibility through instances of reference
- This maintains the distinction between higher and lower orders

3. Signs and Objects

p and -p function as objects
q and -q function as their signs
This mirrors Peirce's semiotic theory:
- Distinction between object and referring sign
- Difference necessary for identity (S. Alexander)
Psychological parallel:
- Memory (p) vs. act of recollection (q)
- Past event vs. present recall

4. The Role of Exportation

Formula: (p ^ q) → r = p → (q ^ r)

Shows how we refer to things mediately
Conjunction of higher and lower order truths (p ^ q) implies r
R transforms conjunction into a constant
Example: Thinking a thought:
- Initial conjunction of thought with awareness
- Recognition of this as a mode or state
- The Recognition itself is 'determinate' and/or 'objective'
- The awareness is a 'sense of awareness'

5. Practical Applications and Husserl's Concepts

Parallels Husserl's noesis/noema distinction
Circular nature of reference:
- Object thought = act of thinking itself
- Circularity necessary for higher processes
Applies to all objects of thought:
- Reference act is circular
- Object nature remains distinct

6. Logical Rules as Inferences

Presupposition of reference includes any and all logical rules
Rules are can be 'inferences' as well as 'object' judgements, I may, in analyzing a term through exportation, presuppose its 'general' validity, in this respect, 'Tautology' for me is 'assertoric', because it's that which I validate the exportation process. But in this interpretation, I recognize the role of exportation in dictating 'Tautology' as an assertion. I'm thus compelled, should I choose a particular rule as 'assertoric' to consider how this 'assertoricity' is reflected from the perspective of that which it asserts. It locks me into a 'commutative' relation in which the reprecussion of inference give rise to more complex interpretations.
Example with Quranic word 'Ard':
- As judgement: refers to plane of existence
- As inference: valid proposition open to analysis
- Distance from immediate meaning allows multiple interpretations

Hermeneutics: Applying Mathematical Logic To The Quran (Part 3)