ArXe Syllogistics: Formal Foundations

View: Analysis of ArXe Syllogistics: A guide to understanding the system

1. Structure of Valid Syllogism

Definition

A valid syllogism is an Aristotelian syllogism with certain restrictions to resolve ambiguities.
What it seeks to resolve:

Interpretations that lead to forms of Platonism:
— Propositional realism: the presumption that the truth of propositions exists independently of judgment
— Magical realism: the presumption that syllogism allows ontological determinations or determinations about reality
— Fusion between being and existence: The presumption that these are the same concepts that actually distinguish themselves from each other
— Negation of essence along with existence

Scope

The syllogism does not define a new ontology nor does it make determinations about the reality of facts.
Its scope is to determine the internal coherence of propositions.

** Note:
Unlike classical logic, ArXe distinguishes essence from existence

In classical logic, what is necessarily exists and vice versa
In ArXe, what is does not necessarily exist, nor is it necessarily true, although what exists necessarily is
In ArXe, falsity or contradiction have essence, that is, they are.

This means that in ArXe, for example, ¬P negates existence or truth, but not essence

A valid syllogism in ArXe consists of exactly:
** Definition of subject:

Everything sayable that can be expressed in a single term or word

Universal Premise:

Universal subject (positive, not explicitly negated)

— Valid: All men, No X
— Invalid: All non-men, No ¬X
— Valid: All immortal, No X
— Valid: All non-mortal, No X

Predicate that affirms, possibly negating, or negates, possibly affirming, something about the subject
All men [if] are bald
No bird [does not] fly
Why is “immortal” valid and “non-mortal” is not?
If in the syllogism “non-mortal” and “immortal” are interchangeable with the same meaning
then “non-mortal” would be valid, but then why not say “immortal” directly?
This prevents improper use, for example “all non-bird” is not interchangeable with “unbird”, therefore “non-bird” is invalid

Particular Premise:

Particular subject as a case of the universal subject
Predicate that affirms, possibly negating, or negates, possibly affirming, something about the subject
Form: “x is a case of S”

Conclusion:

The particular inherits the predicate of the universal
Form: “x is (or is not) P”

2. Explicit Form

If [we hold as true] that all S is (possibly not being) P
And [we hold as true] that x is (possibly not being) a case of S
[It is implied that] x is (possibly not being) P

Classic example:

Universal: All men are mortal
Particular: Socrates is a case of man
Conclusion: Socrates is mortal

Extended form:

If [we hold as true] that all men [if] are mortal
And [we hold as true] that Socrates [if] is a case of man
[It is implied that] Socrates [if] is mortal

3. Nature of Propositions

Fundamental principle:

Propositions are assertions such that, if they are held as true (they may not be held as such and the syllogism would lack value), the truth of the conclusion is implied from them.

Characteristics:

Conditionality: Truth is hypothetical, not categorical
- The syllogism does not assert: “All men ARE mortal” (ontological)
- The syllogism says:
  “IF we say that all men are mortal
  AND we say that Socrates is a man,
  THEN we say that Socrates is mortal” (logical implication)
Acceptance: The premises can be “held as true” or not
- If they are accepted: the conclusion follows necessarily
- If they are not accepted: the syllogism lacks value
- But the structure remains valid regardless
Coherence, not reality: The syllogism concerns the consistency of the proposition, not the reality of facts

4. Meaning of “Being a Case Of”

Definition:

“x is a case of S” means:

“we affirm that x is a case of that which we have affirmed that in all cases is S”

Clarification:

It is NOT set membership: x ∈ {S}
It is NOT instantiation of Platonic essence
It IS a propositional relation: x satisfies the condition that defines S in the universal proposition

Example:

“Socrates is a case of man” = “Socrates is a case of that which in all cases is man”

This means that Socrates satisfies the condition of the universal in the proposition.

5. The Truth Contained in Propositions

Principle of containment:

The truth of the conclusion is contained in the propositions.

This implies:

The syllogism does not generate new truth — it only makes explicit the truth already contained
The syllogism is extraction, not construction
The conclusion was already implicit in the combination of premises

Analogy:

Like opening a box: the conclusion was already inside the premises.

6. Why There Is NO Inverse Syllogism

Invalid form:

✗ INVALID:
Particular: Socrates is mortal
Universal: [???]
Conclusion: All men are mortal

Reason:

The truth of the universal CANNOT be derived from the particular because:

The particular is to the universal what the part is to the whole
Only the universal contains the particular, not the reverse
The direction is unidirectional: universal → particular → conclusion

7. Why Only Two Propositions

Apparently valid form (INVALID):

✗ INVALID as a single syllogism:
Universal 1: All men are mortal
Universal 2: All mortals are corporeal
Particular: Socrates is a man
Conclusion: Socrates is corporeal

Reason:

This is not one syllogism — it is two chained syllogisms:

Syllogism 1:

Universal: All men are mortal
Particular: Socrates is a man
Conclusion: Socrates is mortal

Syllogism 2:

Universal: All mortals are corporeal
Particular: Socrates is mortal [conclusion from previous]
Conclusion: Socrates is corporeal

Principle:

Truth is contained in each pair of propositions, not in sets of three or more.

A syllogism has exactly:

One universal (contains the general truth)
One particular (identifies the case)
One conclusion (extracts the contained truth)

Necessary structure:

There cannot be:

A single proposition (there is no inference)
Three or more simultaneous propositions (they are chained syllogisms, not a single syllogism)

8. Ontology-Syllogism Separation

Within the syllogism:

There are only conditional implications: If P₁ and P₂, then C
Propositional coherence is preserved
Existence or factual reality is not asserted

Outside the syllogism:

ArXe can make primitive ontological assertions
Example: “T₀ is contradictory act” (not syllogistically derived)
These assertions are not grounded in syllogism

Principle of separation:

The syllogism never grounds ontology — it only preserves coherence between accepted propositions.

9. Contrast with Other Systems

ArXe vs. ZF (Zermelo-Fraenkel)

Aspect	ZF	ArXe
Structure	Posits existences axiomatically	Derives through syllogism from universals
Justification	Without universal containing them	Every conclusion contained in universal
Propositions	Quantifiers without clear syllogistics	Exactly two propositions
Ontology	Confuses coherence with existence	Separates syllogism from ontology

ArXe vs. Aristotle

Aspect	Aristotle	ArXe
“Case of”	Ambiguous (membership? instantiation?)	Defined: satisfying the universal’s condition
Truth	Ambiguous (formal? ontological?)	Explicit: propositional conditional
Universal	Ambiguous (does it exist? is it mental?)	Propositional, not ontological

10. Synthesis

Fundamental principles of ArXe syllogistics:

✅ Truth is contained in the propositions (no new truth is generated)
✅ There is no valid inverse syllogism (particular does not contain universal)
✅ Only two propositions per syllogism (more than two = chained syllogisms)
✅ The syllogism preserves propositional coherence, does not assert reality
✅ “Being a case of” is satisfying the condition of the universal in the proposition
✅ Truth is conditional, not categorical
✅ The syllogism does not ground ontology — they are separate domains

Validity of the syllogism:

A syllogism is valid if and only if:

It has exactly one universal premise
It has exactly one particular premise that is a case of the universal
The conclusion attributes to the particular the predicate of the universal
The structure is: If P₁ and P₂, then C

Nature of the syllogism:

The syllogism is extraction of already contained truth, not construction of new truth.

The conclusion was already implicit in the premises — the syllogism only makes it explicit.

11. Methodological Consequences

For ArXe as a theory:

Every derived assertion must be expressible as a valid syllogism
Every primitive assertion (not derived) must be declared as ontological
Logical coherence is not confused with ontological truth
Nothing is posited that is not a case of a prior universal or ontological primitive

Criterion of theoretical validity:

A theory is syllogistically valid if:

Its derivations respect the form of the syllogism (2 propositions, 1 conclusion)
It does not invert the direction (particular → universal)
It does not confuse propositional coherence with assertion of existence
It explicitly declares its ontological primitives

Conclusion

ArXe syllogistics establishes a rigorous framework for reasoning where:

The formal structure is impeccable (two propositions, universal→particular direction)
The ontology-logic separation is clear (the syllogism does not create existence)
Truth is conditional and contained (not generated, but extracted)
It seeks to avoid propositional realism (the validity of propositions is independent of judgment)
It seeks to avoid Platonism (there exists a reality for propositions distinct from the reality of those who formulate them)
The truth of propositions exists independently of the sensible world

This avoids:

The problems of ZF (postulation without justification)
The ambiguities of Aristotle (nature of the universal, of “case of”)
The Platonist confusion (formal coherence ≠ ontological existence)

And provides:

Solid foundation for theoretical derivations
Clear criterion of inferential validity
Explicit separation between the derived and the primitive

12. Dialectical Contrast with Classical Systems of Logic

On the Principle of Bivalence
The syllogism in ArXe does not produce ontological truth, but preserves inferential coherence within a conditional framework.
In this way:
It does not affirm what is true of the world,
But what would be coherent if a certain set of propositions is assumed.