“Classical logic” does not name a single thing — it is a label covering at least four distinct projects, with distinct commitments: the propositional and predicate calculus (heir to Boole), Frege’s logicism, Cantor’s set theory, and Russell’s logic (types, descriptions). ArXe differs from each for a different reason, not just one. This document separates them and tests them with concrete syllogisms, verified with finite models, not merely argued.
1. The four projects “classical” blends together, and the axis of difference with each
| Project | Characteristic commitment | Axis of difference with ArXe |
|---|---|---|
| Propositional/predicate calculus (Boole and successors) | Formal from the outset, uncommitted to reality | No conflict of foundation — the conflict lies in the later use of the inclusive OR (Jevons) and in truth-in-all-models semantics, not in Boole himself |
| Frege | Logicism + Third Realm (abstract objects with their own, independent existence) | ArXe rejects the ontological commitment — existence is never granted by default, only dialogically |
| Cantor | The absolute infinite exists “in the mind of God”; the transfinite hierarchy as objective reality | The same rejection of ontological commitment, applied to mathematical existence in general |
| Russell | Theory of descriptions: existence is asserted within the logical form itself (∃x...) when analyzing statements with definite subjects |
ArXe never lets logical form alone imply existence — a term’s existence must be posited separately, explicitly, never smuggled in through the grammar of the sentence |
2. Three axes of difference, verifiable with concrete syllogisms
Axis A — Truth-conditional vs. dialogical
Classical logic defines validity as “true in all models.” ArXe defines it as “a winning strategy exists in the dialogue” (Felscher 1985) — they coincide in most cases, but the starting point differs: one asks about the reality of the models, the other about what can be upheld before an interlocutor.
Axis B — Bivalence vs. availability of three states
Classical logic requires every proposition to be true or false, with no third option. ArXe uses a third state — undefined — for what has not yet been accepted or rejected dialogically. This is not a third truth-value in the style of traditional three-valued logics — it is the absence of an assignment, until the dialogue produces one.
Axis C — Existence imposed by form vs. existence only dialogical
This is the axis best tested with syllogisms, and where the contrast is sharpest — because it has a deceptive variant that must be carefully separated out.
3. Gallery of syllogisms, verified
Barbara (AAA, Figure 1) — compatible with ArXe without any adjustment
All M is P
All S is M
∴ All S is PThis fits directly into the form ArXe requires (“A is a case of B which in every case is C”): the minor premise identifies S as a case of M, the major states what M is in every case. Neither classical logic nor ArXe needs any additional assumption here — both validate it the same way.
Darapti (AAI, Figure 3) — invalid for ArXe for TWO distinct reasons, not one
All M is P
All M is S
∴ Some S is PVerified by brute force: without requiring M to have at least one element, this mood fails in 9 of 25 possible models. Requiring M to be non-empty, it fails in 0 of 9. Darapti’s validity in traditional logic depends on a premise that was never stated — that class M is not empty.
Beyond this, there is a second, structural problem: M is the subject in both premises (“all M is P,” “all M is S”) — no premise says “something is a case of M.” There is no “A is a case of B, B is C” chain — there are two statements about the same class, combined by overlap. ArXe rejects Darapti twice over: it presupposes an unstated existence, and it does not even have the causal-chain form Axiom 1 requires.
Felapton (EAO, Figure 3) — the same double problem as Darapti
No M is P
All M is S
∴ Some S is not PVerified: an identical pattern to Darapti — 9 of 25 fail without M’s existence, 0 of 9 fail when it is required. Same double reason for exclusion: unstated existence, and M as subject in both premises.
Disamis (IAI, Figure 3) — invalid for ArXe, but for only one reason, not two
Some M is P
All M is S
∴ Some S is PVerified: this mood does not need M’s existence to be imposed — it fails 0 of 11 times either way. The reason is that the premise “some M is P” already carries the existence of at least one case, with no need for an extra assumption. Here ArXe does not object on grounds of unstated existence — it objects only on grounds of form: M is still the subject in both premises, and no “is a case of” ever chains A to B. It is invalid for ArXe for a purely structural reason, not because of hidden existential import.
Datisi (AII, Figure 3) — the same pattern as Disamis, mirrored
All M is P
Some M is S
∴ Some S is PVerified: 0 of 11 fail with or without imposed existence — the particular premise already carries its own existence. Excluded from ArXe on grounds of form alone, like Disamis.
Bocardo (OAO, Figure 3) — the same pattern
Some M is not P
All M is S
∴ Some S is not PVerified: 0 of 11 fail in both cases. Excluded from ArXe on grounds of form alone.
Ferison (EIO, Figure 3) — closes out the Figure 3 list
No M is P
Some M is S
∴ Some S is not PVerified: 0 of 11 fail in both cases. Excluded from ArXe on grounds of form alone — no hidden existential import.
4. Contrast table — verified summary
| Mood | Figure | Valid in classical logic outright? | Needs unstated existence of M? | Has ArXe chain form? | Valid for ArXe? |
|---|---|---|---|---|---|
| Barbara | 1 | Yes | No | Yes | Yes |
| Darapti | 3 | Only with existential import | Yes | No | No (double reason) |
| Felapton | 3 | Only with existential import | Yes | No | No (double reason) |
| Disamis | 3 | Yes | No | No | No (form only) |
| Datisi | 3 | Yes | No | No | No (form only) |
| Bocardo | 3 | Yes | No | No | No (form only) |
| Ferison | 3 | Yes | No | No | No (form only) |
The lesson of this table: modern classical logic (without existential import, an inheritance from Boole) already fixed the Darapti/Felapton problem on its own — but it fixed it with a technical stipulation (reading “all A is B” as a material conditional, vacuously true if A is empty), not for the same reason as ArXe. ArXe rejects all six moods of Figure 3 alike, including the four that modern classical logic already accepts without reservation — because ArXe’s reason (requiring the explicit causal-chain form, A is a case of B) is stricter than merely avoiding existential import. Agreeing with classical logic on Darapti’s outcome does not imply agreeing on the foundation — and on Disamis, Datisi, Bocardo, and Ferison, ArXe does not even agree on the outcome.
5. Frege, Cantor, Russell — where each critique lands, without blending them
- Frege: the Third Realm is the same move Darapti makes in miniature — granting existence (to abstract objects, to class M) without anyone having posited it dialogically. ArXe’s critique of Frege is the general philosophical version of what the Darapti verification shows in concrete syllogistic form.
- Cantor: the same mechanism, applied to the existence of the actual infinite — “it exists in the mind of God” is, structurally, the same unstated existential import that broke Darapti, transposed from a finite class M to the transfinite totality.
- Russell: the theory of descriptions solves the problem of “the present King of France” by analyzing it as an explicit existential claim (
∃x, x is king of France, and x is unique...) — unlike Frege and Cantor, Russell does make the existence explicit rather than silently assuming it. On this point Russell is closer to ArXe’s standard than Frege or Cantor — the disagreement with Russell is finer-grained: for ArXe, even an existence made explicit within the logical form itself is not enough — it must be accepted dialogically, as a step of its own, not derived automatically from the grammatical analysis of the sentence.
6. Status
Verified by brute force: Barbara and the six moods of Figure 3 (Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison), with and without existential import. Pending, should the gallery be extended: Figure 2 (Cesare, Camestres, Festino, Baroco) and the full Figure 4 — it has already been established that Figure 2 reduces to Figure 1 via conversion prior to the dialogue (syllogism document, §3), so its ArXe status is expected to match Barbara/Celarent once converted; not yet verified specifically for Festino and Baroco.