In a conversation where this paper had just been summarized, on The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction (PNC).
Write a fictional debate about PNC with characters exhibiting the views and strategies in this paper.
“Gamma Night at the Logic Café” — a fictional debate on PNC
Cast
- Ari — the calm supporter (Aristotelian at heart)
- Dee — the detractor (friendly dialetheist, flirts with Heraclitean moods)
- Fierce — the fierce supporter (thinks debate is pointless)
- Demo — the demonstrator (yearns to prove PNC)
- Method — the methodologist (chooses logics by theory-choice virtues)
- Mod — the moderator
Scene 1: Opening claims
Mod: Welcome to Gamma Night. One principle, five temperaments. Two minutes each. Fierce?
Fierce: Thanks. Here’s my position: the law of non-contradiction is more certain than anything you could use to argue for or against it. Therefore, a “debate” about it is either question-begging or idle. I abstain.
Dee: I don’t. Some contradictions are true. Sometimes reality bites back and we must say: A and ¬A. Meaning doesn’t explode; we just grow up.
Demo: I’ll do better than allegiance or rebellion: I’ll derive non-contradiction from principles we all accept. No mystery, just proof.
Method: And I’ll evaluate all this like any other scientific quarrel: simplicity, explanatory power, conservativeness, fruitfulness. Whichever logic best scores wins—even if it tolerates a few contradictions.
Ari: I’m after a quiet middle. Strong versions of PNC are overstatements. But there’s a minimal semantic backbone we all rely on, even Dee: call it Semantic Exclusion—an expression can’t bear mutually exclusive semantic statuses at once when we’re using it to rule something out. I’ll argue this modest core is presupposed by everyone here whenever they mean to exclude anything at all.
Scene 2: The common ground problem
Fierce: You’re already presupposing the very principle in dispute. That’s not debate; that’s catechism.
Ari: Not quite. I’m not presupposing a grand, global PNC. I’m pointing to the shared practice in this room: when you state your view, you aim to say this and not also its incompatible. You may philosophize about contradictions later, but to get your thesis on the table, you treat its incompatible as not asserted by you. That local exclusion is enough common ground to talk.
Dee: I can say, “Some contradictions are true,” and also say, “That sentence is false,” without combusting. Why must I accept exclusion?
Ari: Because you want your thesis to do work. If your key claim is true only together with its negation, it doesn’t rule out anything. Suppose you tell us: “All and only contradictions are true.” Do you mean that by itself? If yes, you’ve excluded its negation. If no, then the thesis fails to exclude the case where some non-contradictions are true—undermining the point of your slogan.
Dee: I can accept that in some contexts, I assert a claim without its negation. But I won’t universalize it.
Ari: Perfect. That “some contexts” is exactly the minimal core I’m defending.
Fierce: Hmph. You’ve smuggled in enough to talk. Fine. Talk.
Scene 3: Demonstration on the chalkboard
(Demo strides to the board.)
Demo: Consider the tautological Law of Identity: if α then α. Using a rather modest modal principle, it follows that it’s impossible that (α and ¬α). Voilà—PNC.
Ari: Before the voilà, one question: do you mean to prove only “It’s impossible that (α and ¬α)”, and not also its negation?
Demo: Of course. Otherwise it isn’t a proof; it’s noise.
Ari: Then your proof presupposes exclusion: the conclusion is meant to hold without its incompatible. And each step is meant to transmit truth without also transmitting its incompatible. That’s the minimal PNC doing backstage duty. Without it, your demonstration could equally well “establish” the conclusion and its negation. In other words, demonstration needs exclusion to be demonstration.
Demo: You’re saying proofs can’t get off the ground without a prohibition on double bookkeeping.
Ari: I’m saying: to prove only one side, you already rely on a norm that bars you from also proving its incompatible. That’s the modest core.
Dee: But I can live with arguments that sometimes yield both. I just won’t call them “proofs” in your absolutist sense.
Ari: Then we agree: to call it a proof—as you just insisted—we rely on the exclusion norm.
(Demo lowers the chalk. Nods, thoughtful.)
Scene 4: The Heraclitean interlude
(Dee grins, adopts a more radical tone.)
Dee: Fine, let’s go full river-talk. Heraclitean mode: things are in flux; only contradictions capture them. So: all and only contradictions are true.
Ari: Do you intend that statement to be true without its negation?
Dee: Ideally, yes.
Ari: Then you’ve conceded the minimal exclusion again. If instead you say it’s true only with its negation, the thesis fails to exclude theories where non-contradictions are true—deflating the Heraclitean claim. Either way, your ability to state the thesis in a way that rules out rivals depends on the minimal core.
Mod: So the radical view either self-limits into the minimal core or dissolves into triviality.
Dee: I’ll take the self-limiting route. I want to exclude some rivals.
Ari: Fruitful! We’ve identified exactly where exclusion is indispensable—and how far it extends.
Scene 5: Method in practice
Method: Let’s change lens. Suppose we weigh logics by simplicity, explanatory unification, and conservativeness. Paraconsistent systems score well on semantic paradoxes; classical logic scores elsewhere. We choose pragmatically.
Ari: Agreed—provided your virtues actually exclude their opposites when you deliberate. When you say, “Simplicity counts in favor,” you don’t mean, “Simplicity and anti-simplicity count equally.” Your ranking presumes that a virtue rules out the vice when they’re genuinely incompatible in a given comparison.
Method: I need only practical exclusion, local to the choice problem.
Ari: Exactly the minimal core again. Your methodology presupposes: in this evaluative act, some properties can’t co-obtain. Otherwise, the choice collapses—every theory both preferable and not.
Dee: I can still say that in other contexts, acceptance and rejection of a claim can both be rational.
Ari: You can; but when you reject a specific claim as part of your method, you don’t also accept it in the very same act. That’s the exclusion structure surfacing in practice.
Method: I’ll grant that. My stance survives: the core is local and defeasible, not metaphysical thunder.
Ari: That’s all I’ve argued for.
Scene 6: Back to Fierce
Fierce: You’re all far too cheerful. Even if you share this minimal exclusion, the moment you ask for reasons for it, the circle tightens. You can’t defend the core without using it.
Ari: Correct. First principles aren’t demonstrated from deeper ones; they’re made manifest through practice. What we’ve done is show two things:
- Possibility — Debate can proceed because the minimal core is already operative in everyone’s speech and method.
- Fruitfulness — The real action lies in scope: which expressions and properties fall under exclusion? Truth/falsity? Acceptance/rejection? Vagueness? Semantic paradoxes?
Fierce: Hm. You’ve lowered the bar from a grand defense to a diagnosis of how we’re already playing the game. That I can live with.
Scene 7: Closing moves — scope disputes
Dee: My closing line: I accept the minimal exclusion when I’m aiming to rule something out. But I still claim: in semantic paradoxes and certain border cases, some A ∧ ¬A are true. The core doesn’t forbid that, it just constrains how I state and use them.
Demo: I withdraw my promise of a proof from nothing. I now say: our best demonstrations display the core at work. They don’t found it.
Method: I keep my methodology, now explicit about its local exclusions. The debate shifts to whether the virtues favor classical, paraconsistent, or mixed systems for different domains.
Fierce: I concede debate is possible in this trimmed sense. But if any of you try to refute PNC globally, I’m back to abstention.
Ari: And my final word: call the core MSPNC if you like—whenever a theory aims to rule out something, some of its expressions must carry meanings that exclude incompatible statuses in that very act. It’s small, but it’s the hinge on which all our doors swing.
Mod: Verdict: we didn’t settle the universe, but we located the hinge—and agreed it’s there. That counts as philosophy done.
(Lights down. Chalk dust hangs like a quiet truce.)
No comments:
Post a Comment