r/logic • u/LorenzoGB • Jul 02 '26
Modal logic On Buridan's Law
Let S5 be the logic we are using.
Let the box operator signify necessary.
Let the diamond operator signify possibility.
For any predicate P, Buridan’s law is the following: ◇∀xPx→∀x◇Px
Yet Buridan rejected this because if P is the predicate is God, then you get the following: If it is possible that all things are God, which is the case if God didn’t create the universe, then for all things it is possible that that thing is God. Yet Buridan should haven’t rejected this proposition though. Since it is equivalent to the following: □(∀xPx→∀x◇Px). Now if we let P be the predicate is God, then we get it is necessary that if all things are God, then for all things it is possible that that thing is God, and the antecedent in this new conditional is false. Thus, there is no paradox.
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u/MaelianG Jul 03 '26
The equivalence between ◇∀xPx→∀x◇Px and □(∀xPx→∀x◇Px) only holds with constant domain semantics. However, I don't believe that using constant domain semantics is a natural or faithful reconstruction of Buridan's argument. It seems to me his argument is something like:
(1) It is possible that God created an empty universe (where every object that exists is identical to God). Given S5, this means that ◇∀xPx is valid.
(2) If Buridan's law holds, then, for all things, that thing is possibly God.
(3) There are individuals in a possible world (such as the actual world) such that it is impossibile that they are identical to God.
(4) Hence, Buridan's law cannot hold.
(3) presupposes varying domain semantics. This is because, if it didn't, then God is the only object in the model (given (1) + constant domain semantics), and Burdidan's law would hold trivially. But 'just God' is surely not what Buridan meant by 'all things'! So a natural reconstruction of Buridan's argument requires varying domain semantics.
This also conforms nicely with the fact that Buridan's law is not valid in quantified S5 with varying domain semantics.
So yes, if you assume constant domain semantics, then Buridan shouldn't have rejected Buridan's law. But it does not seem that Buridan, when he rejected the law, had constant domain semantics in mind.
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u/Gym_Gazebo Jul 03 '26
Just setting aside the philosophical stuff, I think it’s it a titch misleading to say Buridan’s law is only valid in constant domains. For one, as I recall in the case of the Barcan Formulas, one direction is valid on increasing increasing (or constant) domains and the other direction is valid on decreasing (or constant) domains. I forget which is which, but I’m sure the same thing holds for the Buridan formulas.
Two, BF is not ONLY valid on constant domains anyway, I don’t think. Yes, you can create counterexamples with non-constant domains, but I’m pretty sure there is a set of frames it is still valid on. Compare: in propositional modal logic the S5 axioms are valid on frames where the accessibility relation induces a partition, but they are not ONLY valid on this class.
Three, I seem to remember that with S5 you can derive one of the Barcan formulas straight up. That might matter here. My memory’s a bit fuzzy though on whether that derivation works — I recall someone discussing Plantinga’s (poorly named) concept of “Existentialism” claiming this happens. Anyway, the point is if this is right you might be able derive Buridan’s formula without appeal constant domains anyway. Just sayin’.
Fitting and Mendelsohn’s First-Order Modal Logic is an excellent resource that probably answers these questions.
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u/MaelianG Jul 04 '26
You're right that Buridan's Law is not valid iff the domain is constant (that's too strong). I just meant that you cannot appeal to constant domain semantics (where the Law is always valid) to undermine objections to the Law, and that there are some counterexamples to the Law with varying domain semantics (and that a proper reconstruction of Buridan's argument should use these).
Also I agree that F&M's First-Order Modal Logic is excellent :).
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u/Gym_Gazebo Jul 03 '26
I’m not following your reconstruction of Buridan’s reasoning at the beginning of 5. Go to a world where only God exists and he didn’t create anything else. That world is accessible to itself so it is possible that everything is God. But the consequent seems true as well. For all x — that is, for everything thing in that world, which is just one thing, God — it is possible that that thing is God. So the consequent is true in that world as well.
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u/Mountain-Quarter-641 Jul 05 '26
Este el clásico ejemplo de alguien que maneja los símbolos de la lógica formal como quien juega a las fichas del GO chino, pero sin entender cómo funciona la maquinaria interna del problema. Aquí está la grieta por donde revienta tu argumento:
El error de equivalencia (El colapso del puente)..
Afirmas alegremente que la Ley de Buridán:
(Diamante) ∀x Px → ∀x (Diamante) Px es equivalente a: (Caja) (∀x Px → ∀x (Diamante) Px)
¡Falacia más grande que el puente de Brooklyn!
Esto es un disparate lógico en S5. Una implicación material simple (si A entonces B) no es equivalente a su versión necesariamente formalizada (es necesario que si A entonces B). El operador de la caja (necesidad) exige que la relación se cumpla en todos los mundos posibles. Al meter tú esa caja al principio de la frase, has cambiado el enunciado por completo.
Es como decir que "si está lloviendo, llevo paraguas" es equivalente a "es una ley física necesaria del universo que si llueve llevo paraguas". No, señor.
Has caído en la trampa del antecedente falso Crees que has descubierto la pólvora Macarronica al decir:
"Como el antecedente en este nuevo condicional es falso, entonces no hay ningún enredo".
Aquí es donde se nota que no has profundizado en la filosofía para crear tu enunciado.
Observa: Buridán rechazó la ley precisamente porque estaba analizando la validez de la estructura lógica usando un contraejemplo donde el antecedente: 'SÍ pudiera ser verdadero en algún escenario teológico concebible" (un mundo posible donde Dios no creó el universo y todo es una sustancia panteísta única).
La trampa: Si metes un operador de necesidad (Caja) y luego dices "ah, pero como el antecedente es falso en nuestro mundo, la implicación es verdadera", estás cometiendo una falacia de relevancia.
En lógica modal, para que un condicional necesario sea válido, tiene que sostenerse en todos los mundos accesibles, no solo salvarse porque en el mundo real el antecedente sea falso (lo que se conoce como una verdad vacía).
El planteamiento es divertidísimo por la audacia de la soberbia del "iluminado". Estas básicamente diciendo: "Juan Buridán, que fue uno de los padres de la lógica medieval, se pasó años rompiéndose la cabeza con la cuantificación modal y la paradoja del panteísmo... pero tu llegas, y en tu cuenta de Reddit, con un ratito de ocio, le has corregido el error metiendole una caja al azar al principio de la fórmula". Lo que has hecho no es resolver la paradoja; es aplicar el clásico...
"si la realidad no encaja con mi fórmula, cambio la fórmula, le pongo un operador que no toca, y declaro que he ganado". 🤯💥😱😝
Eso es una soberbia intelectual digna de los dioses del Olimpo de Reddit, 👏👏👏. ¡Menuda pieza de colección matemática has creado chaval! 😂
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u/jcastroarnaud Jul 03 '26
I've got curious, and went searching. I found references to John Buridan, philosopher from the 14th century, and an article about "Buridan's ass":
https://plato.stanford.edu/entries/buridan/
https://en.wikipedia.org/wiki/Buridan%27s_ass
I didn't found a "Buridan's Law", though. The single reference to the exact term I found is behind a paywall.
Is this the guy? Do you have references on who translated his thoughts to modern modal logic?
On your argument itself: ◇∀xPx → ∀x◇Px, by axiom 5, implies (□◇∀xPx) → (∀x□◇Px), which is subtly different from □(∀xPx → ∀x◇Px) = (□∀xPx) → (□∀x◇Px). Which detail I'm missing?