Are the propositions "Everything happens for a reason", and "Nothing happens without a reason" logically equivalent?
We may introduce two predicates, thinghood, τ(x), and happening for a reason, ρ(x). Then we can translate the statements into the standard first-order language as follows:
‘Everything happens for a reason’
∀x(τ(x) → ρ(x)) ↔ ∀x(¬τ(x) ∨ ρ(x))
‘Nothing happens without a reason’
¬∃x(τ(x) ∧ ¬ρ(x)) ↔ ∀x(¬τ(x) ∨ ρ(x))
We see that they are logically equivalent. However, the translation hinges on the idea of thinghood, and the related issue of quantifying over absolute generality is a matter of metaphysical dispute.
1: Everything happens for a reason 2:nothing happens without a reason.
In 1, maybe Reason is there but nothing happens but in 2 there has to be a reason for things to happen.
I definitely see a difference between the two. So I translated the propositions into French and,again, perceived a difference between the two: 1 (il y a une raison pour tout) has a connotation of explanation. No randomness but 2 (rien ne se passe sans une bonne raison) indicates that there has to be a reason for things to happen.