Realize that all statements made in a logical proof will end up becoming a logically-equivalent statement (or a statement that is always assumed to be true no matter what condition it ends up being at the end).
An example of this approach is logical behaviourism, which maintains that statements about mental events and states are logically equivalent to statements which, while typically much more complicated, are wholly about observable behaviour in varying kinds of circumstances.
Universal quantifications ∀xFx are logically equivalent to negative statements of the form ∼∃x∼Fx. Since the latter are negative, they're true, if they are, only by default.
This is logically equivalent to the statement, that if there is no such path from j to i, then A ˜ ij = 0.
As discussed above (see History of analytic philosophy: Bertrand Russell, Russelll held that definite descriptions are not genuinely referring expressions, as are logically proper names, and that sentences containing them are logically equivalent to complex general statements containing existential and universal quantifiers.
Teller's method exploits the fact that when abnormic laws are rewritten in certain logically equivalent ways, the resulting statements must then also count as abnormic laws.
On this view, the sentence "Venus is the morning star" is logically equivalent to the complex statement "(i) There is a morning star, (ii) there is at most one morning star, and (iii) if anything is a morning star, then it is Venus".
Then because the former statements are logically equivalent to them, the optimalists surmise that universal quantifications are true by default too (Mellor 2003: 214; Simons 2008: 14 5).
Fundamental or "original" nomological statements are those logically equivalent to a true sentence (with no terms that "essentially refer" to particulars) in prenex form with at least one universal quantifier and such that no logically stronger sentence in the same vocabulary is true.
But to be prosperous, one doesn't have to live long because a conditional statement is not logically equivalent to its converse.
