### the necessity of the past redux.

Let PAST be the conjunction of every true proposition about the past. The

(NP) PAST.

Since the necessity operator ‘’ is equivalent to ‘~◊~’ (i.e., it is not possible that not), the truth of NP is equivalent to

(NP’) ~◊~PAST.

If NP’ is true, it follows that there are no contingent truths about the past, since for P to be contingent, P must be both

(P) Atra was on the mat,

P could have been false. In other symbols:

(PC) P ∧ ◊~P.

If (PC) is true, then (NP’) is false, for (~◊~P ∧ ◊~P) is a formal contradiction. Ergo, etc.

*necessity of the past*thesis says that the past is necessary. In other symbols:(NP) PAST.

Since the necessity operator ‘’ is equivalent to ‘~◊~’ (i.e., it is not possible that not), the truth of NP is equivalent to

(NP’) ~◊~PAST.

If NP’ is true, it follows that there are no contingent truths about the past, since for P to be contingent, P must be both

*not necessary*and*not impossible*. But surely there is at least one contingent truth about the past. For example, though it’s true that(P) Atra was on the mat,

P could have been false. In other symbols:

(PC) P ∧ ◊~P.

If (PC) is true, then (NP’) is false, for (~◊~P ∧ ◊~P) is a formal contradiction. Ergo, etc.

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