The core structure of a Benardete paradox typically involves an infinite sequence of entities or events (This sequence often has no first member (a "beginningless set") or an infinitely dense arrangement), a condition that each entity/event satisfies if and only if no earlier entity/event in the sequence satisfies it. This creates a self-referential or dependent relationship and a resulting contradiction or impossibility. When analyzing the scenario, one is led to conclude that the event in question must occur, but also that it cannot occur, or that no individual entity can perform its designated action.
The "Grim Reaper" Paradox (a popular variant)
One of the most famous and illustrative versions is the Grim Reaper paradox:
Imagine an infinite sequence of "Grim Reapers," each assigned a unique natural number (Reaper 1, Reaper 2, Reaper 3, and so on). Each Reaper is set to kill a specific person (let's call him Fred) at a designated time.
The times are arranged such that Reaper 1 is to kill Fred at 1:00 PM. Reaper 2 is t kill Fred at 12:30 PM. Reaper 3 is to kill Fred at 12:15 PM. And so on, ad infinitum, with each subsequent Reaper assigned a time halfway between the previous Reaper's time and noon. (The sequence of times approaches, but never reaches, noon).
The crucial condition is that each Reaper will kill Fred if and only if no earlier Reaper has already killed him.
Now, consider what happens-
Must Fred be killed? Yes. If Fred isn't killed before 1:00 PM, then Reaper 1 will kill him at 1:00 PM (since no earlier Reaper would have killed him). So, someone must kill Fred between noon and 1:00 PM.
Can any specific Reaper kill Fred? Let's assume Reaper 'n' (any specific Reaper) kills Fred. If Reaper 'n' kills Fred, then according to the rule, no earlier Reaper (1, 2, ..., n-1) killed Fred.
But if no earlier Reaper killed Fred, then Reaper (n+1) would have killed him earlier than Reaper 'n', since Reaper (n+1) is set to act before Reaper 'n'. This contradicts the assumption that Reaper 'n' killed Fred first.
This line of reasoning applies to every Reaper. No matter which Reaper you pick, you can always show that if that Reaper were to act, an earlier Reaper would have already acted, or should have acted.
Therefore, we arrive at a contradiction: Fred must be killed, but no individual Reaper can be the one to kill him.
The "Gods and the Traveler" Paradox (original Benardete version)
Another version from Benardete's original work involves a man trying to walk a mile.
A god is positioned at the 1/2 mile mark, ready to throw up a wall if the man reaches that point. A second god is at the 1/4 mile mark, ready to throw up a wall if the man reaches that point. A third god is at the 1/8 mile mark, and so on, ad infinitum, with gods at every point 1/2^n mile from the start.
The condition is that each god will throw up a wall if the man reaches their designated point AND no earlier god has already thrown up a wall.
The paradox arises because if the man tries to take even the tiniest step from the starting point, he would pass the infinitely many gods positioned closer and closer to the start. If he were to pass the god at 1/2^n, that god would throw up a wall. But then, to reach that point, he would have had to pass the god at 1/2^(n+1), who would have thrown up a wall even earlier. This suggests that the man cannot even begin to move, as he would be stopped at the very first infinitesimal distance by an infinite series of walls, none of which would actually be thrown up unless he passed the previous one.
Implications and Proposed Solutions from the philosophical perspective.
Benardete paradoxes are significant because they challenge the intuition that any logically coherent scenario involving actual infinities should be metaphysically possible. They are often used to argue for-
A) Causal Finitism: The view that nothing can have infinitely many causes, or that infinite causal regresses are impossible.
B) Temporal Finitism: The idea that time must have a beginning (i.e., an infinite past is impossible).
C) The Unsatisfiable Pair Diagnosis (UPD): This view holds that Benardete paradoxes are inherently logically impossible constructions, and therefore no metaphysical thesis (like finitism) needs to be adopted to avoid them. The conditions set up simply cannot be simultaneously satisfied.
Philosophers continue to debate the proper resolution of these paradoxes, exploring their implications for metaphysics, the nature of time, and the concept of infinity.
1.https://en.wikipedia.org/wiki/Grim_Reaper_paradox?wprov=sfla1
2.https://youtu.be/dOwTikkzhq0?si=Z1hbW0VvGg-E0jmQ
Pratyush Chaudhuri
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