Abstract
It is generally believed that, for a one-off Prisoner’s Dilemma game, it is logical to defect. However, both players cooperating is apparently a better choice than both defecting, hence the dilemma. In this paper, by resorting to Ramsey’s Test, Kripke’s possible world semantics, and Stalnaker/Lewis-style account of conditionals, I show that the first horn of the Prisoner’s Dilemma is an unsound argument. It originates from failing to differentiate between a possible world and a possible set of possible worlds and failing to observe that the set of accessible possible worlds associated with a possible world in general varies from conditional to conditional. This phenomenon can also be illustrated in terms of the recently developed hi-world semantics. Moreover, a meta-argument is constructed to establish the non-existence of a logical argument for defection.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
See Trivers (1971).
See Axelrod (1984).
We shall have more to say about this assumption of awareness later.
Note that if P0, P1 and P2 are understood as ‘a 1 knows that C2 or D2’, ‘if a 1 knows that C2 then she would defect’, and ‘if a 1 knows that D2 then she would defect’ respectively, then [Main] is no longer an instance of Disjunction Elimination.
There are some complications concerning the possible referents of v(C1) and v(D1), and we shall say more about them later.
See Chalmers and Hájek (2007) for a nice illustration of this problem.
See Tsai (2016) for more detailed discussion of this distinction. Note that the terms used there are ‘autistic’ and ‘realistic’ instead.
After all, a 1 by default has no access to the truth of C2. She can at best reason that if C2 then ‘I should defect’. However, ‘I should defect’ is different from ‘I would defect’, as the truth of the former is independent of whether a 1 knows C2 or not, while the latter is.
Some might want to insist that the former is a contingent condition on John himself still. However, if so, then we should at least allow the set of John’s friends to be a contingent set, just as John’s height is a contingent fact. Analogously, for the B1 case, to regard v(D1) > v(C1)—which is clearly a condition on a set of possible worlds rather than a condition on a possible world—as a condition on w itself, we would have to allow for the possibility that w has access to a different set of accessible worlds. This, nevertheless, cannot be done in the usual Kripkean framework. In the Kripkean scheme, once a model is given, the set of accessible worlds is fixed.
Recall that D1 stands for ‘a 1 defects’, and the truth of it certainly depends on who this a 1 is, in particular, whether this a 1 would have v(D1) > v(C1) in her stock of knowledge and whether that knowledge will prompt her to defect.
For instance, imagine that the a 1 is one of us who are pondering the three arguments in question.
To see what the extension I(p) means and to appreciate the subtle difference between I(p) and ||p||M please refer to the “Appendix”.
See the “Appendix” for the definition.
There is a further complication concerning whether we should impose the Existential Import for our conditionals, that is, whether we should impose ◇p into a Ramsey conditional and translate ‘if p then q’ into ◇p ∧ □(p ⊃ q) instead. But, I will ignore the problem here and refer interested readers to Tsai (2016) for more details.
A worth-mentioning fact is that in hi-world semantics, the most unrestrictive mode is that every hi-world takes the form (w 0 , D, P(D), P 2(D), … ). In this case, every hi-world is a sub-hi-world of each other, and then □(p ⊃ q) and □(□p ⊃□q) can be shown to entail each other. In comparison, in the Kripkean semantics, these two formulas are not logically equivalent even in S5.
The author would like to thank an anonymous reviewer for this journal for pointing out that the analysis concerning conditionals in this paper could be developed into other domains as well, in particular, its relation to constructivism could be explored. However, the treatment of this more general subject will have to await another paper.
Here, Modus Ponens and Hypothetical Syllogism (in the order of if A then B, if B then C/ ∴ if A then C) are assumed to be valid argument forms.
As the behaviors of both agents are guided by logic and shaped by public knowledge, if an agent knows A then the other knows it as well. So here we do not need to introduce a subscript to indicate who the knower is.
Note, however, in contrast, if the players happen to be twins that would always produce the same sign, then they can indeed optimize their outcome without resorting to any logical reasoning.
We should take into consideration the possible long term effect that an action can have on P(C2) and P(D2), rather than take them as held constants.
See McElreath and Boyd (2007).
Here we adopt the notion of social evolution only as a means to help explain how one would have calculated in advance, in her mind, what the prospect of her action would be, before she makes her decision. The PD game we are concerned with remains the one-off PD game.
References
Axelrod R (1984) The evolution of cooperation. Penguin, London
Becker O (1952) Untersuchungen úbef den Modalkalkúl. Westkulturverlag Anton Hain, Meisenheim/Glan
Chalmers D, Hájek A (2007) Ramsey + Moore = God. Analysis 67:170–172
McElreath R, Boyd R (2007) Mathematical models of social evolution, a guide for the perplexed. University of Chicago Press, Chicago
Nowak MA, Highfield R (2012) SuperCooperators: altruism, evolution, and why we need each other to succeed. Free Press, New York
Press WH, Dyson FJ (2012) Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent. Proc Natl Acad Sci USA 109:10409–10413
Ramsey FP (1990) General propositions and causality. In: Mellor DH (ed) Philosophical papers. Cambridge University Press, Cambridge, pp 145–163
Stewart AJ, Plotkin JB (2012) Extortion and cooperation in the Prisoner’s Dilemma. Proc Natl Acad Sci USA 109:10134–10135
Trivers RL (1971) The evolution of reciprocal altruism. Q Rev Biol 46:35–57
Tsai C-C (2012) The genesis of hi-worlds: towards a principle-based possible world semantics. Erkenntnis 76(1):101–114
Tsai C-C (2016) Becker, Ramsey, and hi-world semantics. Toward a unified account of conditionals. Croat J Philos 16(1):69–89
Acknowledgments
This work was supported in part by the National Science Council, Taiwan (Grant No. NSC 101-2410-H-715-001-) and the Ministry of Science and Technology, Taiwan (Grant No. MOST 102-2410-H-715-001-MY3).
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
The language L is defined in the usual way. A model M for L consists of a non-empty domain set D together with an interpretation generated by an interpretation function I to be defined below.
-
1.
The atomic truth sets
For each atomic formula p i , I(p i ) ⊆ D.
-
2.
The interpretation ||α|| M of an expression α with respect to M
-
1.
A hi-world s is an element of Π ∞i=0 (P *)i(D), where P is the power set operator and P * is defined by P *(A) = P(A)\{∅} where ∅ is the empty set.
-
2.
A hi-world t is a sub-hi-world of s provided that π i (t) ∈ π i+1(s) for all i ≥ 0, where π i is the projection into the ith component.
-
3.
If α is an atomic formula, then ||α||M = Π ∞i=1 U i, where U 1 = I(α) and U i = (P *)i(D) for i > 1.
-
4.
If α is a formula, then
||□α||M = {s ∈ Π ∞i=0 (P *)i(D)| t ∈ ||α||M for all sub-hi-worlds t of s}
||◇α||M = {s ∈ Π ∞i=0 (P *)i(D)| there is a sub-hi-worlds t of s such that t ∈ ||α||M}
-
5.
|If α and β are formulas, then
||¬α||M = ||α|| cM = Π ∞i = 0 (P *)i(D)\ ||α||M
||α∨β||M = ||α||M∪||β||M
||α ⊃ β||M = ||¬α∨β||M
-
1.
Rights and permissions
About this article
Cite this article
Tsai, Cc. The Prisoner’s Dilemma: From a Logical Point of View. Axiomathes 27, 417–436 (2017). https://doi.org/10.1007/s10516-016-9314-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10516-016-9314-2