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Minds and Machines
Published in: Minds and Machines 4/2017

24-11-2017 | Commentary

Commentary on Bringsjord on P = NP

Published in: Minds and Machines | Issue 4/2017

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The authors use the phrase “solves in a polynomial number of steps” in a polysemous fashion. Specifically, the term “solves” is used in at least two different capacities—with two different types, so to speak. In the first case, the left domain of the solves  predicate is physical phenomena, and in the second case it is Turing machines. The authors need to distinguish these senses, say, by indexing them as \({\textit{solves}}_1\) and \({\textit{solves}}_2\). Doing so would help to bring out the main flaw of their argument, which is this: What it means for a macroscopic physical process to solve a problem in a polynomial number of steps is very different from what it means for a digital computer—such as a Turing machine—to solve a problem in a polynomial number of steps. …
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Footnotes
1
Perhaps all our intuitions are vague and subjective, but more so for modal notions. If determining truth in this world is elusive, how much more elusive must it be to ponder it in “possible” or “possibly possible” worlds? Who is to pronounce what is logically possible or necessary? Kripke has famously argued that it is logically necessary that every person has the parents that he or she has, so that in every possible world you and I have the exact same parents that we have in this world, because that is part of our essence. But different people have radically different intuitions on such matters; how can we adjudicate?
 
2
A number of widely divergent analyses have been offered, but none manages to combine clarity with prima facie plausibility.
 
3
For instance, see Hughes and Cresswell (1996, p. 243).
 
4
We can ignore accessibility relations, since they are not necessary for the semantics of S5.
 
Literature
Fales, E. (1990). Causation and universals. London: Routledge.
Hughes, G. E., & Cresswell, M. J. (1996). A new introduction to modal Logic. London, England: Routledge.CrossRefMATH
Kripke, S. (1963). Semantical considerations on modal logic. Acta Philosophica Fennica, 16, 83–94.MathSciNetMATH
Kripke, S. (1972). Naming and necessity. In D. Davidson & G. Harman (Eds.), Semantics of natural language (pp. 253–355). Dordrecht: Reidel.CrossRef
Shoemaker, S. (1998). Causal and metaphysical necessity. Pacific Philosophical Quarterly, 79, 59–77.CrossRef
Swoyer, C. (1982). The nature of natural laws. Australasian Journal of Philosophy, 60, 203–223.CrossRef
Metadata
Title
Commentary on Bringsjord on P = NP
Publication date
24-11-2017
Publisher
Springer Netherlands
Published in
Minds and Machines / Issue 4/2017
Print ISSN: 0924-6495
Electronic ISSN: 1572-8641
DOI
https://doi.org/10.1007/s11023-017-9452-3

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