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A Dutch book is a situation that allows a smart gambler to place bets in a way that guarantee a profit for him. It can be shown that if a bookmaker follows the rules of Bayesian calculus in the construction of his odds, he can avoid Dutch books.

Is it possible to justify logic in a similar way, i.e. by considering penalties that result for an agent (like a bookmaker) who does not follow the rules of logic? For example, by assigning true for "A" and "A & B", but false to "B".

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    "the dutch book" ???
    – Mauro ALLEGRANZA
    Feb 13, 2017 at 15:58
  • See Principle of explosion.
    – Mauro ALLEGRANZA
    Feb 13, 2017 at 15:59
  • @MauroALLEGRANZA i meant Dutch Book Arguments
    – Koncopd
    Feb 13, 2017 at 16:58
  • @MauroALLEGRANZA i don't see the principle of explosion to be relevant here, because our hypothetical agent should obey the axioms of logic in assigning truth\false values to propositions to get any harm from violating the law of noncontradiction. If it doesn't obey the laws of logic (for example, assigns truth values at random), it has no problems with it.
    – Koncopd
    Feb 13, 2017 at 17:14
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    It is hard to understand what the post means, it reads like a middle of a sentence without beginning or end. "True" and "false" are not probabilities, so what does it have to do with assignment of probabilities, and how is any of it an "argument" for anything?
    – Conifold
    Feb 13, 2017 at 20:10

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In inductive logic/probability theory a set of betting rates is called coherent if it's not open to a sure-loss contract (called a Dutch Book). Interestingly you can prove that:

(1) A set of betting rates is coherent iff it satisfies the rules of probability.

If I understand your question correctly, you are asking if it's possible to prove a similar sort of result for rules of logic: So give somehow an analogous definition of coherence for a set of assignments of truth values to some set of sentences, and then prove that such an assignment is coherent iff it satisfies (in some sense) the rules of logic. "Rules of logic" here should refer to an axiomatization of logic, since in the case of probability it's similarly the axioms that betting rates have to satisfy to avoid a Dutch book. The obvious idea would be to define coherence as follows:

Df. A set of truth assignments to some sentences is coherent iff there is a truth valuation that gives the same truth-values to the sentences as your assignments do.

(The above definition means that coherence guarantees that your truth assignments are not open to a sure error, i.e. you could be right in your assignments of truth values.)

Define a set S of sentences as follow: if your assignment assigns 1 to a formula F, then F∈S, if your assignment assigns 0 to a formula F then not-F∈S. Let's call this set S the set determined by the truth assignments.

It's now easy to show that a set of truth assignments to some sentences is coherent iff the set S determined by those truth assignments is satisfiable. Now since the soundness and completeness theorems tell us that a set S is satisfiable iff S is consistent (relative to the rules/axiomatization of logic), the analogue of theorem (1) is as follows:

(1)* A set of truth assignments is coherent iff the set determined by the truth assignments is consistent (relative to an axiomatization of logic)

You can think of this as a version of the soundness and completeness theorems.

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  • Thanks for expressing my question in a more clear way. However i see a problem in your formulation of the coherence notion for a set of assignments. In dutch book arguments betting rates and losses due to a dutch book aren't directly related to the probability axioms, this argument doesn't seem circular. However it seems that your very definition of coherence is really "A set of truth assignments to some sentences is coherent iff it obeys the axioms of logic", and nothing else, if i understand it correctly. It appeals to the axioms to "defend" these same axioms.
    – Koncopd
    Feb 14, 2017 at 15:29
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    No since my definition of coherence is semantic (there is no mention of axioms in the definition) and the axioms of logic are syntactic, it's a discovery that semantics and syntax agree like this, completeness has to be proved it can't just be assumed, sometimes it can't be proved.
    – Johannes
    Feb 14, 2017 at 15:44
  • I still fail to see the point. Maybe it is a problem with interpretation or lack of knowledge of the subject on my side. You give normative power to the existance of a suitable valuation, which by definition obeys the axioms, doesn't it? And sure-error only means that there is no appropriate valuation, isn't it?
    – Koncopd
    Feb 14, 2017 at 19:11
  • Seems i need to digest the distinction between syntax and semantics.
    – Koncopd
    Feb 15, 2017 at 9:12
  • I actually think I understand what you are asking: basically you are asking a way to justify logic. You assume that the "Dutch book argument" justifies probability axioms in some sense, and seek a similar justification for logic (and think that the argument I gave doesn't give that). There is, I think, a confusion embedded in that way of thinking and it could take some effort to spell it out clearly. I might think about the issue a little later.
    – Johannes
    Feb 15, 2017 at 12:01

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