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Put simply, I want to know whether the principle of bivalence implies the law of excluded middle and the law of non-contradiction, and conversely, whether the law of excluded middle together with the law of non-contradiction implies the principle of bivalence.

The tricky thing for me here is that the laws of excluded middle and non-contradiction state that $P\lor\neg P$ and $\neg(P\land\neg P)$ are theorems respectively, which is a purely syntactic statement. On the contrary, the principle of bivalence states that every proposition is either true or false, but not both, which is a semantic statement. Despite this disparity, I have seen examples of logics which satisfy one of the two laws, but fail to be bivalent (e.g. intuitionist logic and some three-valued logics).

Thus, I am wondering whether a logic which violates one or both of the law of excluded middle and the law of non-contradiction can still have a bivalent semantics, and conversely, whether a logic which satisfies both of the laws must have a bivalent semantics.

As a crude attempt at a counterexample, could I form a logic whose syntax is the same as classical logic, but whose semantics completely ignores the syntactic element of the logic by declaring all propositions to always be both tautologically true as well as false? This potential counterexample begs the question of what actually constitutes a "logic" since I would at least expect that the semantics and syntax of a logic should be related in some way.

Anonymous
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  • How does your counter example satisfy the "not both" part of the bivalence principal? Also, I have an example for you, where the bivalence principal is satisfied, but not the law of excluded middle and non-contradiction: Use the same syntax as propositional logic, but interpret truth as false and false as true. Then, $P\lor \lnot P$ is true in PL, which we interpret as false in L', and $\lnot(P\land \lnot P)$ is also true in PL, but false in L'. However, all statements are either true or false, and not both. – user400188 May 08 '20 at 07:18
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    @user400188 The example I stated was meant to be a potential counterexample to the conjecture that LEM+LNC implies PB since, in my example, bivalence fails and LEM and LNC still hold (since the syntax is the same as classical propositional logic). Your example does satisfy bivalence, but it also satisfies LEM and LNC. LEM does not say "$P\lor\neg P$ is true" but rather "$P\lor\neg P$ is a theorem" (i.e. $\vdash P\lor\neg P$), although that is equivalent if the logic is sound and complete, which your example is not. – Anonymous May 08 '20 at 17:03
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    Thanks for the correction. I now understand your example better too. I am no longer confident enough to give a formal answer, however I do think I have a correct example where bivalence holds, but not LEM and LNC. Take PL, but interpret the syntax of $\land$ as meaning OR, and $\lor$ as meaning AND. Then $P\lor \lnot P$ will no longer be a theorem, and neither will $\lnot(P\land\lnot P)$. This works under the definitions you gave in the comments, however if LEM is the meta statement "P OR not P is a theorem", regardless of how OR is expressed, then it will not. – user400188 May 09 '20 at 03:40
  • I see! That's a very clever example. Thank you! I've thought of a couple more interesting examples as well which I might turn into an answer for this question. – Anonymous May 09 '20 at 17:15

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