## Abstract

Augment the propositional language with two modal operators: □ and ■. Define \(\blacklozenge \) to be the dual of ■, i.e. \(\blacklozenge \equiv \neg \blacksquare \neg \). Whenever (X) is of the form *φ* → *ψ*, let (X\(^{\blacklozenge } \) ) be \(\varphi \rightarrow \blacklozenge \psi \). (X\(^{\blacklozenge } \) ) can be thought of as the modally qualified counterpart of (X)—for instance, under the metaphysical interpretation of \(\blacklozenge \), where (X) says *φ* implies *ψ*, (X\(^{\blacklozenge } \) ) says *φ* implies *possibly**ψ*. This paper shows that for various interesting instances of (X), fairly weak assumptions suffice for (X\(^{\blacklozenge } \)) to imply (X)—so, the modally qualified principle is as strong as its unqualified counterpart. These results have surprising and interesting implications for issues spanning many areas of philosophy.

This is a preview of subscription content, log in to check access.

## References

- 1.
Chandler, H. (1976). Plantinga and the contingently possible.

*Analysis*,*36*(2), 106–109. - 2.
Chellas, B. (1980).

*Modal logic: an introduction*. Cambridge: Cambridge University Press. - 3.
Church, A. (2009). Referee reports on Fitch’s “A definition of value”. In J. Salerno (Ed.)

*New essays on the Knowability Paradox*(pp. 13–20). Oxford: Oxford University Press. - 4.
Fitch, F. (1963). A logical analysis of some value concepts.

*The Journal of Symbolic Logic*,*28*, 135–142. - 5.
Jenkins, C.S. (2009). The mystery of the disappearing diamond. In J. Salerno (Ed.)

*, New essays on the Knowability Paradox*(pp. 302–319). Oxford: Oxford University Press. - 6.
Kvanvig, J. (2006).

*The Knowability Paradox*. Oxford: Oxford University Press. - 7.
Salmon, N. (1989). The logic of what might have been.

*The Philosophical Review*,*98*(1), 3–34. - 8.
Smith, M. (2007). Ceteris paribus conditionals and comparative normalcy.

*Journal of Philosophical Logic*,*36*, 97–121. - 9.
Williamson, T. (1993). Verificationism and non-distributive knowledge.

*Australasian Journal of Philosophy*,*71*(1), 78–86.

## Acknowledgments

I owe thanks to Jeff Russell, Gabriel Uzquiano, Tim Williamson, and an anonymous reviewer for their helpful comments.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix

### Appendix

Theorem 6 shows that given *K**T*_{□}⊕*K*_{■}, (4\(^{\blacklozenge }_{\Box }\)) and (5\(^{\blacklozenge }_{\Box }\)) together give rise to an *S*5 logic for □. We will show that though jointly sufficient, (4\(^{\blacklozenge }_{\Box }\)) and (5\(^{\blacklozenge }_{\Box }\)) are not individually sufficient to give rise to an *S*5 logic for □.

First, given *K**T*_{□}⊕*K*_{■}, (4\(^{\blacklozenge }_{\Box }\)) isn’t strong enough by itself to generate an *S*5 logic for □:

###
**Theorem 20**

*The smallest normal extension of**K**T*_{□}⊕*K*_{■}*containing*\({(4^{\blacklozenge }_{\Box })}\)*does**not extend**S*5_{□}⊕*K*_{■}*.*

### Proof

As before, nothing depends on □ and ■ being distinct operators. So, it suffices to show that the smallest normal monomodal extension of *S*4_{□} containing □*p* →♢□□*p* does not extend *S*5_{□}. And that is obvious, since □*p* →♢□□*p* is already a theorem of *S*4_{□}. □

Now, we show, by a semantic argument, that (5\(^{\blacklozenge }_{\Box }\)) also isn’t strong enough by itself to generate an *S*5 logic for □. First, some preliminaries: A *Kripke frame* is a structure \(\mathfrak {F}=<W, R_{\Box }, R_{\blacksquare }>\), where the domain *W* is a non-empty set, whose elements we shall refer to as ‘worlds’, and *R*_{□}⊆ (*W* × *W*) and *R*_{■}⊆ (*W* × *W*) are binary relations on *W*. A *Kripke model*\(\mathfrak {M}=<\mathfrak {F}, V>\) is a frame with a valuation function *V* which maps each propositional letter to a set of worlds. If \(\mathfrak {M}=<\mathfrak {F}, V>\), we say that \(\mathfrak {M}\) is based on \(\mathfrak {F}\). A *pointed Kripke model*\(<\mathfrak {M},w>\) is a model \(\mathfrak {M}\) together with a world *w* in the domain of \(\mathfrak {M}\) (by the domain of \(\mathfrak {M}\), we mean the domain of the frame on which \(\mathfrak {M}\) is based). Satisfaction in a pointed model is defined:

\(\mathfrak {M}\Vdash \varphi \) iff \(\mathfrak {M},w\Vdash \varphi \) for all *w* in the domain of \(\mathfrak {M}\). And \(\mathfrak {F}\Vdash \varphi \) iff \(\mathfrak {M}\Vdash \varphi \) for every model \(\mathfrak {M}\) based on \(\mathfrak {F}\).

It is easy to show that any frame satisfying the condition in the antecedent of the lemma below validates (5\(^{\blacklozenge }_{\Box }\)):

###
**Lemma 21**

*If*
\(\mathfrak {F}\vDash \forall wv (wR_{\Box } v \rightarrow \exists u (wR_{\blacksquare } u\wedge \forall t(uR_{\Box } t\rightarrow tR_{\Box } v)))\)
*,*
*then*
\(\mathfrak {F}\Vdash \Diamond p\rightarrow \blacklozenge \Box \Diamond p\)
*.*

Thus:

###
**Theorem 22**

*Let**L**be the smallest normal extension of**K**T*_{□}⊕*K*_{■}*containing*\({(5^{\blacklozenge }_{\Box })}\)*.**Then,**L**doesn’t extend**S*4_{□}⊕*K*_{■}*(and thus also doesn’t extend**S*5_{□}⊕*K*_{■}*).*

###
*Proof Sketch*

Consider the model \(\mathfrak {M}\) below, where the arrows represent *R*_{□} (let *R*_{■} be the universal accessibility relation):

*R*_{□} is reflexive, so \(\mathfrak {M}\) is a *K**T*_{□}⊕*K*_{■}-model. Furthermore, checking that the condition in the antecedent of the previous lemma is satisfied is a tedious but straightforward exercise. Thus, the model is an *L*-model. However, \(\mathfrak {M},w\Vdash \Box p\wedge \neg \Box \Box p\), so \(\mathfrak {M}\) is a countermodel to (4_{□}). Thus, *L* doesn’t extend *S*4_{□}⊕*K*_{■} and thus also doesn’t extend *S*5_{□}⊕*K*_{■}. □

## Rights and permissions

## About this article

### Cite this article

San, W.K. Disappearing Diamonds: Fitch-Like Results in Bimodal Logic.
*J Philos Logic* **48, **1003–1016 (2019). https://doi.org/10.1007/s10992-019-09504-0

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Fitch’s Paradox
- Paradox of Knowability
- Modal logic
- Bimodal logic