Guidelines for Final Project

The final project accounts for \(30 \%\) of the grade for this course. It will proceed in two stages.

• You should submit a project proposal by Monday, November 11.

  The proposal should be a brief document, 2 – 5 pages, which provides some background, outlines your chosen question or problem and your own approach to it, and, finally, the expected outcome. The document should cite the most relevant literature.

• You should submit your final project by Wednesday, December 10

  Your final project can consist of a philosophical discussion of some question of issue directly connected to the material we have discussed in the course or, if you prefer, can present a formal result of interest to you. If you choose a philosophical project, your discussion should approach 10 to 15 pages, whereas a technical project should not involve more than 5 to 8 pages.

What follows is a list of sample topics that would be appropriate for a final project for the course. The list is by no means exhaustive. Please feel free to find an alternative project in the area, and I would be happy to discuss it with you. In the meantime, I hope to add more suggestions as we cover more ground.

Sample Philosophical Projects

1. What is the modal logic of metaphysical necessity? Both \(\textsf{S4}\) and \(\textsf{S5}\) have been proposed as candidates, but they remain controversial.

   • One influential argument against the validity of axiom \(4\) when \(\Box\) is interpreted in terms of metaphysical modality is discussed by Nathan Salmon in The Logic of What Might Have Been, Philosophical Review, 1989, 98, 3-34. In that paper, he argues that axiom \(B\) may be true, but neither analytic nor a truth of logic.

   • One recent response to this argument is discussed by Sarah-Jane Leslie in Essence, Plenitude, and Paradox, Philosophical Perspectives 2011, 25, 277-295.

2. A concise formulation of the modal ontological argument is as follows:

   1. Necessarily, if God exists, then God necessarily exists.

   2. God possibly exists.

   3. Therefore, God exists.

   When formalized in the language of modal propositional logic, the argument becomes valid only against the background of certain powerful normal modal systems such as \(\textsf{S5}\).

   On the other hand, there appears to be a parallel argument available for the non-existence of God, which may be paraphrased as follows:

   1. Necessarily, if God exists, then God necessarily exists.

   2. God possibly does not exist.

   3. Therefore, God does not exist.
      

   The argument has recently been called the reverse modal ontological argument for atheism. Both arguments have recently been discussed in Fritz, Lo, Schmidt, Symmetry Lost: A Modal Ontological Argument for Atheism, forthcoming in Nous. The symmetry problem is the observation that the standard reasons offered in support of the second premise of the modal ontological argument appear to support the second premise of the reverse ontological argument. Here are some important questions to consider:

   • Which modal logics validate the modal ontological argument?

   • Which modal logics validate the reverse modal ontological argument?

   • Is there a principled reason to prefer one argument over the other?
     

Sample Technical Projects

1. Find a modal logic that characterizes the class of finite transitive trees. Please note that the modal logic in question may not be canonical in which case the justification of completeness may require a different method.

2. A pure modality is a finite sequence of zero or more of occurrences of \(\Box\) and \(\Diamond\), e.g., \(-\), \(\Box\), \(\Diamond\), \(\Box \Diamond\), \(\Diamond \Diamond\), \(\Diamond \Box\), \(\Box \Diamond \Box\), etc. In Modal Logic for Open Minds, pp. 58-59, Johan van Benthem reports what H. B. Smith, a pioneer of modal logic, regarded as two plausible hypotheses in the 1930s:

   • Distinction. No two distinct pure modalities are provably equivalent.

   • Comparison. Given two pure modalities \(\textsf{O}_1\) and \(\textsf{O}_2\), one entails the other.

   Prove that no normal modal logic can accommodate both hypotheses. That is, if (\Sigma) is a normal modal logic, then either two distinct pure modalities \(\textsf{O}_1\) and \(\textsf{O}_2\) are equivalent, namely,

   \[\Sigma \vdash \textsf{O}_1\varphi \leftrightarrow \textsf{O}_2 \varphi\]

   or there are incomparable pure modalities \(\textsf{O}_1\) and \(\textsf{O}_2\) such that

\[\begin{array}{lll} \Sigma \nvdash \textsf{O}_1 \varphi \to \textsf{O}_2 \varphi & \text{and} & \Sigma \nvdash \textsf{O}_2 \varphi \to \textsf{O}_1 \varphi. \end{array}\]