## Final Report Summary - PCM (THE PROBABILISTIC CONSTRUAL OF MODALITY)

The project is located at the intersection of two areas of inquiry. The first is the study of modal expressions, like "must", "might", and "can", in natural language. This is a central topic in philosophy of language and linguistics, and informs key debates in metaphysics, epistemology, metaethics, and other subfields of philosophy. The second area is the modeling of mental states and rational inference with probabilistic tools. Both in philosophy and in the social sciences, probability is widely used to represent cognitive states, learning, and reasoning. The project's aim has been to investigate their connection. Specifically, the project has investigated the following hypothesis:

Probabilistic Construal of Modality (PCM). Modal expressions in natural language, i.e. expressions such as "must", "should", "can", should be given a semantic analysis that appeals to probability.

The project has aimed at making progress both on a methodological and on a substantive front. On the methodological side, it has aimed at establishing what kind of evidence can confirm or disconfirm the PCM hypothesis. On the substantive side, it has aimed at developing and evaluating new theories of modal expressions that have better empirical coverage than existing theories, and are better integrated with existing models of belief.

The work carried out during the project has been divided into four stages. (1) Investigating what kind of evidence, if any, can confirm or disconfirm the PCM hypothesis. (2) Determine how much evidence exists in favor of the Probabilistic Construal of Modality hypothesis.(3) Determine the impact of the findings of points (1) and (2) on the choice between probabilistic models for attitudes. (4) Determine the implications of the foregoing for the classical debate literature on probabilities of conditionals.

With regard to (1): the PI has individuated five areas of inquiry where modality and probability appear closely linked, and that hence might provide evidence for the PCM hypothesis. These areas are: (i) predictive modality (in particular, the modal "will"), (ii) counterfactual modality and its relationship with causal notions, (iii) pragmatic effects (in particular, so-called free choice effects) associated to epistemic modality, (iv) the logic of epistemic discourse and conditionals, (v) the subjective probabilities assigned to conditional statements.

With regard to (2): the evidence in favor of the PCM hypothesis has been found to be scarce. The only source of evidence that directly supports the presence of probability functions in the semantics is (iii). (The PI shows that a probabilistic semantics for epistemic modals like "might" can be used to easily explain pragmatic effects that are otherwise very hard to account for.) At the same time, the research conducted shows that probability works as an important constraint on the semantics and the logic of modals. If we want to vindicate intuitive assignments of probability to modal claims (for example, the assignment of probability 1/2 to "This fair coin will land tails") our semantics for modality have to be deeply constrained. Findings of this sort have been obtained for the predictive domain (in particular, the modal "will"), the epistemic domain (in particular, the modal "might" and indicative conditionals) and the counterfactual domain (in particular, "if ... would ...} conditionals).

With regard to (3): the standard Bayesian model of probability has proven to be fully adequate for modeling all phenomena in play. Some of the findings concerning the relationship between conditionals and probability point to the idea that a move to a nonstandard probability theory might be needed for assigning probabilities to conditionals. This question falls beyond the direct purview of the project, but the PI will pursue it in further work.

With regard to (4): as for other kinds of modality, the main finding is that assignments of probability deeply constrain the semantic options available for conditionals. In particular, in the last paper i the project, the PI shows that a logical principle required for vindicating correct probability assignments to conditionals (so-called Conditional Excluded Middle) is in tension with the standard truth-conditional view of conditionals.

Overall: the project has brought to light important links between modality and probability, which are going to have substantial impact on a number of ongoing debates about modality. The results have been achieved fully in time, and the number of outcomes produced exceeds the original expectations. The only outstanding work concerns two paper, one of which is undergoing revision after an invitation to revise and resubmit, and one of which is submitted to conferences and will be submitted to journals after receiving feedback. The PI is also developing new lines of research and collaborations that build on the work carried out for the present project. At this stage, it is anticipated that these lines of research will lead to the completion of 4-5 new papers and possibly a book. This testifies the substantial positive impact that the project has had on the PI's career and on his future professional development.

The work that has already appeared is disseminated at the highest levels in the profession. Currently, six papers related to the project are forthcoming or published in leading journals or conference proceedings, a seventh paper is undergoing revisions after an invitation to revise it and resubmit it from a leading journal, and an eight paper is submitted to two conferences. The PI has given or is scheduled to give twenty-six project-related presentations in UK, European, and American universities, and at international conferences. In addition, several of the papers produced are already cited by a diverse array of international researchers.

Probabilistic Construal of Modality (PCM). Modal expressions in natural language, i.e. expressions such as "must", "should", "can", should be given a semantic analysis that appeals to probability.

The project has aimed at making progress both on a methodological and on a substantive front. On the methodological side, it has aimed at establishing what kind of evidence can confirm or disconfirm the PCM hypothesis. On the substantive side, it has aimed at developing and evaluating new theories of modal expressions that have better empirical coverage than existing theories, and are better integrated with existing models of belief.

The work carried out during the project has been divided into four stages. (1) Investigating what kind of evidence, if any, can confirm or disconfirm the PCM hypothesis. (2) Determine how much evidence exists in favor of the Probabilistic Construal of Modality hypothesis.(3) Determine the impact of the findings of points (1) and (2) on the choice between probabilistic models for attitudes. (4) Determine the implications of the foregoing for the classical debate literature on probabilities of conditionals.

With regard to (1): the PI has individuated five areas of inquiry where modality and probability appear closely linked, and that hence might provide evidence for the PCM hypothesis. These areas are: (i) predictive modality (in particular, the modal "will"), (ii) counterfactual modality and its relationship with causal notions, (iii) pragmatic effects (in particular, so-called free choice effects) associated to epistemic modality, (iv) the logic of epistemic discourse and conditionals, (v) the subjective probabilities assigned to conditional statements.

With regard to (2): the evidence in favor of the PCM hypothesis has been found to be scarce. The only source of evidence that directly supports the presence of probability functions in the semantics is (iii). (The PI shows that a probabilistic semantics for epistemic modals like "might" can be used to easily explain pragmatic effects that are otherwise very hard to account for.) At the same time, the research conducted shows that probability works as an important constraint on the semantics and the logic of modals. If we want to vindicate intuitive assignments of probability to modal claims (for example, the assignment of probability 1/2 to "This fair coin will land tails") our semantics for modality have to be deeply constrained. Findings of this sort have been obtained for the predictive domain (in particular, the modal "will"), the epistemic domain (in particular, the modal "might" and indicative conditionals) and the counterfactual domain (in particular, "if ... would ...} conditionals).

With regard to (3): the standard Bayesian model of probability has proven to be fully adequate for modeling all phenomena in play. Some of the findings concerning the relationship between conditionals and probability point to the idea that a move to a nonstandard probability theory might be needed for assigning probabilities to conditionals. This question falls beyond the direct purview of the project, but the PI will pursue it in further work.

With regard to (4): as for other kinds of modality, the main finding is that assignments of probability deeply constrain the semantic options available for conditionals. In particular, in the last paper i the project, the PI shows that a logical principle required for vindicating correct probability assignments to conditionals (so-called Conditional Excluded Middle) is in tension with the standard truth-conditional view of conditionals.

Overall: the project has brought to light important links between modality and probability, which are going to have substantial impact on a number of ongoing debates about modality. The results have been achieved fully in time, and the number of outcomes produced exceeds the original expectations. The only outstanding work concerns two paper, one of which is undergoing revision after an invitation to revise and resubmit, and one of which is submitted to conferences and will be submitted to journals after receiving feedback. The PI is also developing new lines of research and collaborations that build on the work carried out for the present project. At this stage, it is anticipated that these lines of research will lead to the completion of 4-5 new papers and possibly a book. This testifies the substantial positive impact that the project has had on the PI's career and on his future professional development.

The work that has already appeared is disseminated at the highest levels in the profession. Currently, six papers related to the project are forthcoming or published in leading journals or conference proceedings, a seventh paper is undergoing revisions after an invitation to revise it and resubmit it from a leading journal, and an eight paper is submitted to two conferences. The PI has given or is scheduled to give twenty-six project-related presentations in UK, European, and American universities, and at international conferences. In addition, several of the papers produced are already cited by a diverse array of international researchers.