@article{crupi21,
	year = {2021},
	publisher = {Springer Verlag},
	pages = {1--47},
	doi = {10.1007/s11225-021-09949-y},
	number = {1},
	title = {Three Ways of Being Non-Material},
	journal = {Studia Logica},
	author = {Vincenzo Crupi and Andrea Iacona}
}

@article{milne03,
    author = {Milne, Peter},
    title = "{The simplest Lewis-style triviality proof yet?}",
    journal = {Analysis},
    volume = {63},
    number = {4},
    pages = {300-303},
    year = {2003},
    month = {10},
    issn = {0003-2638},
    doi = {10.1093/analys/63.4.300},
    url = {https://doi.org/10.1093/analys/63.4.300},
    eprint = {https://academic.oup.com/analysis/article-pdf/63/4/300/255098/63-4-300.pdf},
}




@article{ciardelli,
author = {Ciardelli, Ivano and Ommundsen, Adrian},
title = {Probabilities of conditionals: updating Adams},
journal = {Noûs},
    year = {2022},
    month = {11},
volume = {n/a},
number = {n/a},
pages = {},
doi = {https://doi.org/10.1111/nous.12437},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/nous.12437},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/nous.12437},
abstract = {Abstract The problem of probabilities of conditionals is one of the long-standing puzzles in philosophy of language. We defend and update Adams' solution to the puzzle: the probability of an epistemic conditional is not the probability of a proposition, but a probability under a supposition. Close inspection of how a triviality result unfolds in a concrete scenario does not provide counterexamples to the view that probabilities of conditionals are conditional probabilities: instead, it supports the conclusion that probabilities of conditionals violate standard probability theory. This does not call into question probability theory per se; rather, it calls for a more careful understanding of its role: probability theory is a theory of probabilities of propositions; but as conditionals do not express propositions, their probabilities are not subject to the standard laws. We argue that both conditional probabilities and probabilities of conditionals are best understood in terms of the dynamics of supposing, modeled as a restriction operation on a probability space. This version of the suppositionalist view allows us to connect Adams' Thesis to the widely held restrictor view of the semantics of conditionals. We address two common objections to Adams' view: that the relevant probabilities are ‘probabilities only in name’, and that giving up conditional propositions puts us at a disadvantage when it comes to interpreting compounds. Finally, we argue that some putative counterexamples to Adams' Thesis can be diagnosed as fallacies of probabilistic reasoning: they arise from applying to conditionals laws of standard probability theory which are invalid for them.}
}


@article{lassiter20,
	number = {9-10},
	author = {Daniel Lassiter},
	year = {2020},
	journal = {Inquiry: An Interdisciplinary Journal of Philosophy},
	pages = {1087--1114},
	publisher = {Routledge},
	title = {What We Can Learn From How Trivalent Conditionals Avoid Triviality},
	doi = {10.1080/0020174x.2019.1698457},
	volume = {63}
}

@article{ibeling19,
author = {Ibeling, Duligur and Icard, Thomas},
year = {2019},
month = {07},
pages = {},
title = {On Open-Universe Causal Reasoning}
}

@article{gilio05,
 ISSN = {00397857, 15730964},
 URL = {http://www.jstor.org/stable/20118621},
 abstract = {We study a probabilistic logic based on the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence). We examine probabilistic conditional knowledge bases associated with imprecise probability assessments defined on arbitrary families of conditional events. We introduce a notion of conditional interpretation defined directly in terms of precise probability assessments. We also examine a property of strong satisfiability which is related to the notion of toleration well known in default reasoning. In our framework we give more general definitions of the notions of probabilistic consistency and probabilistic entailment of Adams. We also recall a notion of strict p-consistency and some related results. Moreover, we give new proofs of some results obtained in probabilistic default reasoning. Finally, we examine the relationships between conditional probability rankings and the notions of g-coherence and g-coherent entailment.},
 author = {Angelo Gilio},
 journal = {Synthese},
 number = {1/2},
 pages = {139--152},
 publisher = {Springer},
 title = {Probabilistic Logic under Coherence, Conditional Interpretations, and Default Reasoning},
 urldate = {2023-02-16},
 volume = {146},
 year = {2005}
}

@article{fusco-forthcoming,
	title = {Dutch-Booking Indicative Conditionals},
	doi = {10.1111/phpr.12910},
	journal = {Philosophy and Phenomenological Research},
	year = {forthcoming},
	author = {Melissa Fusco}
}

@article{frisch94,
title = {Anytime deduction for probabilistic logic},
journal = {Artificial Intelligence},
volume = {69},
number = {1},
pages = {93-122},
year = {1994},
issn = {0004-3702},
doi = {https://doi.org/10.1016/0004-3702(94)90079-5},
url = {https://www.sciencedirect.com/science/article/pii/0004370294900795},
author = {Alan M. Frisch and Peter Haddawy},
abstract = {This paper proposes and investigates an approach to deduction in probabilistic logic, using as its medium a language that generalizes the propositional version of Nilsson's probabilistic logic by incorporating conditional probabilities. Unlike many other approaches to deduction in probabilistic logic, this approach is based on inference rules and therefore can produce proofs to explain how conclusions are drawn. We show how these rules can be incorporated into an anytime deduction procedure that proceeds by computing increasingly narrow probability intervals that contain the tightest entailed probability interval. Since the procedure can be stopped at any time to yield partial information concerning the probability range of any entailed sentence, one can make a tradeoff between precision and computation time. The deduction method presented here contrasts with other methods whose ability to perform logical reasoning is either limited or requires finding all truth assignments consistent with the given sentences.}
}

@article{bochmanlifschitz15, 
    title={Pearl’s Causality in a Logical Setting}, 
    volume={29}, url={https://ojs.aaai.org/index.php/AAAI/article/view/9411}, 
    DOI={10.1609/aaai.v29i1.9411}, 
    abstractNote={ &lt;p&gt; We provide a logical representation of Pearl’s structural causal models in the causal calculus of McCain and Turner (1997) and its first-order generalization by Lifschitz. It will be shown that, under this representation, the nonmonotonic semantics of the causal calculus describes precisely the solutions of the structural equations (the causal worlds of the causal model), while the causal logic from Bochman (2004) is adequate for describing the behavior of causal models under interventions (forming submodels). &lt;/p&gt; }, 
    number={1}, 
    journal={Proceedings of the AAAI Conference on Artificial Intelligence}, 
    author={Bochman, Alexander and Lifschitz, Vladimir}, 
    year={2015}, 
    month={Feb.} }

@article{gilio12,
title = {Generalizing inference rules in a coherence-based probabilistic default reasoning},
journal = {International Journal of Approximate Reasoning},
volume = {53},
number = {3},
pages = {413-434},
year = {2012},
note = {Precisely Imprecise: A Collection of Papers Dedicated to Henry E. Kyburg,Jr},
issn = {0888-613X},
doi = {https://doi.org/10.1016/j.ijar.2011.08.004},
url = {https://www.sciencedirect.com/science/article/pii/S0888613X11001198},
author = {Angelo Gilio},
keywords = {Coherence principle, Probabilistic default reasoning, Generalization of inference rules, Lower and upper bounds, t-Norms, t-Conorms},
abstract = {In this paper we first recall some notions and results on the coherence-based probabilistic treatment of uncertainty. Then, we deepen some probabilistic aspects in nonmonotonic reasoning, by generalizing OR, CM, and Cut rules. We also illustrate the degradation of these inference rules when the number of premises increases. Finally, we show that the lower bounds obtained when applying OR and Quasi-Conjunction inference rules coincide, respectively, with Hamacher and Lukasiewicz t-norms; the upper bounds in both rules coincide with Hamacher t-conorm.}
}

@article{fitelson15,
	journal = {Thought: A Journal of Philosophy},
	title = {The Strongest Possible Lewisian Triviality Result},
	number = {2},
	volume = {4},
	doi = {10.1002/tht3.159},
	pages = {69--74},
	author = {Branden Fitelson},
	year = {2015}
}

@article{fagin90,
title = {A logic for reasoning about probabilities},
journal = {Information and Computation},
volume = {87},
number = {1},
pages = {78-128},
year = {1990},
note = {Special Issue: Selections from 1988 IEEE Symposium on Logic in Computer Science},
issn = {0890-5401},
doi = {https://doi.org/10.1016/0890-5401(90)90060-U},
url = {https://www.sciencedirect.com/science/article/pii/089054019090060U},
author = {Ronald Fagin and Joseph Y. Halpern and Nimrod Megiddo},
abstract = {We consider a language for reasoning about probability which allows us to make statements such as “the probability of E1 is less than 13” and “the probability of E1 is at least twice the probability of E2,” where E1 and E2 are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson's probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satisfiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields.}
}

@article{boricic15,
    author = {Boričić, Marija},
    title = "{Suppes-style sequent calculus for probability logic}",
    journal = {Journal of Logic and Computation},
    volume = {27},
    number = {4},
    pages = {1157-1168},
    year = {2015},
    month = {10},
    abstract = "{In order to treat the deduction relation ⊢ in the context of probabilistic reasoning, we introduce a system LKprob(ε) making it possible to work with expressions of the form Γ⊢nΔ, a generalization of Gentzen's sequents Γ⊢Δ of classical propositional logic LK, with the intended meaning that ‘the probability of the sequent Γ⊢Δ is greater than or equal to 1−nε’, for a given small real ε\\&gt;0 and any natural number n. The system LKprob(ε) can be considered a program inferring a conclusion of the form Γ⊢nA from a finite set of hypotheses of the same form Γi⊢niAi(1≤i≤n). We prove that our system is sound and complete with respect to the Carnap–Popper–type probability models.}",
    issn = {0955-792X},
    doi = {10.1093/logcom/exv068},
    url = {https://doi.org/10.1093/logcom/exv068},
    eprint = {https://academic.oup.com/logcom/article-pdf/27/4/1157/17659760/exv068.pdf},
}

@book{boole16,
	year = {1916},
	publisher = {Open Court},
	title = {George Boole's Collected Logical Works},
	author = {George Boole}
}

@article{hailperin65,
 ISSN = {00029890, 19300972},
 URL = {http://www.jstor.org/stable/2313491},
 author = {Theodore Hailperin},
 journal = {The American Mathematical Monthly},
 number = {4},
 pages = {343--359},
 publisher = {Mathematical Association of America},
 title = {Best Possible Inequalities for the Probability of a Logical Function of Events},
 urldate = {2022-11-21},
 volume = {72},
 year = {1965}
}

@article{hailperin84,
	author = {Theodore Hailperin},
	title = {{Probability logic.}},
	volume = {25},
	journal = {Notre Dame Journal of Formal Logic},
	number = {3},
	publisher = {Duke University Press},
	pages = {198 -- 212},
	year = {1984},
	doi = {10.1305/ndjfl/1093870625},
	URL = {https://doi.org/10.1305/ndjfl/1093870625}
}

@book{hailperin96,
	author = {Theodore Hailperin},
	title = {{Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications}},
	publisher = {Lehigh University Press},
	year = {1996},
}

@inproceedings{nguyen02,
author = {Nguyen, Hung and Mukaidono, Masao and Kreinovich, Vladik},
  booktitle={2002 IEEE World Congress on Computational Intelligence. 2002 IEEE International Conference on Fuzzy Systems. FUZZ-IEEE'02. Proceedings (Cat. No.02CH37291)},
year = {2002},
month = {02},
pages = {530 - 535},
title = {Probability of implication, logical version of Bayes theorem, and fuzzy logic operations},
volume = {1},
isbn = {0-7803-7280-8},
doi = {10.1109/FUZZ.2002.1005046}
}

@article{wagner04,
	ISSN = {00070882, 14643537},
	URL = {http://www.jstor.org/stable/3541627},
	author = {Carl G. Wagner},
	journal = {The British Journal for the Philosophy of Science},
	number = {4},
	pages = {747--753},
	publisher = {[Oxford University Press, The British Society for the Philosophy of Science]},
	title = {Modus Tollens Probabilized},
	urldate = {2022-11-20},
	volume = {55},
	year = {2004}
}

@article{adams65,
	year = {1965},
	journal = {Inquiry: An Interdisciplinary Journal of Philosophy},
	volume = {8},
	author = {Ernest Adams},
	pages = {166--197},
	publisher = {Dordrecht: D. Reidel},
	doi = {10.1080/00201746508601430},
	number = {1-4},
	title = {The Logic of Conditionals}
}

@book{pearl88,
	title={Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference},
	author={Pearl, J.},
	isbn={9781558604797},
	lccn={88013069},
	series={Morgan Kaufmann series in representation and reasoning},
	url={https://books.google.com/books?id=AvNID7LyMusC},
	year={1988},
	publisher={Elsevier Science}
}


@article{pfeiferkleitner06,
author = {Pfeifer, Niki and Kleiter, Gernot},
year = {2006},
month = {01},
pages = {},
title = {Inference in conditional probability logic},
volume = {4},
journal = {Kybernetika}
}
    
@article{duboisprade90,
	title = {The logical view of conditioning and its application to possibility and evidence theories},
	journal = {International Journal of Approximate Reasoning},
	volume = {4},
	number = {1},
	pages = {23-46},
	year = {1990},
	issn = {0888-613X},
	doi = {https://doi.org/10.1016/0888-613X(90)90007-O},
	url = {https://www.sciencedirect.com/science/article/pii/0888613X9090007O},
	author = {Didier Dubois and Henri Prade},
	keywords = {material implication, production rule, conditioning, uncertainty, nonmonotonicity, probability, possibility, belief function},
	abstract = {The concept of conditioning is well known in probability theory, where it is used in artificial intelligence to quantify the uncertainty of rules, but is totally absent from logic, where material implication is widely used to express conditional statements. This problem has intrigued several researchers in the past. This paper is both a survey of works pertaining to the introduction of conditioning relations in logic and a discussion about how these conditioning relations leave some room for non-monotonicity and might be useful in formalizing the concept of production rules in expert systems. Moreover, it is shown how this formal setting leads to generalizing Cox's axiomatic approach to conditional probability. Conditional probability, possibility, and belief functions are then obtained by solving functional equations. They obey the natural requirement that a conditional uncertainty measure be the measure of a conditional, such conditional being expressed as a conditioning relation between propositions that may fail to coincide with material implication.}
}


@article{nilsson86,
title = {Probabilistic logic},
journal = {Artificial Intelligence},
volume = {28},
number = {1},
pages = {71-87},
year = {1986},
issn = {0004-3702},
doi = {https://doi.org/10.1016/0004-3702(86)90031-7},
url = {https://www.sciencedirect.com/science/article/pii/0004370286900317},
author = {Nils J. Nilsson},
abstract = {Because many artificial intelligence applications require the ability to reason with uncertain knowledge, it is important to seek appropriate generalizations of logic for that case. We present here a semantical generalization of logic in which the truth values of sentences are probability values (between 0 and 1). Our generalization applies to any logical system for which the consistency of a finite set of sentences can be established. The method described in the present paper combines logic with probability theory in such a way that probabilistic logical entailment reduces to ordinary logical entailment when the probabilities of all sentences are either 0 or 1.}
}

@article{nilsson93,
title = {Probabilistic logic revisited},
journal = {Artificial Intelligence},
volume = {59},
number = {1},
pages = {39-42},
year = {1993},
issn = {0004-3702},
doi = {https://doi.org/10.1016/0004-3702(93)90167-A},
url = {https://www.sciencedirect.com/science/article/pii/000437029390167A},
author = {Nils J. Nilsson}
}

@inproceedings{greene08,
  title={An overview of conditionals and biconditionals in probability},
  author={Nataniel Greene},
  booktitle={MATH'08: Proceedings of the American Conference on Applied Mathematics},
  pages={171–177},
  year={2008},
}

@article{khoomandelkern18,
    author = {Khoo, Justin and Mandelkern, Matthew},
    title = "{Triviality Results and the Relationship between Logical and Natural Languages}",
    journal = {Mind},
    volume = {128},
    number = {510},
    pages = {485-526},
    year = {2018},
    month = {05},
    issn = {0026-4423},
    doi = {10.1093/mind/fzy006},
    url = {https://doi.org/10.1093/mind/fzy006},
    eprint = {https://academic.oup.com/mind/article-pdf/128/510/485/28656427/fzy006.pdf}
}


@article{wallmannkleiter14,
author = {Wallmann, Christian and Kleiter, Gernot},
year = {2014},
month = {05},
pages = {},
title = {Degradation in probability logic: When more information leads to less precise conclusions},
volume = {2},
journal = {Kybernetika},
doi = {10.14736/kyb-2014-2-0268}
}

@inproceedings{faginhalpern89,
author = {Fagin, Ronald and Halpern, Joseph Y.},
title = {Uncertainty, Belief, and Probability},
year = {1989},
publisher = {Morgan Kaufmann Publishers Inc.},
address = {San Francisco, CA, USA},
abstract = {We introduce a new probabilistic approach to dealing with uncertainty, based on the observation that probability theory does not require that every event be assigned a probability. For a nonmeasurable event (one to which we do not assign a probability), we can talk about only the inner measure and outer measure of the event. Thus, the measure of belief in an event can be represented by an interval (defined by the inner and outer measure), rather than by a single number. Further, this approach allows us to assign a belief (inner measure) to an event E without committing to a belief about its negation E (since the inner measure of an event plus the inner measure of its negation is not necessarily one). Interestingly enough, inner measures induced by probability measures turn out to correspond in a precise sense to Dempster-Shafer belief functions. Hence, in addition to providing promising new conceptual tools for dealing with uncertainty, our approach shows that a key part of the important Dempster-Shafer theory of evidence is firmly rooted in classical probability theory.},
booktitle = {Proceedings of the 11th International Joint Conference on Artificial Intelligence - Volume 2},
pages = {1161–1167},
numpages = {7},
location = {Detroit, Michigan},
series = {IJCAI'89}
}

@article{frechet35,
author = {Frechet, Maurice},
journal = {Fundamenta Mathematicae},
keywords = {Probability theory, statistics, operations research},
language = {fre},
number = {1},
pages = {379-387},
title = {Generalisation du theoreme des probabilites totales},
url = {http://eudml.org/doc/212798},
volume = {25},
year = {1935},
}
