85:
decision on guilt, which in turn is not the same as assigning a numerical probability to the commission of the crime, and deciding whether it is above a numerical threshold of guilt. The verdict on a single suspect may be guilty or not guilty with some uncertainty, just as the flipping of a coin may be predicted as heads or tails with some uncertainty. Given a large collection of suspects, a certain percentage may be guilty, just as the probability of flipping "heads" is one-half. However, it is incorrect to take this law of averages with regard to a single criminal (or single coin-flip): the criminal is no more "a little bit guilty" than predicting a single coin flip to be "a little bit heads and a little bit tails": we are merely uncertain as to which it is. Expressing uncertainty as a numerical probability may be acceptable when making scientific measurements of physical quantities, but it is merely a mathematical model of the uncertainty we perceive in the context of "common sense" reasoning and logic. Just as in courtroom reasoning, the goal of employing uncertain inference is to gather evidence to strengthen the confidence of a proposition, as opposed to performing some sort of probabilistic entailment.
84:
More precisely, in evidentiary logic, there is a need to distinguish the objective truth of a statement from our decision about the truth of that statement, which in turn must be distinguished from our confidence in its truth: thus, a suspect's real guilt is not necessarily the same as the judge's
180:
involved in the given logical sentences. A binomial opinion applies to a single proposition and is represented as a 3-dimensional extension of a single probability value to express probabilistic and epistemic uncertainty about the truth of the proposition. For the computation of derived opinions
208:
can be used to obtain a logic in which the models are the probability distributions and the theories are the lower envelopes. In such a logic the question of the consistency of the available information is strictly related with the one of the coherence of partial probabilistic assignment and
250:
and sentences. The rules of deduction and induction incorporate this uncertainty, thus side-stepping difficulties in purely
Bayesian approaches to logic (including Markov logic), while also avoiding the paradoxes of
141:" was first used by Jon Von Neumann in a series of Cal Tech lectures 1952 and 1956 paper "Probabilistic logics and the synthesis of reliable organisms from unreliable components", and subsequently in a paper by
862:
736:
Riveret, R.; Baroni, P.; Gao, Y.; Governatori, G.; Rotolo, A.; Sartor, G. (2018), "A Labelling
Framework for Probabilistic Argumentation", Annals of Mathematics and Artificial Intelligence, 83: 221–287.
931:
266:, various formal frameworks have been put forward. The framework of "probabilistic labellings", for example, refers to probability spaces where a sample space is a set of labellings of
62:
There are numerous proposals for probabilistic logics. Very roughly, they can be categorized into two different classes: those logics that attempt to make a probabilistic extension to
811:
270:. In the framework of "probabilistic argumentation systems" probabilities are not directly attached to arguments or logical sentences. Instead it is assumed that a particular subset
93:
Historically, attempts to quantify probabilistic reasoning date back to antiquity. There was a particularly strong interest starting in the 12th century, with the work of the
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121:(whereby probabilism was used to give support to almost any statement at all, it being possible to find an expert opinion in support of almost any proposition.).
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of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in
181:
based on a structure of argument opinions, the theory proposes respective operators for various logical connectives, such as e.g. multiplication (
73:
That the concept of probability can have different meanings may be understood by noting that, despite the mathematization of probability in the
109:(the idea that it is always safe to follow the established rules of doctrine or the opinion of experts, even when they are less probable), the
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remains, to this very day, entirely unused in criminal courtrooms, when evaluating the "probability" of the guilt of a suspected criminal.
945:
54:, are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals.
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that represents the state of knowledge that a rational agent has about the world. Probabilities are then defined over the resulting
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2016 IEEE International
Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016)
70:, and those that attempt to address the problems of uncertainty and lack of evidence (evidentiary logics).
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243:
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Williamson, J., 2002, "Probability Logic," in D. Gabbay, R. Johnson, H. J. Ohlbach, and J. Woods, eds.,
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with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the
764:," ISIPTA'05, 4th International Symposium on Imprecise Probabilities and Their Applications: 193-202.
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224:
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Truth, Possibility and
Probability: New Logical Foundations of Probability and Statistical Inference
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when the probabilities of all sentences are either 0 or 1. This generalization applies to any
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Representing and reasoning with
Probabilistic Knowledge. A Logical Approach to Probabilities
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Below is a list of proposals for probabilistic and evidentiary extensions to classical and
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A Mathematical Theory of Hints. An
Approach to the Dempster–Shafer Theory of Evidence
672:
Jøsang, A. and McAnally, D., 2004, "Multiplication and
Comultiplication of Beliefs,"
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486:
311:
256:
198:
773:
751:. Vol. 425 in Lecture Notes in Economics and Mathematical Systems. Springer Verlag.
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495:
169:
51:
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Hájek, A., 2001, "Probability, Logic, and
Probability Logic," in Goble, Lou, ed.,
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524:
189:), division (UN-AND) and co-division (UN-OR) of opinions, conditional deduction (
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932:
Handbook of the Logic of
Argument and Inference: the Turn Toward the Practical
686:
969:
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546:
194:
150:
841:
255:. The implementation of PLN attempts to use and generalize algorithms from
101:(so that two half-proofs are sufficient to prove guilt), the elucidation of
924:. PhD thesis, Faculty of Philosophy, University of Groningen, Netherlands.
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for which the consistency of a finite set of sentences can be established.
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246:(PLN) add an explicit confidence ranking, as well as a probability to
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346:, respectively. Degrees of support can be regarded as non-additive
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The
Science of Conjecture: Evidence and Probability before Pascal
318:. This induces two distinct probability measures with respect to
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Towards a Unifying Theory of Logical and Probabilistic Reasoning
118:
883:
Jaynes, E., 1998, "Probability Theory: The Logic of Science",
869:
Haenni, H., Romeyn, JW, Wheeler, G., and Williamson, J. 2011.
660:
Subjective Logic: A formalism for reasoning under uncertainty
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466:
appears to be a generalized form of probabilistic reasoning.
958:
393:
360:
350:, which generalizes the concepts of ordinary logical
324:
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276:
413:). Mathematically, this view is compatible with the
866:. Number 166 in Mathematics Studies. North-Holland.
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Ruspini, E.H., Lowrance, J., and Strat, T., 1992, "
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Journal of Multiple-Valued Logic and Soft Computing
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949:: Probabilistic Logic And Probabilistic Networks
905:The Logical Foundations of Statistical Inference
204:The approximate reasoning formalism proposed by
871:Probabilistic Logics and Probabilistic Networks
704:Generalising Bayes' Theorem in Subjective Logic
816:International Journal of Approximate Reasoning
674:International Journal of Approximate Reasoning
837:. CSLI Publications (Univ. of Chicago Press).
651:
644:Nilsson, N. J., 1986, "Probabilistic logic,"
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367:
687:Conditional Reasoning with Subjective Logic
878:The Blackwell Guide to Philosophical Logic
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617:
917:, Cambridge: Cambridge University Press.
696:
614:
432:) as a general notion for both logical
14:
968:
911:Kyburg, H. E. & C. M. Teng, 2001.
796:: CS1 maint: archived copy as title (
960:The Society for Imprecise Probability
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444:by considering an epistemic operator
168:The central concept in the theory of
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887:and Cambridge University Press 2003.
745:Kohlas, J., and Monney, P.A., 1995.
310:involved in the sentences defines a
157:, which reduces to ordinary logical
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852:Logical Foundations of Probability
825:
812:Understanding evidential reasoning
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440:. The idea is to augment standard
25:
1002:
939:
627:, 2001 The Johns Hopkins Press,
954:Subjective logic demonstrations
896:Probability and Inductive Logic
721:Inferences in Probability Logic
478:Statistical relational learning
458:of all propositional sentences
854:. University of Chicago Press.
730:
259:, subject to these extensions.
13:
1:
834:A Primer of Probability Logic
710:, Baden-Baden, Germany, 2016.
607:
601:Upper and lower probabilities
145:published in 1986, where the
426:probabilities of probability
348:probabilities of provability
244:Probabilistic Logic Networks
97:, with the invention of the
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571:Probabilistic argumentation
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314:over the corresponding sub-
264:probabilistic argumentation
10:
1007:
424:also defines non-additive
44:computational complexities
225:maximum entropy principle
922:Bayesian Inductive Logic
557:Probabilistic soft logic
231:assign probabilities to
976:Probabilistic arguments
725:Artificial Intelligence
676:, 38(1), pp.19-51, 2004
663:. Springer Verlag, 2016
646:Artificial Intelligence
576:Probabilistic causation
430:epistemic probabilities
385:posterior probabilities
178:propositional variables
36:probabilistic reasoning
18:Probabilistic reasoning
920:Romeiyn, J. W., 2005.
693:, 15(1), pp.5-38, 2008
552:Probabilistic database
501:Dempster–Shafer theory
464:Dempster–Shafer theory
415:Dempster–Shafer theory
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253:Dempster–Shafer theory
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902:Kyburg, H. E., 1974.
516:Imprecise probability
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344:degree of possibility
333:
305:
285:
217:Markov logic networks
185:), comultiplication (
117:, and the scandal of
68:Markov logic networks
935:. Elsevier: 397–424.
908:, Dordrecht: Reidel.
840:Bacchus, F., 1990. "
831:Adams, E. W., 1998.
506:Fréchet inequalities
491:Bayesian probability
422:evidential reasoning
391:
358:
322:
294:
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268:argumentation graphs
233:finite state machine
219:implement a form of
111:case-based reasoning
107:Catholic probabilism
991:Formal epistemology
981:Non-classical logic
914:Uncertain Inference
685:Jøsang, A., 2008, "
596:Uncertain inference
586:Scientific evidence
581:Probabilistic proof
529:Non-monotonic logic
442:propositional logic
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338:, which are called
221:uncertain inference
139:probabilistic logic
28:Probabilistic logic
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719:Gerla, G., 1994, "
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450:epistemic universe
436:(provability) and
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487:Bayesian networks
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131:predicate logic
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240:Ben Goertzel
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143:Nils Nilsson
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40:truth tables
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873:, Springer.
858:Chuaqui, R.
702:A. Jøsang.
657:A. Jøsang.
562:Probability
539:Probabilism
511:Fuzzy logic
438:probability
206:fuzzy logic
95:Scholastics
970:Categories
899:Macmillan.
848:Carnap, R.
783:2006-06-18
608:References
543:Half-proof
434:entailment
352:entailment
213:phenomena.
211:Dutch book
159:entailment
155:entailment
137:The term "
99:half-proof
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947:Progicnet
316:σ-algebra
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893:, 1970.
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850:, 1950.
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471:See also
174:opinions
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