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| The aim of a '''probabilistic logic''' (also '''probability logic''' and '''probabilistic reasoning''') is to combine the capacity of [[probability theory]] to handle uncertainty with the capacity of [[deductive logic]] to exploit structure. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logics attempt to find a natural extension of traditional logic truth tables: the results they define are derived through probabilistic expressions instead. A difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as those of [[Dempster-Shafer theory]]. The need to deal with a broad variety of contexts and issues has led to many different proposals.
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| ==Historical context==
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| 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 [[logical consequence|logical entailment]], such as [[Markov logic network]]s, and those that attempt to address the problems of uncertainty and lack of evidence (evidentiary logics).
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| That probability and uncertainty are not quite the same thing may be understood by noting that, despite the mathematization of probability in the [[Age of Enlightenment|Enlightenment]], mathematical [[probability theory]] remains, to this very day, entirely unused in criminal courtrooms, when evaluating the "probability" of the guilt of a suspected criminal.<ref name=franklin>James Franklin, ''The Science of Conjecture: Evidence and Probability before Pascal'', 2001 The Johns Hopkins Press, ISBN 0-8018-7109-3</ref>
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| More precisely, in evidentiary logic, there is a need to distinguish the truth of a statement from the confidence in its truth: thus, being uncertain of a suspect's guilt is not the same as assigning a numerical probability to the commission of the crime. A single suspect may be guilty or not guilty, just as a coin may be flipped heads or tails. 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", just as a single coin flip is "a little bit heads and a little bit tails": we are merely uncertain as to which it is. Conflating probability and uncertainty may be acceptable when making scientific measurements of physical quantities, but it is an error, 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.
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| 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 [[Scholastics]], with the invention of the [[half-proof]] (so that two half-proofs are sufficient to prove guilt), the elucidation of [[moral certainty]] (sufficient certainty to act upon, but short of absolute certainty), the development of [[Catholic probabilism]] (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 [[case-based reasoning]] of [[casuistry]], and the scandal of [[Laxism]] (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.).<ref name=franklin/>
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| ==Modern proposals==
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| Below is a list of proposals for probabilistic and evidentiary extensions to classical and predicate logic.
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| * The term "''probabilistic logic''" was first used in a paper by [[Nils Nilsson (researcher)|Nils Nilsson]] published in 1986, where the [[truth value]]s of sentences are [[probabilities]].<ref name="N86">Nilsson, N. J., 1986, "Probabilistic logic," ''Artificial Intelligence'' 28(1): 71-87.</ref> The proposed semantical generalization induces a probabilistic logical [[entailment]], which reduces to ordinary logical [[entailment]] when the probabilities of all sentences are either 0 or 1. This generalization applies to any [[logical system]] for which the consistency of a finite set of sentences can be established.
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| * The central concept in the theory of [[subjective logic]]<ref name="JO1">Jøsang, A., 2001, "A logic for uncertain probabilities," ''[[International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems]]'' 9(3):279-311.</ref> are ''opinions'' about some of the [[propositional variable]]s 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 various degrees of ignorance about the truth of the proposition. For the computation of derived opinions based on a structure of argument opinions, the theory proposes respective operators for various logical connectives, such as e.g. multiplication ([[Logical conjunction|AND]]), comultiplication ([[Logical disjunction|OR]]), division (UN-AND) and co-division (UN-OR) of opinions <ref name="JO4">Jøsang, A. and McAnally, D., 2004, "Multiplication and Comultiplication of Beliefs," ''International Journal of Approximate Reasoning'', 38(1), pp.19-51, 2004</ref> as well as conditional deduction ([[Modus ponens|MP]]) and abduction ([[Modus tollens|MT]]).<ref name="JO8">Jøsang, A., 2008, "Conditional Reasoning with Subjective Logic," ''Journal of Multiple-Valued Logic and Soft Computing'', 15(1), pp.5-38, 2008.</ref>
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| * Approximate reasoning formalism proposed by [[fuzzy logic]] can be used to obtain a logic in which the models are the probability distributions and the theories are the lower envelopes.<ref name="G94">Gerla, G., 1994, "Inferences in Probability Logic," ''Artificial Intelligence'' 70(1–2):33–52.</ref> 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 therefore with [[Dutch book]] phenomenon.
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| * [[Markov logic network]]s implement a form of [[uncertain inference]] based on the [[maximum entropy principle]]—the idea that probabilities should be assigned in such a way as to maximize entropy, in analogy with the way that [[Markov chain]]s assign probabilities to [[finite state machine]] transitions.
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| * Systems such as [[Pei Wang]]'s [[Non-Axiomatic Reasoning System]] (NARS) or [[Ben Goertzel]]'s [[Probabilistic Logic Network]]s (PLN) add an explicit confidence ranking, as well as a probability to [[atomic formula|atoms]] 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 [[Dempster-Shafer theory]]. The implementation of PLN attempts to use and generalize algorithms from [[logic programming]], subject to these extensions.
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| * In the theory of [[probabilistic argumentation]],<ref name="KM95">Kohlas, J., and Monney, P.A., 1995. ''A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence''. Vol. 425 in Lecture Notes in Economics and Mathematical Systems. Springer Verlag.</ref><ref name="HO5">Haenni, R, 2005, "Towards a Unifying Theory of Logical and Probabilistic Reasoning," ISIPTA'05, 4th International Symposium on Imprecise Probabilities and Their Applications: 193-202. [http://www.iam.unibe.ch/~run/papers/haenni05d.pdf]</ref> probabilities are not directly attached to logical sentences. Instead it is assumed that a particular subset <math>W</math> of the variables <math>V</math> involved in the sentences defines a [[probability space]] over the corresponding sub-[[σ-algebra]]. This induces two distinct probability measures with respect to <math>V</math>, which are called ''degree of support'' and ''degree of possibility'', respectively. Degrees of support can be regarded as non-additive ''probabilities of provability'', which generalizes the concepts of ordinary logical [[entailment]] (for <math>V=\{\}</math>) and classical [[posterior probability|posterior probabilities]] (for <math>V=W</math>). Mathematically, this view is compatible with the [[Dempster-Shafer theory]].
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| * The theory of [[evidential reasoning]]<ref name="RLS92">Ruspini, E.H., Lowrance, J., and Strat, T., 1992, "Understanding evidential reasoning," ''International Journal of Approximate Reasoning'', 6(3): 401-424.</ref> also defines non-additive ''probabilities of probability'' (or ''epistemic probabilities'') as a general notion for both logical [[entailment]] (provability) and [[probability]]. The idea is to augment standard [[propositional logic]] by considering an epistemic operator '''K''' that represents the state of knowledge that a rational agent has about the world. Probabilities are then defined over the resulting ''epistemic universe'' '''K'''''p'' of all propositional sentences ''p'', and it is argued that this is the best information available to an analyst. From this view, [[Dempster-Shafer theory]] appears to be a generalized form of probabilistic reasoning.
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| ==Possible application areas==
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| *[[Argumentation theory]]
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| *[[Artificial intelligence]]
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| *[[Artificial general intelligence]]
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| *[[Bioinformatics]]
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| *[[Formal epistemology]]
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| *[[Game theory]]
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| *[[Philosophy of science]]
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| *[[Psychology]]
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| *[[Statistics]]
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| ==See also==
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| * [[Statistical relational learning]]
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| * [[Bayesian inference]], [[Bayesian networks]], [[Bayesian probability]]
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| * [[Cox's theorem]]
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| * [[Dempster-Shafer theory]]
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| * [[Fréchet inequalities]]
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| * [[Fuzzy logic]]
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| * [[Imprecise probability]]
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| * [[Logic]], [[Deductive logic]], [[Non-monotonic logic]]
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| * [[Possibility theory]]
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| * [[Probabilism]], [[Half-proof]], [[Scholasticism]]
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| * [[Probabilistic database]]
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| * [[Probability]], [[Probability theory]]
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| * [[Probabilistic argumentation]]
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| * [[Reasoning]]
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| * [[Subjective logic]]
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| * [[Uncertainty]]
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| * [[Uncertain inference]]
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| * [[Upper and lower probabilities]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * Adams, E. W., 1998. ''A Primer of Probability Logic''. CSLI Publications (Univ. of Chicago Press).
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| * Bacchus, F., 1990. "Representing and reasoning with Probabilistic Knowledge. A Logical Approach to Probabilities". The MIT Press.
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| * [[Rudolf Carnap|Carnap, R.]], 1950. ''Logical Foundations of Probability''. University of Chicago Press.
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| * Chuaqui, R., 1991. ''Truth, Possibility and Probability: New Logical Foundations of Probability and Statistical Inference''. Number 166 in Mathematics Studies. North-Holland.
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| * Haenni, H., Romeyn, JW, Wheeler, G., and Williamson, J. 2011. ''Probabilistic Logics and Probabilistic Networks'', Springer.
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| * Hájek, A., 2001, "Probability, Logic, and Probability Logic," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic'', Blackwell.
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| * [[Henry E. Kyburg, Jr.|Kyburg, H. E.]], 1970. ''Probability and Inductive Logic'' Macmillan.
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| * Kyburg, H. E., 1974. ''The Logical Foundations of Statistical Inference'', Dordrecht: Reidel.
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| * Kyburg, H. E. & C. M. Teng, 2001. ''Uncertain Inference'', Cambridge: Cambridge University Press.
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| * Romeiyn, J. W., 2005. ''Bayesian Inductive Logic''. PhD thesis, Faculty of Philosophy, University of Groningen, Netherlands. [http://www.philos.rug.nl/~romeyn/paper/2005_romeijn_-_thesis.pdf]
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| * Williamson, J., 2002, "Probability Logic," in D. Gabbay, R. Johnson, H. J. Ohlbach, and J. Woods, eds., ''Handbook of the Logic of Argument and Inference: the Turn Toward the Practical''. Elsevier: 397-424.
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| == External links ==
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| * [http://www.kent.ac.uk/secl/philosophy/jw/2006/progicnet.htm ''Progicnet'': Probabilistic Logic And Probabilistic Networks]
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| * [http://www.unik.no/people/josang/sl/ Subjective logic demonstrations]
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| * [http://www.sipta.org/ ''The Society for Imprecise Probability'']
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| [[Category:Probabilistic arguments]]
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| [[Category:Non-classical logic]]
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| [[Category:Formal epistemology]]
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