Probabilistic logic

Probabilistic logic(alsoprobability logicandprobabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logictruth tableswith probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply thecomputational complexitiesof their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion inDempster–Shafer theory.Source trust and epistemic uncertainty about the probabilities they provide, such as defined insubjective logic,are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals.

Logical background

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 tological entailment,such asMarkov logic networks,and those that attempt to address the problems of uncertainty and lack of evidence (evidentiary logics).

That the concept of probability can have different meanings may be understood by noting that, despite the mathematization of probability in theEnlightenment,mathematicalprobability theoryremains, to this very day, entirely unused in criminal courtrooms, when evaluating the "probability" of the guilt of a suspected criminal.

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 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.

Historical context

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 theScholastics,with the invention of thehalf-proof(so that two half-proofs are sufficient to prove guilt), the elucidation ofmoral certainty(sufficient certainty to act upon, but short of absolute certainty), the development ofCatholic 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), thecase-based reasoningofcasuistry,and the scandal ofLaxism(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.).

Modern proposals

Below is a list of proposals for probabilistic and evidentiary extensions to classical andpredicate logic.

• The term "probabilistic logic"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 byNils Nilssonpublished in 1986, where thetruth valuesof sentences areprobabilities.The proposed semantical generalization induces a probabilistic logicalentailment,which reduces to ordinary logicalentailmentwhen the probabilities of all sentences are either 0 or 1. This generalization applies to anylogical systemfor which the consistency of a finite set of sentences can be established.
• The central concept in the theory ofsubjective logicisopinionsabout some of thepropositional variablesinvolved 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 based on a structure of argument opinions, the theory proposes respective operators for various logical connectives, such as e.g. multiplication (AND), comultiplication (OR), division (UN-AND) and co-division (UN-OR) of opinions,conditional deduction (MP) and abduction (MT).,as well asBayes' theorem.
• The approximate reasoning formalism proposed byfuzzy logiccan 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 therefore withDutch bookphenomena.
• Markov logic networksimplement a form ofuncertain inferencebased on themaximum entropy principle—the idea that probabilities should be assigned in such a way as to maximize entropy, in analogy with the way thatMarkov chainsassign probabilities tofinite state machinetransitions.
• Systems such asBen Goertzel'sProbabilistic Logic Networks(PLN) add an explicit confidence ranking, as well as a probability toatomsand 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 ofDempster–Shafer theory.The implementation of PLN attempts to use and generalize algorithms fromlogic programming,subject to these extensions.
• In the field ofprobabilistic argumentation,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 ofargumentation graphs.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$W$of the variables$V$involved in the sentences defines aprobability spaceover the corresponding sub-σ-algebra.This induces two distinct probability measures with respect to$V$,which are calleddegree of supportanddegree of possibility,respectively. Degrees of support can be regarded as non-additiveprobabilities of provability,which generalizes the concepts of ordinary logicalentailment(for$V=\{\}$) and classicalposterior probabilities(for$V=W$). Mathematically, this view is compatible with theDempster–Shafer theory.
• The theory ofevidential reasoningalso defines non-additiveprobabilities of probability(orepistemic probabilities) as a general notion for both logicalentailment(provability) andprobability.The idea is to augment standardpropositional logicby considering an epistemic operatorKthat represents the state of knowledge that a rational agent has about the world. Probabilities are then defined over the resultingepistemic universeKpof all propositional sentencesp,and it is argued that this is the best information available to an analyst. From this view,Dempster–Shafer theoryappears to be a generalized form of probabilistic reasoning.

See also

• Statistical relational learning
• Bayesian inference,Bayesian networks,Bayesian probability
• Cox's theorem
• Dempster–Shafer theory
• Fréchet inequalities
• Fuzzy logic
• Imprecise probability
• Logic,Deductive logic,Non-monotonic logic
• Possibility theory
• Probabilism,Half-proof,Scholasticism
• Probabilistic database
• Probabilistic soft logic
• Probability,Probability theory
• Probabilistic argumentation
• Probabilistic causation
• Probabilistic proof
• Scientific evidence
• Subjective logic
• Uncertain inference
• Upper and lower probabilities

Further reading

• Adams, E. W., 1998.A Primer of Probability Logic.CSLI Publications (Univ. of Chicago Press).
• Bacchus, F., 1990. "Representing and reasoning with Probabilistic Knowledge. A Logical Approach to Probabilities".The MIT Press.
• Carnap, R.,1950.Logical Foundations of Probability.University of Chicago Press.
• Chuaqui, R.,1991.Truth, Possibility and Probability: New Logical Foundations of Probability and Statistical Inference.Number 166 in Mathematics Studies. North-Holland.
• Haenni, H., Romeyn, JW, Wheeler, G., and Williamson, J. 2011.Probabilistic Logics and Probabilistic Networks,Springer.
• Hájek, A., 2001, "Probability, Logic, and Probability Logic," in Goble, Lou, ed.,The Blackwell Guide to Philosophical Logic,Blackwell.
• Jaynes, E., 1998, "Probability Theory: The Logic of Science",pdfand Cambridge University Press 2003.
• Kyburg, H. E.,1970.Probability and Inductive LogicMacmillan.
• Kyburg, H. E., 1974.The Logical Foundations of Statistical Inference,Dordrecht: Reidel.
• Kyburg, H. E. & C. M. Teng, 2001.Uncertain Inference,Cambridge: Cambridge University Press.
• Romeiyn, J. W., 2005.Bayesian Inductive Logic.PhD thesis, Faculty of Philosophy, University of Groningen, Netherlands.[1]
• 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.

External links

• Progicnet:Probabilistic Logic And Probabilistic Networks
• Subjective logic demonstrations
• The Society for Imprecise Probability