Probabilistic Logic

Information about Probabilistic Logic

The aim of a probabilistic logic (or probability logic) 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. The difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components.

Proposals

There are numerous proposals for probabilistic logics:

The term "probabilistic logic" was first used in [N86], where the truth values of sentences are probabilities. 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.
In the theory of probabilistic argumentation [KM95,H05], probabilities are not directly attached to logical sentences. Instead it is assumed that a particular subset $\text{[math]}$ of the variables $\text{[math]}$ involved in the sentences defines a probability space over the corresponding sub-σ-algebra. This induces two distinct probability measures with respect to $\text{[math]}$ , 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 $\text{[math]}$ ) and classical posterior probabilities (for $\text{[math]}$ ). Mathematically, this view is compatible with the Dempster-Shafer theory.
The theory of evidential reasoning [RLS90] 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 Kp 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.
Approximate reasoning formalism proposed by fuzzy logic is used in [G94] to obtain a logic in which the models are the probability distributions and the theories are the lower envelopes. 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.
The central concept in the theory of subjective logic [J01] are opinions about some of the propositional variables involved in the given logical sentences. An opinion is a two-dimensional extension of a single probabiliy value to express various degrees of ignorance. For the computation of overall opinions with repsect to some query variables, the theory proposes respective operators for various logical connectives.

Possible application areas

References

[A98] E. W. Adams, 1998. A Primer of Probability Logic. CSLI Publications (Univ. of Chicago Press).
[C37] Rudolf Carnap, 1937. Logical Foundations of Probability. University of Chicago Press.
[C91] Chuaqui, R., 1991. Truth, Possibility and Probability: New Logical Foundations of Probability and Statistical Inference. Number 166 in Mathematics Studies. North-Holland.
[G94] Gerla, G., 1994, "Inferences in Probability Logic," Artificial Intelligence 70(1–2):33–52.
[H05] 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
Hajek, Alan, 2001, "Probability, Logic, and Probability Logic," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic, Blackwell.
[J01] A. Jøsang, A., 2001, "A logic for uncertain probabilities," International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9(3):279-311.
[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.
[K70] Henry Kyburg, 1970. Probability and Inductive Logic Macmillan.
[K74] H. E. Kyburg, 1974. The Logical Foundations of Statistical Inference, Dordrecht: Reidel.
[KT01] H. E. Kyburg and C. M. Teng, 2001. Uncertain Inference, Cambridge: Cambridge University Press.
[N86] Nilsson, N. J., 1986, "Probabilistic logic," Artificial Intelligence 28(1): 71-87.
[R05] Romeijn, J. W., 2005. Bayesian Inductive Logic. PhD thesis, Faculty of Philosophy, University of Groningen, Netherlands. http://home.medewerker.uva.nl/j.w.romeijn/bestanden/omslag%20proefschrift%20e-versie.pdf
[RLS92] Ruspini, E.H., Lowrance, J., and Strat, T., 1992, "Understanding evidential reasoning," International Journal of Approximate Reasoning, 6(3): 401-424.
[W02] 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

Probability theory is the branch of mathematics concerned with analysis of random phenomena.^[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
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In philosophical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it "naturally" occurs.

Natural deductive logic

One version of natural deductive logic has no axioms. System L, developed by E.J.
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Probability is the likelihood that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of
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entailment (or logical implication) is a relation between sets of formulae such that, if A and B are sets of formulae of a formal language, then A entails B if and only if every model (or interpretation) that makes all the members of A true, makes at least one of the members of B
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Probabilistic argumentation is a general theory of reasoning under uncertainty and ignorance. It combines the fields of probability theory and deductive logic, making it a probabilistic logic .
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In probability theory, the definition of the probability space is the foundation of probability theory. It was introduced by Kolmogorov in the 1930s. For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.
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The posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned when the relevant evidence is taken into account.
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The Dempster-Shafer theory is a mathematical theory of evidence^[1] based on belief functions and plausible reasoning, which is used to combine separate pieces of information (evidence) to calculate the probability of an event.
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Evidential reason or Evidential reasoning may refer to:

Probabilistic logic, a combination of the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure

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In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules
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Fuzzy Logic may refer to:

Fuzzy Logic (album), the debut album by the Super Furry Animals
Fuzzy logic, an application of fuzzy set theory

For the music album, see Fuzzy Logic (album)

Fuzzy logic
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In gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, regardless of the outcome of the gamble. It is associated with probabilities implied by the odds not being coherent.
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<noinclude></noinclude>Subjective logic is a type of probabilistic logic that explicitly takes uncertainty and belief ownership into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and incomplete knowledge
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Argumentation theory, or argumentation, embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion. It studies rules of inference, logic, and procedural rules in both artificial and real world settings.
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artificial intelligence (or AI) is "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions which maximizes its chances of success.
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Bioinformatics and computational biology involve the use of techniques including applied mathematics, informatics, statistics, computer science, artificial intelligence, chemistry, and biochemistry to solve biological problems usually on the molecular level.
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Formal epistemology is a subdiscipline of epistemology that utilizes formal methods from logic, probability theory and computability theory to elucidate traditional epistemic problems.
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Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose.
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Philosophy of science is the study of assumptions, foundations, and implications of science. The philosophy of science may be divided into two areas: Epistemology of science and metaphysics of science.
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Psychology (from Greek: Literally "talk about the soul" (from logos)) is both an academic and applied discipline involving the scientific study of mental processes and behavior.
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
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Rudolf Carnap (May 18, 1891, Ronsdorf, Germany – September 14, 1970, Santa Monica, California) was an influential philosopher who was active in Europe before 1935 and in the United States thereafter.
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Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. The name "Bayesian" comes from the frequent use of Bayes' theorem in the inference process.
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A Bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic independencies. For example, a Bayesian network can be used to calculate the probability of a patient having a specific disease, given the
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Bayesian probability is an interpretation of the probability calculus which holds that the concept of probability can be defined as the degree to which a person (or community) believes that a proposition is true.
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