Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
s with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the
computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in
Dempster–Shafer theory
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and ...
. Source trust and epistemic uncertainty about the probabilities they provide, such as defined in
subjective logic
Subjective logic is a type of probabilistic logic that explicitly takes epistemic uncertainty and source trust into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and relatively unreli ...
, 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 to
logical entailment, such as
Markov logic network A Markov logic network (MLN) is a probabilistic logic which applies the ideas of a Markov network to first-order logic, defining probability distributions on possible worlds on any given domain.
History
In 2002, Ben Taskar, Pieter Abbeel and ...
s, 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 the
Enlightenment, mathematical
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
remains, to this very day, entirely unused in criminal courtrooms, when evaluating the "probability" of the guilt of a suspected criminal.
[James Franklin, ''The Science of Conjecture: Evidence and Probability before Pascal'', 2001 The Johns Hopkins Press, .]
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 the
Scholastics
Scholasticism was a medieval European philosophical movement or methodology that was the predominant education in Europe from about 1100 to 1700. It is known for employing logically precise analyses and reconciling classical philosophy and C ...
, with the invention of the
half-proof (so that two half-proofs are sufficient to prove guilt), the elucidation of
moral certainty
Moral certainty is a concept of intuitive probability. It means a very high degree of probability, sufficient for action, but short of absolute or mathematical certainty.
Origins
The notion of different degrees of certainty can be traced back to a ...
(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
Case-based reasoning (CBR), broadly construed, is the process of solving new problems based on the solutions of similar past problems.
In everyday life, an auto mechanic who fixes an engine by recalling another car that exhibited similar sympto ...
of
casuistry
Casuistry ( ) is a process of reasoning that seeks to resolve moral problems by extracting or extending abstract rules from a particular case, and reapplying those rules to new instances. This method occurs in applied ethics and jurisprudence. ...
, 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.).
[
]
Modern proposals
Below is a list of proposals for probabilistic and evidentiary extensions to classical and predicate logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables ove ...
.
* The term "''probabilistic logic''" was first used by Jon Von Neumann in a series of Caltech
The California Institute of Technology (branded as Caltech) is a private university, private research university in Pasadena, California, United States. The university is responsible for many modern scientific advancements and is among a small g ...
lectures 1952 and 1956 paper "Probabilistic logics and the synthesis of reliable organisms from unreliable components", and subsequently in a paper by Nils Nilsson published in 1986, where the truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...
s of sentences are probabilities
Probability is a branch of mathematics and statistics concerning Event (probability theory), events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probab ...
. The proposed semantical generalization induces a probabilistic logical entailment
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid l ...
, which reduces to ordinary logical entailment
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid l ...
when the probabilities of all sentences are either 0 or 1. This generalization applies to any logical system
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
for which the consistency of a finite set of sentences can be established.
* Gaifman and Snir have developed a globally consistent and empirically satisfactory unification of classic probability theory and first-order logic that is suitable for inductive reasoning. Their theory assigns probabilities or degrees of beliefs to sentences consistent with the knowledge base (probability 1 for facts and axioms), consistent with the standard (Kolmogorov) probability axioms
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-worl ...
and logical deduction, and allows ( Bayesian) inductive reasoning
Inductive reasoning refers to a variety of method of reasoning, methods of reasoning in which the conclusion of an argument is supported not with deductive certainty, but with some degree of probability. Unlike Deductive reasoning, ''deductive'' ...
and learning in the limit. Most importantly, unlike most alternative proposals, it allows confirmation of universally quantified
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by ev ...
hypotheses. The theory has also been extended to higher-order logic. Both solutions are purely theoretical but have spawned practical approximations.
* The central concept in the theory of subjective logic
Subjective logic is a type of probabilistic logic that explicitly takes epistemic uncertainty and source trust into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and relatively unreli ...
[A. Jøsang. ]
Subjective Logic: A formalism for reasoning under uncertainty
'. Springer Verlag, 2016 is ''opinions'' about some of the propositional variable
In mathematical logic, a propositional variable (also called a sentence letter, sentential variable, or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building ...
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 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 as Bayes' theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For exampl ...
.
* The approximate reasoning formalism proposed by fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
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 to that of the coherence of partial probabilistic assignment and therefore with Dutch book
In decision theory, economics, and probability theory, the Dutch book arguments are a set of results showing that agents must satisfy the axioms of rational choice to avoid a kind of self-contradiction called a Dutch book. A Dutch book, somet ...
phenomena.
* Markov logic network A Markov logic network (MLN) is a probabilistic logic which applies the ideas of a Markov network to first-order logic, defining probability distributions on possible worlds on any given domain.
History
In 2002, Ben Taskar, Pieter Abbeel and ...
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
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
s assign probabilities to finite-state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number o ...
transitions.
* Systems such as Ben Goertzel's Probabilistic Logic Networks (PLN) add an explicit confidence ranking, as well as a probability to atoms
Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
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 theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and ...
. The implementation of PLN attempts to use and generalize algorithms from logic programming
Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical form, representing knowledge about some problem domain. Computation is performed by applyin ...
, subject to these extensions.
* In the field of probabilistic 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 of argumentation graphs. In the framework of "probabilistic argumentation systems"[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.[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. probabilities are not directly attached to arguments or logical sentences. Instead it is assumed that a particular subset of the variables involved in the sentences defines a probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
over the corresponding sub-σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
. This induces two distinct probability measures with respect to , 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
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid l ...
(for ) and classical posterior probabilities (for ). Mathematically, this view is compatible with the Dempster–Shafer theory
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and ...
.
* The theory of evidential reasoning also defines non-additive ''probabilities of probability'' (or ''epistemic probabilities'') as a general notion for both logical entailment
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid l ...
(provability) and probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
. The idea is to augment standard propositional logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
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
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and ...
appears to be a generalized form of probabilistic reasoning.
See also
* Statistical relational learning
Statistical relational learning (SRL) is a subdiscipline of artificial intelligence and machine learning that is concerned with domain models that exhibit both uncertainty (which can be dealt with using statistical methods) and complex, relational ...
* Bayesian inference
Bayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian infer ...
, Bayesian network
A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Whi ...
, Bayesian probability
Bayesian probability ( or ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quant ...
* Cox's theorem
Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of pr ...
* Fréchet inequalities
* Imprecise probability
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Ther ...
* Non-monotonic logic
A non-monotonic logic is a formal logic whose entailment relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences, i.e., a kind of inference in which reasoners draw tentative concl ...
* Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessa ...
* Probabilistic database Most real databases contain data whose correctness is uncertain. In order to work with such data, there is a need to quantify the integrity of the data. This is achieved by using probabilistic databases.
A probabilistic database is an uncertain da ...
* Probabilistic soft logic
Probabilistic Soft Logic (PSL) is a statistical relational learning (SRL) framework for modeling probabilistic and relational domains.
It is applicable to a variety of machine learning problems, such as collective classification, entity reso ...
* Probabilistic causation Probabilistic causation is a concept in a group of philosophical theories that aim to characterize the relationship between cause and effect using the tools of probability theory. The central idea behind these theories is that causes raise the proba ...
* Uncertain inference
* Upper and lower probabilities
References
{{reflist
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"
pdf
and Cambridge University Press 2003.
* Kyburg, H. E., 1970.
Probability and Inductive Logic
' Macmillan.
* 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
* 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
Subjective logic demonstrations
''The Society for Imprecise Probability''
Probabilistic arguments, logic
Non-classical logic
Scientific method
Formal epistemology