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=                        Probabilistic logic                         =
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                            Introduction
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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 of formal argument. 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 in evidence-based subjective
logic. The need to deal with a broad variety of contexts and issues
has led to many different proposals.


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

That probability and uncertainty are not quite the same thing may be
understood by noting that, despite the mathematization of probability
in the 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.

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" than 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.

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.).


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

* The term "'probabilistic logic'" was first used in a paper by Nils
Nilsson published in 1986, 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.
* The central concept in the theory of subjective logic are 'opinions'
about some of the propositional variables 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 (AND), comultiplication (OR), division (UN-AND) and
co-division (UN-OR) of opinions  as well as conditional deduction (MP)
and abduction (MT).
* 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. 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.
* Markov logic networks 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 chains assign probabilities to finite state machine
transitions.
* Systems such as Pei Wang's Non-Axiomatic Reasoning System (NARS) or
Ben Goertzel's Probabilistic Logic Networks (PLN) add an explicit
confidence ranking, as well as a probability to 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.
* In the theory of probabilistic argumentation, probabilities are not
directly attached to logical sentences. Instead it is assumed that a
particular subset W of the variables V involved in the sentences
defines a probability space over the corresponding sub-�-algebra. This
induces two distinct probability measures with respect to V, 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 V=\{\}) and classical posterior probabilities (for
V=W). Mathematically, this view is compatible with the Dempster-Shafer
theory.
* The theory of evidential reasoning 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.


                     Possible application areas
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*Argumentation theory
*Artificial intelligence
*Artificial general intelligence
*Bioinformatics
*Formal epistemology
*Game theory
*Philosophy of science
*Psychology
*Statistics
*Life


                              See also
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* 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 proof
* Subjective logic
* Uncertain inference
* Upper and lower probabilities


                          Further reading
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* Adams, E. W., 1998. '[https://philpapers.org/rec/ADAAPO-2 A Primer
of Probability Logic]'. CSLI Publications (Univ. of Chicago Press).
* Bacchus, F., 1990. "[https://dl.acm.org/citation.cfm?id=914650
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.
'[https://books.google.com/books?hl=en&lr=&id=D5aCKWMtrtQC&oi=fnd&am
p;pg=PP1&dq=%22Truth,+Possibility+and+Probability:+New+Logical+Foundations+o
f+Probability+and+Statistical+Inference%22&ots=dzloxitdW3&sig=ekP_NPlUTd
nFJLF4yOpM6qgPdSs#v=onepage&q=%22Truth%2C%20Possibility%20and%20Probability%
3A%20New%20Logical%20Foundations%20of%20Probability%20and%20Statistical%20Infere
nce%22&f=false
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",
[http://bayes.wustl.edu/etj/prob/book.pdf pdf] and Cambridge
University Press 2003.
* Kyburg, H. E., 1970. '[https://philpapers.org/rec/KYBPAI Probability
and Inductive Logic]' Macmillan.
* Kyburg, H. E., 1974.
'[https://books.google.com/books?hl=en&lr=&id=uFVtCQAAQBAJ&oi=fnd&am
p;pg=PP8&dq=%22The+Logical+Foundations+of+Statistical+Inference%22+kyburg&am
p;ots=a0m49xG5Vt&sig=cgDXfw19FujcuIq-bz_OOYLxicc
The Logical Foundations of Statistical Inference]', Dordrecht: Reidel.
* Kyburg, H. E. & C. M. Teng, 2001.
'[https://books.google.com/books?hl=en&lr=&id=CicvkPZhohEC&oi=fnd&am
p;pg=PR11&dq=%22Uncertain+Inference%22+Teng&ots=JjnRNjQ4i6&sig=JBV9_
yWGmiAlrl2EXylLEJ9GRrk
Uncertain Inference]', Cambridge: Cambridge University Press.
* 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]
* Williamson, J., 2002, "Probability Logic," in D. Gabbay, R. Johnson,
H. J. Ohlbach, and J. Woods, eds.,
'[https://books.google.com/books?hl=en&lr=&id=c-bthgk3DQAC&oi=fnd&am
p;pg=PP1&dq=%22Handbook+of+the+Logic+of+Argument+and+Inference:+The+turn+tow
ards+the+practical%22&ots=yrjiTlKa-O&sig=9ToPqYX6B2Fzn-GhYVJm33IUeB8#v=o
nepage&q=%22Handbook%20of%20the%20Logic%20of%20Argument%20and%20Inference%3A
%20The%20turn%20towards%20the%20practical%22&f=false
Handbook of the Logic of Argument and Inference: the Turn Toward the
Practical]'. Elsevier: 397-424.


                           External links
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*
[https://web.archive.org/web/20070930031527/http://www.kent.ac.uk/secl/philosoph
y/jw/2006/progicnet.htm
'Progicnet': Probabilistic Logic And Probabilistic Networks]
* [http://www.unik.no/people/josang/sl/ Subjective logic
demonstrations]
* [http://www.sipta.org/ 'The Society for Imprecise Probability']


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