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An Axiomatic Characterization of Causal Counterfactuals
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Galles and Pearl and Halpern have given a complete axiomatization of structural counterfactuals, embracing both recursive and non-recursive models.
“Galles and Pearl [1] and Halpern [2] have given a complete axiomatization of structural counterfactuals, embracing both recursive and non-recursive models”
From The Deductive Approach to Causal Inference (2014) · cites this paper as [1]
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“A systematic analysis of the syntax and semantics of the two notational systems reveals that they are logically equivalent (Galles and Pearl 1998; Halpern 1998); a theorem in one is a theorem in the other, and an assumption in one has a parallel interpretation in the other.”
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“Remarkably, formal analysis proves the exact opposite: structural and causal parameters are one and the same thing, and they should always be interpreted as effects (Galles and Pearl, 1998; see Section 4).”
BackgroundSelf-citationGraphs, Causality, and Structural Equation Models
PEARL, J. (1998) · Sociological Methods & Research · doi:10.1177/0049124198027002004 · Cited by 341
Self-citationInterventionist counterfactuals
Briggs, R. (2012) · Philosophical Studies · doi:10.1007/s11098-012-9908-5 · Cited by 242
“Galles and Pearl (1998) have proposed a partial account of the truth conditions of counterfactuals; similar accounts can be found in (Hiddleston 2005) and (Halpern 2000). Woodward (2003, p. 137) argues that the causal modeling account disagrees with Lewis’s account in one crucial respect: it assigns different truth co…”
BackgroundMethodInfluentialElwert, F. (2013) · Handbooks of Sociology and Social Research · doi:10.1007/978-94-007-6094-3_13 · Cited by 325
Identifiability of path-specific effects
Avin, C., Shpitser, I., & Pearl, J. (2005) · eScholarship (California Digital Library) · Cited by 215
Self-citationHalpern, J. Y. (2000) · Journal of Artificial Intelligence Research · doi:10.1613/jair.648 · Cited by 199
Probabilities of causation: Bounds and identification
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Self-citationApplying the structural causal model framework for observational causal inference in ecology
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“Not surprisingly, the consistency rule articulated in (1) can be shown to be among those theorems.(9,10) This agreement between two diverse accounts of counterfactuals is not coincidental; the structural account can be given a “closest-world” interpretation, provided worlds that share identical histories are deemed eq…”
BackgroundSelf-citationComplete Identification Methods for the Causal Hierarchy
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“For a more comprehensive discussion, see Galles and Pearl [1998] and Halpern [1998, 2000].”
BackgroundMethodModernizing the Bradford Hill criteria for assessing causal relationships in observational data
Cox, L. A. (2018) · Critical Reviews in Toxicology · doi:10.1080/10408444.2018.1518404 · Cited by 107
“…generally exogeneity constraints specify that some variables have values derived from (i.e. endogenously determined by) the values of others; the values of variables that are not derived from others are determined from outside (i.e. exogenous to) the system being examined (Galles and Pearl 1998).”
BackgroundInfluentialProbabilities Of Causation: Three Counterfactual Interpretations And Their Identification
Pearl, J. (1999) · Synthese · doi:10.1023/a:1005233831499 · Cited by 100
“Glymour (1998) analyzes this example with the aim of identifying the probability P(q) (Cheng’s “causal power”) which coincides with PS (see Lemma 19.”
InfluentialSelf-citationBayesianism and Causality, or, Why I am Only a Half-Bayesian
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Self-citationStructural Counterfactuals: A Brief Introduction
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Self-citationCauses of Effects and Effects of Causes
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“This counterfactual quantity, which Robins and Greenland (1989) named ‘‘probability of causation’’ (PC) and Pearl (2000a:296) named ‘‘probability of necessity’’ (PN), to be distinguished from two other nuances of ‘‘causation,’’ captures the ‘‘but for’’ criterion according to which judgment in favor of a plaintiff shou…”
BackgroundSelf-citationRobins, J. M., & Greenland, S. (2000) · Journal of the American Statistical Association · doi:10.1080/01621459.2000.10474214 · Cited by 69
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“This they refuse to do citing theology (Rubin, 2010) and lack of evidence (Imbens, 2010; Pearl, 2009b); at the price of 6 The potential outcome language has been shown to be logically equivalent to the language of nonparametric structural equations (Galles and Pearl, 1998; Halpern, 1998); a theorem in one is a theorem…”
Self-citationInterventions in Premise Semantics
Santorio, P. (2016) · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · Cited by 56
FROM CAUSAL MODELS TO COUNTERFACTUAL STRUCTURES
Halpern, J. Y. (2012) · The Review of Symbolic Logic · doi:10.1017/s1755020312000305 · Cited by 55
“A proof in AX consists of a sequence of formulas inL, each of which is either an axiom in AX or 3In [Halpern 2000], following [Galles and Pearl 1998], the context was included in the formula.”
BackgroundMethodInfluentialRigour versus the need for evidential diversity
Cartwright, N. D. (2021) · Synthese · doi:10.1007/s11229-021-03368-1 · Cited by 53
“Consider comments from Joshua 6 For more on the relations between (formalised) counterfactual approaches and POEs see Galles and Pearl (1998) and Halpern (1998).”
BackgroundWhat Counterfactuals Can Be TestedPreprint
Shpitser, I., & Pearl, J. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1206.5294 · Cited by 48
Self-citationEvaluating Mode Effects in Mixed-Mode Survey Data Using Covariate Adjustment Models
Vannieuwenhuyze, J., Loosveldt, G., & Molenberghs, G. (2014) · Journal of Official Statistics · doi:10.2478/jos-2014-0001 · Cited by 39
“For that reason, these data lines are called counterfactual (Galles and Pearl 1998, Greenland et al. 1999), but these counterfactuals are, nevertheless, important for the estimation of population means, selection effects, and measurement effects, as will be shown below.”
BackgroundAxiomatizing Causal ReasoningPreprint
Halpern, J. Y. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.cs/0005030 · Cited by 39
“Both the language and models I use are based on those considered by Galles and Pearl (1997, 1998).”
BackgroundMethodInfluentialMatching on Balanced Nonlinear Representations for Treatment Effects Estimation
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Detecting Latent Heterogeneity
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Self-citationEffects of cognitive training on the structure of intelligence
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“…mathematically equivalent to counterfactual or potential-outcomes frameworks more commonly used in randomized controlled trials in their underlying assumptions and implications (Galles & Pearl, 1998). treatment effects, despite the fact that they are between-subjects’ estimates (Weinberger, 2015).”
BackgroundCausal inference in travel demand modeling (and the lack thereof)
Brathwaite, T., & Walker, J. L. (2017) · Journal of Choice Modelling · doi:10.1016/j.jocm.2017.12.001 · Cited by 37
“However, the estimating expressions derived from the “do-calculus” operations on causal graphs have already been shown to sufficient for consistently estimating causal effects (Galles and Pearl, 1998; Huang and Valtorta, 2006).”
BackgroundA Lewisian Logic of Causal Counterfactuals
Zhang, J. (2011) · Minds and Machines · doi:10.1007/s11023-011-9261-z · Cited by 31
“…of (hypothetical) interventions, was vigorously developed and defended, based on Pearl’s seminal and influential work on causal modeling (Pearl 1995, 1998, 2009).1 Logics of such causal counterfactuals have also been studied from an axiomatic perspective (Galles and Pearl 1998; Halpern 2000).”
InfluentialInstrumental Variables: Response to Angrist and Imbens
Heckman, J. J. (1999) · The Journal of Human Resources · doi:10.2307/146419 · Cited by 31
Conditional logic of actions and causation
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“Axiom (MP ) (A > B) → (A → B) has been widely used implicitly or explicitly in the literature on causality [29, 26, 6, 45, 15].”
BackgroundMethodLocal Characterizations of Causal Bayesian Networks
Bareinboim, E., de Brito, C. E. F., & Pearl, J. (2012) · Lecture notes in computer science · doi:10.1007/978-3-642-29449-5_1 · Cited by 30
Self-citationThe algorithmization of counterfactuals
Pearl, J. (2011) · Annals of Mathematics and Artificial Intelligence · doi:10.1007/s10472-011-9247-9 · Cited by 30
“J. Pearl (B) Computer Science Department, University of California Los Angeles (UCLA), Los Angeles, CA, 90095-1596, USA e-mail: judea@cs.ucla.edu Yet, not many students of conditionals asked the next question: How do we, humans, reach such consensus?”
BackgroundInfluentialSelf-citationCausal Inference from Series of Events
Pötter, U. (2001) · European Sociological Review · doi:10.1093/esr/17.1.21 · Cited by 29
The Deductive Approach to Causal Inference
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“(1) In words: The counterfactual Yx(u) in model M is defined as the solution for Y in the “modified” submodel Mx. Galles and Pearl (1998) and Halpern (1998) have given a complete axiomatization of structural counterfactuals, embracing both recursive and non-recursive models (see also Pearl, 2009b,…”
BackgroundSelf-citationUnit Selection Based on Counterfactual Logic
Li, A., & Pearl, J. (2019) · International Joint Conference on Artificial Intelligence · doi:10.24963/ijcai.2019/248 · Cited by 27
Self-citationProbabilities of Causation: Three Counterfactual Interpretations and Their Identification
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Self-citationShen, E., Chou, C., Pentz, M. A., & Berhane, K. T. (2014) · Multivariate Behavioral Research · doi:10.1080/00273171.2014.904221 · Cited by 21
“…effects can be accomplished nonparametrically without the use of sequential ignorability by employing the recursive rule (Bollen, 1989) along with assuming linearity in covariates and parameters for a continuous outcome, no confounding, and no X-by-M interaction (Galles & Pearl, 1998; Pearl 2011).”
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“See Galles and Pearl [1998], Definitions 2 and 3 and Footnote 2, also Pearl [2009] Definitions 7.”
BackgroundTableau calculus for preference-based conditional logics
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“An attractive logic of this kind is the one characterizing the class of recursive causal models [7, 9, 10].”
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“…of causality with similar scope, i.e., Rubin’s potential outcomes framework (Rubin, 2005); however, since the two theories are equivalent in the sense that all the axioms of one theory can be derived from the axioms of the other (Galles and Pearl, 1998), the choice is only a matter of convenience.”
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Didelez, V., & Sheehan, N. A. (2007) · Cited by 17
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“Since intervening on Y has no causal effect on X (Galles and Pearl 1998), it is sufficient to show that for any SCM M ∈ M , one could construct a discrete SCM N ∈ N so that The construction procedure is described as follows.”
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“For LTC, causation has no bearing on similarity, whereas when SCM is expressed in similarity terms, similarity is based on causal relationships represented in causal models (Galles and Pearl 1998).”
BackgroundInfluentialCausal Inference from Big Data: Theoretical Foundations and the Data-fusion Problem
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“Such data are called counterfactuals or potential outcomes (Rubin, 1974; Rosenbaum & Rubin, 1983; Holland, 1988; Galles & Pearl, 1998).”
BackgroundRelative Mode Effects on Data Quality in Mixed-Mode Surveys by an Instrumental Variable
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Mental Causation: A Counterfactual Theory
Kroedel, T. (2019) · Cited by 12
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“Drawing on Gren (2001), the willingness to pay for nutrient load reduction in the Baltic Sea is set to be Rn = =C 6.”
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“Additionally, this semantics can be characterized by a complete set of axioms [22, 23], which can be used as inference rules in the analysis.”
Self-citationThinking About Causation: A Causal Language with Epistemic Operators
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Huber, F. (2014) · Journal of Philosophical Logic · doi:10.1007/s10992-014-9314-x · Cited by 10
“Therefore several authors – Boutilier [4], Darwiche & Pearl [11], and many others (see [23, 24]) – have proposed various constraints in addition to the AGM constraints to guide iterated belief revisions.”
BackgroundExplaining 3D Semantic Segmentation Through Generative AI ‐Based Counterfactuals
Rozajac, D., Lukač, N., Schweng, S., Gollob, C., Nothdurft, A., Stampfer, K., Ser, J. D., & Holzinger, A. (2025) · Expert Systems · doi:10.1111/exsy.70163 · Cited by 11
Observing interventions: a logic for thinking about experiments
Barbero, F., Schulz, K., Velázquez–Quesada, F. R., & Xie, K. (2022) · Journal of Logic and Computation · doi:10.1093/logcom/exac011 · Cited by 10
Causal counterfactuals are not interventionist counterfactuals
Fisher, T. (2016) · Synthese · doi:10.1007/s11229-016-1183-0 · Cited by 10
“Judea Pearl has given perhaps the most influential presentation of a strictlyinterventionist semantics (Galles and Pearl 1998; Pearl 2000).”
BackgroundConditional and Preferential Logics: Proof Methods and Theorem Proving
Pozzato, G. L. (2010) · Cited by 10
Causality from bottom to top: a survey
Weinberg, A. I., Premebida, C., & Faria, D. R. (2025) · Machine Learning · doi:10.1007/s10994-025-06855-5 · Cited by 17
“The notions of interventions and counterfactuals have been rigorously established, clarifying assumptions in ob-3 servational studies [40, 41].”
Gras, N., Dutrénit, G., & Vera‐Cruz, M. (2019) · Innovation and Development · doi:10.1080/2157930x.2019.1567817 · Cited by 9
“See Galles and Pearl (1998) and Halpern (2000) for an axiomatization of this approach, and Cartwright and Reiss (2004) for a comprehensive critique.”
BackgroundFrom Verification to Causality-based ExplicationsPreprint
Christel Baier, A., Dubslaff, C., Funke, F., Jantsch, S., Majumdar, R., Piribauer, J., & Ziemek, R. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2105.09533 · Cited by 9
On the Substitution of Identicals in Counterfactual Reasoning*
Kocurek, A. W. (2019) · Noûs · doi:10.1111/nous.12273 · Cited by 8
Communicating more clearly about deaths caused by air pollution
Cox, L. A. (2019) · Global Epidemiology · doi:10.1016/j.gloepi.2019.100003 · Cited by 9
Uniform Labelled Calculi for Conditional and Counterfactual Logics
Girlando, M., Negri, S., & Sbardolini, G. (2019) · Lecture notes in computer science · doi:10.1007/978-3-662-59533-6_16 · Cited by 8
Comparing Rubin and Pearl’s causal modelling frameworks: a commentary on Markus (2021)
Weinberger, N. (2022) · Economics and Philosophy · doi:10.1017/s0266267121000353 · Cited by 8
“Galles and Pearl (1998) discuss Lewis ’ counterfactual semantics in isolation from his philosophical commitments, and prove that were one (contra Lewis) to interpret claims about the closeness of possible worlds in terms of interventions on variables in causal models, doing so would require no…”
BackgroundA new characterization of the experimental implications of causal Bayesian networks
Tian, J. W., & Pearl, J. (2002) · eScholarship (California Digital Library) · doi:10.5555/777092.777181 · Cited by 8
Self-citationInterventionist Counterfactuals on Causal Teams
Barbero, F., & Sandu, G. (2019) · Electronic Proceedings in Theoretical Computer Science · doi:10.4204/eptcs.286.2 · Cited by 8
“Galles&Pearl ([6]) and Halpern ([10]) provide an axiom system for (a case slightly more general than) recursive structural equation models.”
BackgroundConditional Independence for Iterated Belief Revision
Kern-Isberner, G., Heyninck, J., & Beierle, C. (2022) · International Joint Conference on Artificial Intelligence · doi:10.24963/ijcai.2022/373 · Cited by 9
Reflections on Heckman and Pinto's Causal Analysis After Haavelmo""
Pearl, J. (2013) · doi:10.21236/ada594324 · Cited by 8
“What regression coefficient in Model 2 would reflect the test devised in (e)?”
BackgroundSelf-citationPearl before economists: the book of why and empirical economics
Huntington‐Klein, N. (2022) · Journal of Economic Methodology · doi:10.1080/1350178x.2022.2088085 · Cited by 8
“Any theorem provable in one system is also provable in the other (Galles & Pearl, 1998).”
BackgroundPermissible idealizations for the purpose of prediction
Strevens, M. (2020) · Studies in History and Philosophy of Science Part A · doi:10.1016/j.shpsa.2020.09.009 · Cited by 7
“That is not Hempel’s symmetry thesis, but it has much of its character. The grand old philosopher of science was onto something after all. To conclude, a few words on the purpose of idealization. I have already sketched one such function: to simplify models in order to make them easier to use. Potochnik (2017) has sug…”
BackgroundTemporal Causal Modelling on Large Volume Enterprise Data
Mohan, R., Chaudhury, S., & Lall, B. (2021) · IEEE Transactions on Big Data · doi:10.1109/tbdata.2021.3053879 · Cited by 7
“Having obtained the skeleton and the sepset from adjacency search algorithm [3] we start with orientation of the edges to obtain Complete Partially Directed Acyclic Graph (CPDAG).”
MethodInfluentialCausal counterfactuals without miracles or backtracking
Gallow, J. D. (2022) · Philosophy and Phenomenological Research · doi:10.1111/phpr.12925 · Cited by 7
“Theories of causal counterfactuals like these have been explored by Jackson (1977), Galles & Pearl (1998), Woodward (2003), Kment (2006), Briggs (2012), Huber (2013), and Hiddleston (2005), among others.”
BackgroundCounterlegal dependence and causation’s arrows: causal models for backtrackers and counterlegals
Fisher, T. (2016) · Synthese · doi:10.1007/s11229-016-1189-7 · Cited by 7
“The last few decades have seen a number of authors advance semantic analyses of counterfactuals that make use of causal models (e.g., Galles and Pearl 1998; Pearl 2000; Halpern 2000; Hiddleston 2005; Briggs 2012).”
The Structure Model Interpretation of Wright’s NESS Test
Baldwin, R. A., & Neufeld, E. M. (2003) · Lecture notes in computer science · doi:10.1007/3-540-44886-1_4 · Cited by 7
“Pearl [7,3, 4 ,9] proposes a mathematical language of (graphical) causal models employing structural equations for formalizing counterfactual and causal concepts.”
BackgroundOn Open-Universe Causal ReasoningPreprint
Ibeling, D., & Icard, T. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1907.02170 · Cited by 7
“Theoretical work on causal representation nearly always assumes this restriction (Galles and Pearl, 1998; Halpern, 2000; Spirtes et al., 2000; Zhang, 2013; Pearl, 2009).”
BackgroundMethodResultInfluentialCausality-enhanced system reliability and safety analysis: An overview
Zheng, S., Pan, K., Chen, Y., Liu, J., & Zio, E. (2025) · Reliability Engineering & System Safety · doi:10.1016/j.ress.2025.112109 · Cited by 6
Graphical models, potential outcomes and causal inference: Comment on Linquist and Sobel
Pearl, J. (2011) · NeuroImage · doi:10.1016/j.neuroimage.2011.06.007 · Cited by 6
Self-citationImproving causal determination
Cox, L. A. (2019) · Global Epidemiology · doi:10.1016/j.gloepi.2019.100004 · Cited by 6
Characterizing Counterfactuals and Dependencies over (Generalized) Causal Teams
Barbero, F., & Yang, F. (2022) · Notre Dame Journal of Formal Logic · doi:10.1215/00294527-2022-0017 · Cited by 6
Mode Effects on Variances, Covariances, Standard Deviations, and Correlations
Vannieuwenhuyze, J. (2015) · Journal of Survey Statistics and Methodology · doi:10.1093/jssam/smv009 · Cited by 6
A Characterization of Interventional Distributions in Semi-Markovian Causal Models
Tian, J. W., Kang, C., & Pearl, J. (2006) · eScholarship (California Digital Library) · Cited by 6
Self-citationOn the Role of Counterfactuals in Inferring Causal Effects of TreatmentsPreprint
Kluve, J. (2001) · SSRN Electronic Journal · doi:10.2139/ssrn.283448 · Cited by 6
“The other way around: Quantum mechanics did force philosophers to re-think possible theories of causation and 4 The formal equivalence of POM and recursive SEMs has been established by Galles and Pearl (1998).”
The mathematics of causal inference
Pearl, J. (2011) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/2020408.2020416 · Cited by 6
“Galles and Pearl (1998) and Halpern (1998) have given a complete axiomatization of structural counterfactuals, embracing both recursive and nonrecursive models (see also Pearl, 2009b, Chapter 7).”
BackgroundSelf-citationCounterfactuals as modal conditionals, and their probability
Rosella, G., Flaminio, T., & Bonzio, S. (2023) · Artificial Intelligence · doi:10.1016/j.artint.2023.103970 · Cited by 5
“This connection between counterfactuals and causality has been further investigated to explain causation [37,45], develop a logic for causal reasoning [3,7,28,22], and explore (recursive) causal models [29].”
BackgroundInfluentialCausal modeling semantics for counterfactuals with disjunctive antecedents
Rosella, G., & Sprenger, J. (2023) · Annals of Pure and Applied Logic · doi:10.1016/j.apal.2023.103336 · Cited by 5
A Review of Causal Methods for High-Dimensional Data
Berkessa, Z. A., Läärä, E., & Waldmann, P. (2024) · IEEE Access · doi:10.1109/access.2024.3524261 · Cited by 5
Measuring and mitigating biases in motor insurance pricing
Moriah, M., Vermet, F., & Arthur Charpentier, A. (2024) · European Actuarial Journal · doi:10.1007/s13385-024-00390-8 · Cited by 6
“…m is considered weakly fair if it fulfills the following conditions: These definitions provide a framework for categorizing models as either strongly fair or weakly fair based on their compliance with different fairness criteria, namely demographic parity, equalized odds, and predictive parity.”
Causal Models and the Ambiguity of Counterfactuals
Lee, K. Y. (2015) · Lecture notes in computer science · doi:10.1007/978-3-662-48561-3_18 · Cited by 5
Metadata
- Authors
- Galles, D., & Pearl, J.
- Affiliations
- University of California, Los Angeles (US)
- Venue
- Foundations of Science
- Published
- 1998-01
- Volume
- 3
- Issue
- 1
- Pages
- 151-182
- ISSN
- 1233-1821 (ISSN-L)
- 1572-8471
- OpenAlex
- W1576238869
- Cited by
- 192
- References
- 49
- Citations by year
- 2026: 4 · 2025: 14 · 2024: 11 · 2023: 9 · 2022: 19 · 2021: 13 · 2020: 10 · 2019: 25 · 2018: 5 · 2017: 4 · 2016: 6 · 2015: 10 · 2014: 8 · 2013: 7 · 2012: 10
- FWCI
- 7.48
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- Locations
- Foundations of Science (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Physical Sciences › Computer Science › Artificial Intelligence › Logic, Reasoning, and Knowledge
- Social Sciences › Arts and Humanities › History and Philosophy of Science › Philosophy of Biology and Scientific Explanation
- Keywords
- counterfactuals, composition properties, reversibility, feedback, probabilistic graphical models, causal inference, axiomatization
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