---
title: "The Handling of Loops in Talmudic Logic, with Application to Odd"
description: "The test builds on Caminada's Socratic-discussion procedure, which reasons backwards along attack arrows, and a tableaux… 15 references."
authors: ["Michael Abraham", "Dov M. Gabbay", "Uri J. Schild"]
year: 2018
venue: "EPiC series in computing"
source: "https://doi.org/10.29007/gkg6"
doi: "10.29007/gkg6"
licence: "Author/repository copy, free to read, no open licence — read only (no licence on the copy: author copy, read only; copy from easychair.org)"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/abraham-2018-handling-loops-talmudic-logic/"
---
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# The Handling of Loops in Talmudic Logic, with Application to Odd and Even Loops in Argumentation

**Authors:** [Michael Abraham](https://md.research.elamur.ai/free/en/authors/michael-abraham/), [Dov M. Gabbay](https://md.research.elamur.ai/free/en/authors/dov-m-gabbay/) and [Uri J. Schild](https://md.research.elamur.ai/free/en/authors/uri-j-schild/) · **Year:** 2018 · **Venue:** [EPiC series in computing](https://md.research.elamur.ai/free/en/venues/epic-series-in-computing/) ([EPiC series in computing 2018](https://md.research.elamur.ai/free/en/venues/epic-series-in-computing/2018/)) · **Licence:** Author/repository copy, free to read, no open licence — read only (no licence on the copy: author copy, read only; copy from easychair.org)

**Links:** [Publisher's page](https://doi.org/10.29007/gkg6) · [PDF](https://easychair.org/publications/open/v) · [DOI 10.29007/gkg6](https://doi.org/10.29007/gkg6) · [OpenAlex](https://openalex.org/W2406883936)

**Citations:** 8 citations (OpenAlex, 2026-10-14)

## Summary
The paper looks at how Talmudic reasoning handles cyclic attacks in abstract argumentation networks (S, R). Odd cycles are the hard case. Under standard Dung semantics they usually leave every argument undecided, yet a court still has to reach a ruling. The authors take a principle from Rabbi Shimon Shkop: if assuming an argument is accepted forces the conclusion that it is rejected, then that argument is struck out. They implement this with an added annihilator node. The test builds on Caminada's Socratic-discussion procedure, which reasons backwards along attack arrows, and a tableaux algorithm is given in an appendix. They also add time stamps to arguments and attacks, so that only the newest argument that closes a loop is tested. The motivating case is a conditional divorce that creates an odd loop between three marriage-related claims. Applying the principle rejects the final, self-undermining marriage and leaves a definite extension. The approach is positioned against CF2 and Stage semantics. It gives a principled, history-aware way out of deadlocks where standard semantics returns only "undecided."

## Key points
- Basic contrast: a two-node mutual attack has three complete extensions ({a}, {b}, ∅), while a three-node cycle has only ∅.
- The paper lists three ways to use valuations V : S → U:
  - (a) build V into the extension definition (Amgoud, Bench-Capon), which Caminada criticises;
  - (b) compute extensions first, then pick among them with V (the Talmud; Wakaki 2010);
  - (c) apply V to the internal parts of arguments (Prakken 2010).
- Shkop temporal frames are N = (S, R, T), where T time-stamps both nodes and attacks. The Shkop test for an argument is run on the sub-network that exists at that argument's time.
- Worked examples:
  - the Jane/John/Terry/Frank divorce, where the March 31 marriage to Frank is the argument that gets annihilated;
  - a gift of a computer conditional on later work, where a sale that would cancel its own precondition is disallowed.
- Appendix A's tableaux can check whether several nodes can sit together in one admissible set. Appendix B models future-conditional actions with backward causality as state revision.

## Interesting ideas
- Self-defeat test: tentatively accept a claim, propagate the consequences, and reject the claim if it ends up contradicting itself. This yields a decision in exactly the cases where neutral semantics give only "undecided."
- Time-stamped conflict resolution: when a loop appears, test only the most recently added element, because earlier steps were reasonable given what was known at the time they happened.
- Incremental networks (S_n ⊆ S_{n+1}): resolve loops step by step as each new argument arrives, rather than searching a whole tangled graph for a global fix.
- Conditional actions whose validity depends on a future event (backward causality) are modelled as precondition/postcondition pairs with later cancellation.
- Make the structural reason for a verdict explicit, here as an annihilator node, instead of silently deleting an attack edge.

## For Mova

How these ideas could serve Mova, our small open language model — its files, stages and tests:

- **I1** (world): When the world builder meets cyclic or self-cancelling states in a story, it could run this check and keep an explicit trace of which assumption collapsed.
- **I2** (world): Contradictory facts from later in a narrative could be the ones retracted first, with event order serving as an explainable tie-breaker.
- **I3** (learn): Self-induced repair rules could be added one at a time, and any new rule that creates a contradiction cycle with existing rules would be rejected.
- **I4** (answer): Questions about tales involving bargains or promises ("if you do X by Saturday") could be answered by checking whether a later action makes an earlier transfer void.
- **I5** (frames): When a frame is rejected for being pragmatically self-undermining, attach an explicit "defeated-by-self" marker so the explanation chain stays inspectable.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

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## Cite

Cite as: Michael Abraham, Dov M. Gabbay and Uri J. Schild. 2018. The Handling of Loops in Talmudic Logic, with Application to Odd and Even Loops in Argumentation. EPiC series in computing.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@inproceedings{abraham2018handling,
  title = {{The Handling of Loops in Talmudic Logic, with Application to Odd and Even Loops in Argumentation}},
  author = {Michael Abraham, Dov M. Gabbay and Uri J. Schild},
  booktitle = {EPiC series in computing},
  year = {2018},
  doi = {10.29007/gkg6},
  url = {https://doi.org/10.29007/gkg6}
}
```

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## References

15 references, 1 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

1. M. Abraham, D. Gabbay, and U. Schild. Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal-Vachomer) using Matrix Abduction. Studia Logica, vol 92 , No 3, pp 281–364, 2009. — [Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction](https://md.research.elamur.ai/free/en/papers/abraham-2009-analysis-talmudic-argumentum-fortiori/)
2. M. Abraham, G. Hazut, D. Gabbay, Maruvka and U. Schild. Logical Analysis of the Talmudic Rule of General and Specific (Klal-u-Prat). Special issue on Judaic Logic edited by A Schumann, Journal of the History and Philosophy of Logic, Vol 32 issue 1, pp 47–62, 2011. [find](https://scholar.google.com/scholar?q=M%2E+Abraham%2C+G%2E+Hazut%2C+D%2E+Gabbay%2C+Maruvka+and+U%2E+Schild%2E+Logical+Analysis+of+the+Talmudic+Rule+of+General+and+Specific+%28Klal%2Du%2DPrat%29%2E+Special+issue+on+Judaic+Logic+edited+by+A+Schumann%2C+Journal+of+the+)
3. M. Abraham, D. Gabbay, and U. Schild. Obligations and Prohibitions in Talmudic Deontic Logic. In G. Governatori and G. Sartor, eds., DEON 2010, LNAI 6181, pp. 166–178, 2010. M. Abraham, D. Gabbay and U. Schild The Handling of Loops in Talmudic Logic . . . [find](https://scholar.google.com/scholar?q=M%2E+Abraham%2C+D%2E+Gabbay%2C+and+U%2E+Schild%2E+Obligations+and+Prohibitions+in+Talmudic+Deontic+Logic%2E+In+G%2E+Governatori+and+G%2E+Sartor%2C+eds%2E%2C+DEON+2010%2C+LNAI+6181%2C+pp%2E+166%E2%80%93178%2C+2010%2E+M%2E+Abraham%2C+D%2E+Gabbay+and+)
4. M. Abraham, D. Gabbay, and U. Schild. Contrary to Time Conditionals in Talmudian Law. Journal of Artificial Intelligence and Law, 20:2, 145–179, 2012. Submitted to J of AI and Law. [find](https://scholar.google.com/scholar?q=M%2E+Abraham%2C+D%2E+Gabbay%2C+and+U%2E+Schild%2E+Contrary+to+Time+Conditionals+in+Talmudian+Law%2E+Journal+of+Artificial+Intelligence+and+Law%2C+20%3A2%2C+145%E2%80%93179%2C+2012%2E+Submitted+to+J+of+AI+and+Law%2E)
5. M. Abraham, D. Gabbay, and U. Schild. Obligations and Prohibitions in Talmudic Deontic Logic. Journal of Artificial Intelligence and Law, 19:2, 117–148, 2011. [find](https://scholar.google.com/scholar?q=M%2E+Abraham%2C+D%2E+Gabbay%2C+and+U%2E+Schild%2E+Obligations+and+Prohibitions+in+Talmudic+Deontic+Logic%2E+Journal+of+Artificial+Intelligence+and+Law%2C+19%3A2%2C+117%E2%80%93148%2C+2011%2E)
6. D. Gabbay. Equational Approach to Argumentation Networks. Argument and Computation, 3:2-3, 87-142, 2012. [find](https://scholar.google.com/scholar?q=D%2E+Gabbay%2E+Equational+Approach+to+Argumentation+Networks%2E+Argument+and+Computation%2C+3%3A2%2D3%2C+87%2D142%2C+2012%2E)
7. S. A. Gaggl and S. Woltran. Strong Equivalence for Argumentation Semantics Based on Conflict-Free Sets, slides of lecture presented in ECSQARU 2011. [find](https://scholar.google.com/scholar?q=S%2E+A%2E+Gaggl+and+S%2E+Woltran%2E+Strong+Equivalence+for+Argumentation+Semantics+Based+on+Conflict%2DFree+Sets%2C+slides+of+lecture+presented+in+ECSQARU+2011%2E)
8. H. Prakken. An abstract framework for argumentation with structured arguments Argument & Computation, 2010. [find](https://scholar.google.com/scholar?q=H%2E+Prakken%2E+An+abstract+framework+for+argumentation+with+structured+arguments+Argument+%26+Computation%2C+2010%2E)
9. M. Caminada. Preferred Semantics as Socratic Discussion, [http://homepages.abdn.ac.uk/martin.caminada/pages/publications/preferred_game.pdf](http://homepages.abdn.ac.uk/martin.caminada/pages/publications/preferred_game.pdf), 2010. [↗](http://homepages.abdn.ac.uk/martin.caminada/pages/publications/preferred_game.pdf)
10. T. Wakaki. Preference-based Argumentation Capturing Prioritized Logic Programming, ArgMAS, 2010. [find](https://scholar.google.com/scholar?q=T%2E+Wakaki%2E+Preference%2Dbased+Argumentation+Capturing+Prioritized+Logic+Programming%2C+ArgMAS%2C+2010%2E)
11. P. Baroni and M. Giacomin. Solving semantic problems with odd-length cycles in argumentation. In Proceedings of the 7th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2003), pp. 440–451. LNAI 2711, Springer-Verlag, Aalborg, Denmark, 2003. [find](https://scholar.google.com/scholar?q=P%2E+Baroni+and+M%2E+Giacomin%2E+Solving+semantic+problems+with+odd%2Dlength+cycles+in+argumentation%2E+In+Proceedings+of+the+7th+European+Conference+on+Symbolic+and+Quantitative+Approaches+to+Reasoning+with+Un)
12. G. A. Bodanza and F. A. Tohmé. Two approaches to the problems of self-attacking arguments and general odd-length cycles of attack. Journal of Applied Logic, 7, 403–420, 2009. [find](https://scholar.google.com/scholar?q=G%2E+A%2E+Bodanza+and+F%2E+A%2E+Tohme%CC%81%2E+Two+approaches+to+the+problems+of+self%2Dattacking+arguments+and+general+odd%2Dlength+cycles+of+attack%2E+Journal+of+Applied+Logic%2C+7%2C+403%E2%80%93420%2C+2009%2E)
13. P. Baroni, M. Giacomi and G. Guida. SCC-recursiveness: a general schema for argumentation semantics. Artificial Intelligence, 168 (1-2):162–210, 2005. [find](https://scholar.google.com/scholar?q=P%2E+Baroni%2C+M%2E+Giacomi+and+G%2E+Guida%2E+SCC%2Drecursiveness%3A+a+general+schema+for+argumentation+semantics%2E+Artificial+Intelligence%2C+168+%281%2D2%29%3A162%E2%80%93210%2C+2005%2E)
14. Bart Verheij. A Labelling approach to to the computation of credulous acceptance in argumentation. In Proceedings IJCAI’07. Proceedings of the 20th international joint conference on Artifical intelligence, 623–628. Morgan Kaufmann Publishers Inc. San Francisco, CA, USA, 2007. [find](https://scholar.google.com/scholar?q=Bart+Verheij%2E+A+Labelling+approach+to+to+the+computation+of+credulous+acceptance+in+argumentation%2E+In+Proceedings+IJCAI%E2%80%9907%2E+Proceedings+of+the+20th+international+joint+conference+on+Artifical+intellig)
15. Martin Caminada. A Labelling Approach for Ideal and Stage Semantics. Argument and Computation, 2 (1): 1–21, 2011. M. Abraham, D. Gabbay and U. Schild [find](https://scholar.google.com/scholar?q=Martin+Caminada%2E+A+Labelling+Approach+for+Ideal+and+Stage+Semantics%2E+Argument+and+Computation%2C+2+%281%29%3A+1%E2%80%9321%2C+2011%2E+M%2E+Abraham%2C+D%2E+Gabbay+and+U%2E+Schild)

*Updated 4 November 2026*