Sorry, no online dictionaries contain the exact phrase multi-premise syllogisms.
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Reverse dictionary results:
1. ergo
2. logic
3. polysyllogism
4. syllogistic
5. syllogistical
6. brandish
7. brandishing
8. ratiocinate
9. sorites
10. syllogise
11. syllogization
12. syllogize
13. abduction
14. antecedent
15. antilogism
16. argument
17. aristotelian logic
18. assumption
19. bambara
20. barbara
21. baroko
22. bocardo
23. conclusion
24. consequence
25. demonstration
26. dilemma
27. disjunctive
28. elenchus
29. enthymeme
30. epicheirema
31. epichirema
32. episyllogism
33. expository
34. extreme
35. fallacy
36. figure
37. figuring
38. formalism
39. granted
40. hence

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