Arthur N. Prior on the Labours of Ł3 Conjunctions

ABSTRACT

In ‘Many-valued Logics’, a lecture broadcast over New Zealand's public radio in 1957, Arthur N. Prior (1914–1969) complained that conjunctions are put ‘to something like forced labour’ in Łukasiewicz's three-valued semantics, Ł3. In this paper, we discuss what Prior might have meant by this.

KEYWORDS:

• Ł3 semantics
• inferentialism
• self-contradiction
• logical conjunctions
• truth-functionality

Acknowledgments

We thank Hazel T. Biana, Ben Blumson, Brian Garrett, and the editor and referees of this journal for the useful suggestions that greatly improved this paper.

Notes

1 A portion of Prior's lecture can be accessed through http://www.ngataonga.org.nz/collections/catalogue/catalogue-item?record_id=168375. The lecture starts at the 15:17 mark.

2 Prior 1957, p. 717.

3 Prior 1957, p. 718.

4 Prior 1957, p. 718.

5 Prior 1957, p. 718.

6 Prior 1957, p. 718.

7 Compare the discussion here with Priest 2008, pp. 122–124, and Sider 2010, pp. 75–77.

8 The designated values are the only values preserved in valid arguments, see Priest 2008, p. 121.

9 A corresponding definition of logical truths in Ł3 can be derived from this.

10 The idea of truth-functionality employed here follows the standard textbook definition as seen, for example, in Beall and Logan 2017, p. 27, and Sider 2010, p. 84.

11 Prior 1960.

12 For purposes of an interpretation of this logic, we do not need to distinguish those atomic sentences that are necessarily true or necessarily false.

13 Prior 1962, p. 5.

14 We follow Prior's practice of using Polish (prefix) notation here. As such, ‘Kpq’ is the prefix notation for ‘$p\wedge q$’. For a discussion of Prior's preference for Polish notation, see Joaquin 2022.

15 Prior 1967, p. 136.