*Maureen T. Carroll and Steven T. Dougherty*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.003.0013
- Subject:
- Mathematics, History of Mathematics

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the ...
More

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the surprisingly interesting problem of arranging thirty-six officers of six different ranks and regiments. In his 1782 paper on the subject, Euler begins with the thirty-six-officer problem and ends with a conjecture about the possible sizes of Graeco-Latin squares. The chapter first explains the rules for a game based on Euler's work, and then analyzes it from a game-theoretic perspective to determine winning and drawing strategies. Along the way, the chapter explains Euler's connection to the story.Less

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the surprisingly interesting problem of arranging thirty-six officers of six different ranks and regiments. In his 1782 paper on the subject, Euler begins with the thirty-six-officer problem and ends with a conjecture about the possible sizes of Graeco-Latin squares. The chapter first explains the rules for a game based on Euler's work, and then analyzes it from a game-theoretic perspective to determine winning and drawing strategies. Along the way, the chapter explains Euler's connection to the story.

*Neil Tennant*

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780198777892
- eISBN:
- 9780191823367
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198777892.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

We compare Tarski’s notion of logical consequence (preservation of truth) with that of Prawitz (transformability of warrants for assertion). The latter is our point of departure for a definition of ...
More

We compare Tarski’s notion of logical consequence (preservation of truth) with that of Prawitz (transformability of warrants for assertion). The latter is our point of departure for a definition of consequence in terms of the transformability of truthmakers (verifications) relative to all models. A sentence’s Tarskian truth-in-M coincides with its having an M-relative truthmaker. An M-relative truthmaker serves as a winning strategy or game plan for player T in the ‘material game’ played on that sentence against the background of the model M. We enter conjectures about soundness and completeness of Classical Core Logic with respect to the notion of consequence that results when the domain is required to be decidable. We consider whether the truthmaker semantics threatens a slide to realism. We work with examples of core proofs whose premises are given M-relative truthmakers; and show how these can be systematically transformed into a truthmaker for the proof’s conclusion.Less

We compare Tarski’s notion of logical consequence (preservation of truth) with that of Prawitz (transformability of warrants for assertion). The latter is our point of departure for a definition of consequence in terms of the transformability of truthmakers (verifications) relative to all models. A sentence’s Tarskian truth-in-M coincides with its having an M-relative truthmaker. An M-relative truthmaker serves as a winning strategy or game plan for player T in the ‘material game’ played on that sentence against the background of the model M. We enter conjectures about soundness and completeness of Classical Core Logic with respect to the notion of consequence that results when the domain is required to be decidable. We consider whether the truthmaker semantics threatens a slide to realism. We work with examples of core proofs whose premises are given M-relative truthmakers; and show how these can be systematically transformed into a truthmaker for the proof’s conclusion.