For Want of an 'And': A Puzzle about Non-Conservative Extension

History and Philosophy of Logic 26 (3):229-266 (2005)
Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free formula. Thus it would seem that any detour going via formulas with conjunction can be avoided, which raises a puzzle: how is this consistent with the point from Section 1? Sections 3 and 4 raise and discuss this puzzle. In fact, the puzzle turns out on closer inspection not to be so puzzling after all, but it does serve as a convenient centrepiece around which to organize a discussion of the phenomenon illustrated by the Bull?Prior example. Section 5 notes that Prior's observation can be extended to the case of the result of adding disjunction to Bull's logic, while Section 6 includes some further remarks aimed at diagnosing one source of possible residual puzzlement. A subtext of our discussion?spanning several of the notes?is that this work by Bull and Prior has been overlooked, their results having to be rediscovered, by many algebraists and logicians in more recent years
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DOI 10.1080/01445340412331325140
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References found in this work BETA
Timothy Smiley (1962). The Independence of Connectives. Journal of Symbolic Logic 27 (4):426-436.
A. N. Prior (1964). Two Additions to Positive Implication. Journal of Symbolic Logic 29 (1):31-32.

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Citations of this work BETA
Lloyd Humberstone (2013). Replacement in Logic. Journal of Philosophical Logic 42 (1):49-89.
Lloyd Humberstone (2014). Prior’s OIC Nonconservativity Example Revisited. Journal of Applied Non-Classical Logics 24 (3):209-235.

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