Abstract
I argue in favour of object languages of logics to be diversely-generated, that is, not having identical (or equivalent) immediate sub-formulas. In addition to diversely-generated object languages constituting a more appropriate abstraction of the use of sentential connectives in natural language, I show that such language lead to a simplifications w.r.t. some specific issues: the identity of proofs, the factual equivalence (in logics of grounding) and the Mingle axiom in Relevance logics. I also point out that some of the properties of classical logic based on freely-generated object languagest.
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Notes
For better readability, I use ‘\(*\)’ in infix notation.
The context is quantum physics propositions, one dealing with the position of some corpuscle and the other with its speed.
Known also as structurality.
All natural language expressions are displayed in the san-serif font, alwyas mentioned, not used.
There is an alternative point of view, according to which ‘\(\rhd \)’ is itself a sentential operator that can be iterated when constructing grounding judgement. This view gives rise to meta-grounding (see, for Example Rabin and Rabern 2016). I will not explore this other view in more detail, as it is orthogonal to the issue of diversification.
I take here grounds to constitute what is known as full and immediate grounds. The issue of a partial ground is orthogonal to the current concerns.
The coinciding with the notation usually used for the meaning of \(\varphi \) is not accidental.
As a referee observed, this observation depends on the set of (functionally complete) connectives used to present classical propositional logic. It holds for the presentation with \(\{\lnot , \wedge , \vee , \supset \}\) used here, but would not hold for a presentation with \(\{\supset ,\bot \}\), as \(\bot \) is not contingent.
Note that this is independent of the number of truth values employed.
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Francez, N. Diversification of Object-Languages for Propositional Logics. J of Log Lang and Inf 27, 193–203 (2018). https://doi.org/10.1007/s10849-018-9266-6
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DOI: https://doi.org/10.1007/s10849-018-9266-6