Abstract
In this paper I lay some of the groundwork for a naturalistic, empirically oriented view of logic, attributing the special status of our knowledge of logic to the power of stipulation and expressing the stipulations that constitute the vocabulary of formal logic by rules of inference. The stipulation hypothesis does nothing by itself to explain the usefulness of logic. However, though I do not argue for it here, I believe the selective adoption and application of stipulations can. My concern here is with an issue that has already received a good bit of attention: it seems that we are free to make whatever stipulations we care to make, but we also know that logical stipulations must be carefully constrained, to avoid trivialization, as well as subtler impositions on the already established inferential practices to which we apply our logical vocabulary. I propose three increasingly stringent criteria that fully conservative extensions of a language should meet, and apply them to evaluate three symmetrical, multiple-conclusion logics. A new result, proven first for classical multiple-conclusion logics and then modified and extended to all reflexive, monotonic, and transitive consequence relations, undergirds the focus on proof-theoretic approaches to the consequence relation I adopt here.
Andrew Tedder was my research assistant when the research presented here was begun. He provided both LaTeX expertise and helpful discussions during early stages of the project, and was coauthor of two conference presentations of that work at the Society for Exact Philosophy meetings of 2012 and 2013.
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Notes
- 1.
This is not to say that some refinements of that practice may not result: as principles of arithmetic are applied, new and reliable ways of determining counts can emerge and become part of the practice. But the practice of counting predates the emergence of formal theories of arithmetic; when we add a theory of arithmetic to our rules for counting and comparing quantities of things, we want the stipulations that theory embodies to be conservative: it should not conflict with our established counting practices.
- 2.
Failed because iteration of such a connective to form n-ary ‘disjunctions’ gives a formula which is true iff the number of sentences ‘disjoined’ is odd, while a natural exclusive disjunction produces a sentence true iff exactly one of the disjoined sentences is true.
- 3.
In conversation.
- 4.
In legal and scientific contexts, the use of natural language is more regimented, and often highly redundant; I take this as evidence of efforts to achieve more uniform, shared understanding by means of regimentation and repetition.
- 5.
The inference here is from the denial of “I see cups or saucers” to the denial of “I see cups” and “I see saucers.” In classical logic we can translate such reasoning into one that runs from assertions of negations as premises to a disjunction of negations that expresses our conclusion as an assertion. However, such translations are awkward, and may simply fail to work if we are dealing with a nonclassical negation.
- 6.
Similarly, the laws of classical physics are time-symmetric, but this symmetry allows for substantial matter-of-fact asymmetries in the course of actual events.
- 7.
Raatikainen does recognize the potential of an appeal to multiple-conclusion logic here. His response is to raise concerns about how one could rule out the possibility of sentences being both true and false; we will not pursue his discussion any further here, though I do not see how this response helps to support a semantic as opposed to a proof-theoretic perspective on logic, since paraconsistent logics provide both proof theory and semantics tolerating such assignments.
- 8.
This use of ‘transverse’ derives from the use of hypergraph transverses in the semantics of weakly aggregative logics [13, pp. 50–51].
- 9.
The template is applied on the left by conjoining the edges and then disjoining the conjunctions, and on the right by disjoining the edges and then conjoining the disjunctions.
- 10.
This is the usual formal understanding of a conservative extension [1].
- 11.
Intuitively, this constraint rejects a bivalence condition; see again [6, p. 221f]. I propose to reconcile this condition with the bivalence of Scott-Lindenbaum semantics by suggesting that we take a sentence p to be ‘unsettled’ at a 1 / 0 valuation v iff \(v(p)=0\) and there is a valuation \(v*\) that monotonically extends the set of sentences assigned the value 1 at v such that \(v*(p) = 1\).
- 12.
This condition is imposed because the consequence relation of L may not be determined by valuations of L that are both 1 and 0-maximal, where a valuation V is 1 (0) maximal iff there is no other valuation \(V'\) that gives the value 1 (0) to a proper superset of those assigned 1 (0) by V. For example, a language interpreted in epistemic terms might interpret 1 as known and 0 as not known for a non-omniscient being. Since the maximal ‘known’ assignments overlap all the maximal ‘not-known’ assignments, the consequence relation is not determined by what we might call the ‘dual-maximal’ assignments.
- 13.
This is why the familiar classical tautologies and contradictions always get the values 1 and 0, respectively: every extension of a partial assignment to a complete one winds up imposing 1 (0) as the value of a tautology (contradiction), even when some or all of its atoms wind up as ‘gaps.’
- 14.
A cut-free proof theory for bi-intuitionistic logic can be found in [4].
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Brown, B. (2015). Stipulation and Symmetrical Consequence. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_16
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