Abstract
Most scientific theories are globally inconsistent. Chunk and Permeate is a method of rational reconstruction that can be used to separate, and identify, locally consistent chunks of reasoning or explanation. This then allows us to justify reasoning in a globally inconsistent theory. We extend chunk and permeate by adding a visually transparent way of guiding the individuation of chunks and deciding on what information permeates from one chunk to the next. The visual representation is in the form of bundle diagrams. We then extend the bundle diagrams to include not only reasoning in the presence of inconsistent information or reasoning in the logical sense of deriving a conclusion from premises, but more generally reasoning in the sense of trying to understand a phenomenon in science. This extends the use of the bundle diagrams in terms of the base space and the fibres. We then apply this to a case in physics, that of understanding binding energies in the nucleus of an atom using together inconsistent models: the liquid drop model and the shell model. We draw some philosophical conclusions concerning scientific reasoning, paraconsistent reasoning, the role of logic in science and the unity of science.
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Change history
14 October 2018
The original version of the book was published with incorrect given and surname for author “M. del Rosario Martínez-Ordaz” in matadata for chapters “The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction” and “Keeping Globally Inconsistent Scientific Theories Locally Consistent” have been corrected. The correction chapters and book have been updated with the changes.
Notes
- 1.
Of course, we do not mean the ‘West’ in the geographical sense. We mean it in the sense of a cultural tradition in science.
- 2.
Not all scientists have a thorough training in formal logic, so they might not be acquainted with the terms ‘classical logic’ or ‘constructive logic’, nevertheless, these are accepted as the cannons of reasoning and rationality by those who do have some training in logic, and their authority is accepted, for reasons of division of labour. It is less fashionable now for scientists to study formal logic. It was more common in the past.
- 3.
Classical and constructive approaches or reasoning styles are the most common in mathematical and scientific practice. We shall discuss an important limitation of the formal representation of such reasoning in the next section.
- 4.
Here we shall focus only on global and local analyses of (in)consistency regarding specific bodies of knowledge. In Sect. 3 we shall introduce the notions of global and local consistency and their representation using bundle diagrams. In Sect. 6.2 we shall briefly introduce the corresponding characterisations of global (in)consistency and local consistency when dealing with inconsistencies in empirical sciences.
- 5.
It is not completely effective, and can only be made so under quite rigid and formal circumstances. See the conclusion for more details about these limitations. Nevertheless, even as an almost effective method, or even as a heuristic in informal reasoning, it will be quite useful.
- 6.
We thank an anonymous reviewer for reminding us that there are supporters of trivialism, and those who think that we can reason sensibly even in a trivial setting. In fact, this is almost what we show using Chunk and Permeate. ‘Almost’ means that there is some ambiguity as to what this means. See some remarks in the conclusion for elaboration. Since our concern is with present-day practice and reasoning in science, we maintain that at present there are no trivialists in science.
- 7.
Some relevant and paraconsistent logicians claim that such mathematicians are actually, as manifested in their reasoning behaviour, relevant or paraconsistent reasoners. This nuance will be addressed in the conclusion.
- 8.
The logicians and mathematicians who disagree with this, who think that ex contradictione quodlibet proofs are invalid, are relevant logicians or paraconsistent logicians. The philosophical difference is that relevant logicians insist on there being a relevant connection between premises and conclusion, paraconsistent logicians think that we can reason coherently with contradictions, or through a contradiction, and they model such reasoning. Briefly, in a paraconsistent logic, while you can derive an infinite number of formulas, as you can from any formula in any logic with a minimum set of inference rules, you cannot derive very much of interest from a contradiction. It is treated as a logical singularity. From \(p \wedge \lnot p\), you can derive p, \(\lnot p\), by \(\wedge \)-elimination, then by \(\wedge \)-introduction, you can derive \((p \wedge \lnot p) \wedge p\) and so on, with double negation introduction you could derive \(\lnot \lnot p\)... The point is that you cannot get to an arbitrary q.
Logically, what distinguishes relevant from other paraconsistent logics is that relevant logicians, as part of the bigger substructural tradition, restrict some structural rules rather than operational ones. Non-relevant paraconsistent logicians change the behavior of the connectives (especially negation) while preserving the full set of structural rules of the language. This guarantees that they stay as close to classical logic as possible (i.e. Priest’s LP).
This second way of putting the distinction reveals an important bias in this paper and for Chunk and Permeate in general: it appeals to a specific kind of logician/mathematician/scientist. Martínez-Ordaz would say that this particular kind of reasoner is one who admits that classical logic is along the right lines and is a good starting point and possibly thinks that formal representations of relevant reasoning sacrifice too much or change the reasoning too much. Chunk and Permeate then appeals to: classical, constructive and some (non-relevant) paraconsistent reasoners (those who think that inconsistency toleration is alright but we should nevertheless reason as consistently as possible).
- 9.
Later in the paper, we shall see that this is exactly what Abramsky recommends.
- 10.
Not all mathematicians at all times finished their work with proofs. In the past, before the twentieth century in Europe, and in the colonies of the European countries, it became wide-spread in the institution of mathematics that results and ideas had to be proved. This is not the case in every mathematical culture, and it has not always been the case in European-based cultures. This is despite the fact that when detailed proofs were given, the proofs in Euclid set the standard for rigour of proof.
- 11.
The ‘none’ is meant as a challenge. The authors know of none that has been published, but of course some might have slipped into the published cannon.
- 12.
The difference is this: if they are using a paraconsistent logic, then they have recourse to a formal representation of the reasoning in the proof. If they are ‘reasoning paraconsistently’ then this is a looser notion. They are reasoning in such a way as to entertain and recognise contradictions but avoid trivialism. Here is the rub: which formal theory best represents their reasoning is usually ambiguous. Their reasoning is represented by a class of formal theories. They are reasoning in the spirit of paraconsistent reasoning in the sense of exercising damage control on the inconsistency.
- 13.
Even though, in principle, one could also have relevant or paraconsistent reasoning within a chunk, we ignore this possibility here out of the respect for the prevailing claims beliefs and practices of working mathematicians. See Priest (2015) for an example of paraconsistent logics within chunks.
- 14.
An interesting question is whether we can use the chunk and permeate strategy on an ex contradictione quodlibet proof. Of course we can, in two different ways: one is to preserve classical validity, so the proof just is a demonstration that anything (written correctly in the formal language) can be derived from inconsistent premises. So the whole proof is one chunk. The second way is to separate the negated reductio inference from the double negation elimination, thus ‘preserving’ consistency within each chunk. A negated reductio inference is one where we conclude the negation of the hypothesis as opposed to the opposite of the hypothesis. If we hypothesise ‘q’, and this leads to a contradiction, then we conclude the negation (and opposite) ‘\(\lnot q\)’. If we hypothesise ‘\(\lnot q\)’, we would conclude the negation, (and not the opposite) ‘\(\lnot \lnot q\)’.
- 15.
It would be nice to make these maximal, but to prove that they are might not be possible. Similarly, to give a method for checking for maximal chunks might not be possible. There might be two \( C \& P\) reconstructions that have the same number, or size, of chunks.
- 16.
- 17.
It would make a nice future project to look into the possibility of more rigorously defining the chunks in this way. Moreover, there promises to be some clean ways of working out what information permeates using the definition of complementary sequent and complementary system. See Piazza and Pulcini (2016) for details. We thank Pulcini for the suggestion in private correspondence.
- 18.
If we are doing formal proof theory, then there is no danger of inconsistency. However, here we are thinking in terms of informal proofs or proofs using suppositions. We move from the model theory story to the proof theory story to respect classical reasoning and constructive reasoning, respectively.
- 19.
We do not know if this might also be due to a reflexive iteration that causes what Dummett calls an ‘indefinitely extensible concept’. This might correspond to the idea that we do not know if we are in an inconsistent cycle, so the edges might spiral up, but we have no way of knowing at any one point if we might then be brought down again. A bundle diagram where the edges spiral upwards indefinitely might represent something like a fractal where a new value is generated as a result of both the formula and the last value or last few values. This is all speculation that requires further investigation. We thank Jean-Paul van Bendegem for asking about spiralling edges.
- 20.
For related notions see [1], p.1.
- 21.
For related notions see [1], p.6.
- 22.
Characterized in [1] as: “there is a local assignment which is in the support, but which cannot be extended to a global assignment which is compatible with the support.” (2015, p.6).
- 23.
Schummer ([48], pp. 64–5) argues convincingly that, especially when considering problems in chemistry, we use the term ‘model’ rather than ‘theory’, since this better reflects the practice of chemists when reasoning about phenomena in chemistry. Of course, here we do not mean model in the model theory sense of the term.
- 24.
- 25.
If we were to be normative, or even prescriptive, about science we could disregard scientific practice and force individuation of scientific theories in order to avoid inconsistency within a scientific ‘theory’. We do not propose to do this, since, as we shall see, this would be quite unnatural to the practice, and of rather limited interest.
- 26.
See ([27]) for related notions of problem solving.
- 27.
Even though almost any scientific theory could be fragmented in infinite ways, here we shall focus only on such subdivisions that are compatible with the way in which scientists use their theories in their standard practice. Henceforth, we shall refer to this way of choosing chunks as separating it into meaningful subsets.
- 28.
In what follows, we shall assume that scientific theories are often individuated following specific problem solving considerations, and that this individuation is often in terms of objects, sets of phenomena, sets of forces acting together, or classes of axiomatic theories, among others.
- 29.
For our bundle diagrams these differences would be drawn out by our choice of variables: be they observations, theories or ideas within a theory.
- 30.
The propositions could be empirical assumptions, observational reports, laws, theorems, axioms, etc.
- 31.
Here a and b could be either distinct theories or distinct meaningful subsets of the same theory.
- 32.
Of course, the union of two locally consistent sets of information need not be consistent with each other.
- 33.
- 34.
The formula is based on the DLM and is used to predict binding energies of nuclei. It is also called “Weizsäcker’s formula”.
- 35.
Note that Tb and Fb are mutually exclusive, and the same goes for Tf and Ff. Nonetheless, the following pairs are mutually compatible: Tb and Tf, Tb and Ff, and Tf and Fb.
- 36.
Note that we have changed the color to indicate that we have moved to the next step in the calculations involving nuclear fission for He4 nuclei. As it is in scientific reasoning, the edges have direction, Sc has to be moved into Eoutput before it is possible to make any prediction regarding nuclear fission. This is new (to the bundle diagram construction) but it is inherent to standard scientific reasoning.
- 37.
Sc is kept as true because it is compatible with the empirical assumptions (Einput), the experimental reports (Exp), and the LDM-predictions for fission, and also because it is now part of the target chunk.
- 38.
DLM-assumptions are next taken as false in cases in which the next move is to predict other properties of nuclei, such as spin and parity of nuclei ground states.
- 39.
Arguably, in more Eastern traditions of ‘science’, contradictions are tolerated, (Garfield: Engaging Buddhism). Such Eastern ‘science’ might not be recognised to be science at all a priori because reasoning with contradictions is a priori impossible.
- 40.
Of course, it does not meta-logically have to, but it happens to.
- 41.
We shall show exactly why this is a presumption, and on what it rests.
- 42.
For a short discussion of this issue where the possibility of a some-places contradictory real physical world, see [21].
- 43.
As for the unity of science, so adently pursued up to the beginning of the twentieth century, what has remained is nothing but a guiding principle, unatainable under the increasing fragmentation of the domains of science.
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Acknowledgements
We are grateful to Diderik Batens, Koen Lefever, Elías Okón Gurvich, Paweł Pawłowski and Elisángela Ramírez-Cámara for the feedback offered throughout the development of this research. Special thanks are deserved to the anonymous referees for their comments and suggestions. The first author would like to acknowledge active support from the Vrije Universiteit Brussel and the Research Stays Program (PREI)-UNAM. The second author was supported by the PAPIIT Projects IA401117 “Philosophical Aspects of Contra-Classical Logics” and IA401717 “Pluralism and Normativity in Logic and Mathematics”.
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Friend, M., Martínez-Ordaz, M.d.R. (2018). Keeping Globally Inconsistent Scientific Theories Locally Consistent. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_4
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