Periodic Reporting for period 2 - ConLog (Contradictory Logics: A Radical Challenge to Logical Orthodoxy)
Okres sprawozdawczy: 2023-04-01 do 2024-09-30
The ConLog project investigates logical systems in which contradictions are provable. These are systems in which for some statements not only the statement but also its negation is provable. Such logics are called "negation inconsistent" or "contradictory", and they are required to be non-trivial, i.e. they must not be absolutely inconsistent in the sense of validating arbitrary statements.
The project radically breaks with a time-honored Aristotelian tradition and assumes that it is theoretically rational to work with logics that, while being non-trivial, admit provable contradictions. According to classical logic, contradictions have a trivializing effect on logical inference. With classical logic, any statement whatsoever logically follows from contradictory assumptions: ex contradictione quodlibet. The 20th century already has seen a change of perspective on contradictions, first among a minority of philosophical logicians and later also among computer scientists dealing with the processing of inconsistent information. As of the 1940s, inconsistency-tolerant, “paraconsistent” logics have been developed in which contradictory premises do not trivialize reasoning. Still, these logics do not allow for provable contradictions. More recently it has been observed that there indeed exist non-trivial contradictory logics that are not at all artifacts and have motivations completely independent of exploring the very possibility of non-trivial negation inconsistent logics.
The overall objectives of the project are (i) to investigate the formal properties of various non-trivial contradictory logics and to thereby obtain insights into their negation inconsistency, (ii) to see whether or not the inferential patterns and axiomatic principles that lead to provable contradictions are endorsed by speakers of natural languages, and (iii) to inquire into the consequences working with non-trivial negation inconsistent logics has or may have within the philosophy of logic. To achieve its goals, the ConLog researchers make use of advanced proof-theoretic and model-theoretic methods and techniques, methods from experimental philosophy of language and linguistics, and detailed conceptual analyses.
If the project achieves its aims, our attitude towards contradictions in scientific theories and theory formation will have to be revised or at least heavily nuanced. We have to distinguish the provable contradictions we have to accept from those that indeed hamper our theorizing.
The project radically breaks with a time-honored Aristotelian tradition and assumes that it is theoretically rational to work with logics that, while being non-trivial, admit provable contradictions. According to classical logic, contradictions have a trivializing effect on logical inference. With classical logic, any statement whatsoever logically follows from contradictory assumptions: ex contradictione quodlibet. The 20th century already has seen a change of perspective on contradictions, first among a minority of philosophical logicians and later also among computer scientists dealing with the processing of inconsistent information. As of the 1940s, inconsistency-tolerant, “paraconsistent” logics have been developed in which contradictory premises do not trivialize reasoning. Still, these logics do not allow for provable contradictions. More recently it has been observed that there indeed exist non-trivial contradictory logics that are not at all artifacts and have motivations completely independent of exploring the very possibility of non-trivial negation inconsistent logics.
The overall objectives of the project are (i) to investigate the formal properties of various non-trivial contradictory logics and to thereby obtain insights into their negation inconsistency, (ii) to see whether or not the inferential patterns and axiomatic principles that lead to provable contradictions are endorsed by speakers of natural languages, and (iii) to inquire into the consequences working with non-trivial negation inconsistent logics has or may have within the philosophy of logic. To achieve its goals, the ConLog researchers make use of advanced proof-theoretic and model-theoretic methods and techniques, methods from experimental philosophy of language and linguistics, and detailed conceptual analyses.
If the project achieves its aims, our attitude towards contradictions in scientific theories and theory formation will have to be revised or at least heavily nuanced. We have to distinguish the provable contradictions we have to accept from those that indeed hamper our theorizing.
Main results that have been obtained so far include the following achievements.
1. Non-trivial contradictory logics have been put into place in the broader picture of inconsistency-tolerant paraconsistent logics and set apart from the latter.
2. A particularly natural and well-defined non-trivial contradictory logic is the system C of connexive logic. Deeper insights into the negation inconsistency of C could be obtained by identifying both necessary and sufficient conditions for the provability of contradictions in C.
3. A proof that the valid inferences of C are indeed matched by derivations in proof systems for C was so far available only by means of a translation into a certain positive, negation-free logic. While this is revealing conceptually, there nevertheless was a research gap that could be filled by giving a detailed direct completeness proof for C in a language with the quantifiers "for all" and "for some".
4. Additional motivation for C could be obtained by (a) identifying desirable principles, involving quantifiers, that are not valid in classical logic but valid in C and (b) showing the possibility to add nonstandard quantifiers that coincide with independently introduced non-standard quantifiers in certain logics which only recently have been noticed to be negation-inconsistent.
6. A systematic understanding of the negation inconsistency of certain logics could be could be gained by considering a logical bilateralism that takes proofs and refutations on a par. Such a bilateralism enables the adoption of unfamiliar falsification conditions of complex statements.
7. It could be experimentally verified that not only certain schematic principles but indeed also certain inferential patterns that naturally lead to provable contradictions find and endorsement of natural language speakers.
8. Groundwork could be done for further investigations of the model-theoretic semantics of contradictory logics much weaker than C.
1. Non-trivial contradictory logics have been put into place in the broader picture of inconsistency-tolerant paraconsistent logics and set apart from the latter.
2. A particularly natural and well-defined non-trivial contradictory logic is the system C of connexive logic. Deeper insights into the negation inconsistency of C could be obtained by identifying both necessary and sufficient conditions for the provability of contradictions in C.
3. A proof that the valid inferences of C are indeed matched by derivations in proof systems for C was so far available only by means of a translation into a certain positive, negation-free logic. While this is revealing conceptually, there nevertheless was a research gap that could be filled by giving a detailed direct completeness proof for C in a language with the quantifiers "for all" and "for some".
4. Additional motivation for C could be obtained by (a) identifying desirable principles, involving quantifiers, that are not valid in classical logic but valid in C and (b) showing the possibility to add nonstandard quantifiers that coincide with independently introduced non-standard quantifiers in certain logics which only recently have been noticed to be negation-inconsistent.
6. A systematic understanding of the negation inconsistency of certain logics could be could be gained by considering a logical bilateralism that takes proofs and refutations on a par. Such a bilateralism enables the adoption of unfamiliar falsification conditions of complex statements.
7. It could be experimentally verified that not only certain schematic principles but indeed also certain inferential patterns that naturally lead to provable contradictions find and endorsement of natural language speakers.
8. Groundwork could be done for further investigations of the model-theoretic semantics of contradictory logics much weaker than C.
The results obtained so far clearly go beyond the state of the art in philosophical logic. Non-trivial contradictory logics have been placed on the agenda of non-classical logic, new motivation for certain negation inconsistent logics could be given, novel proof-theoretical techniques have been developed for the classification of provable contradictions, and additional experimental evidence could be gained for the endorsement of inferential patterns that are crucial in obtaining contradictory logics. Substantial groundwork for the model-theoretic study of certain negation inconsistent logics could be completed.
The next steps include further experimental work on inference patterns that lead to or display contradictions. On the proof-theoretic side, it is expected that the methods that have been developed can be applied the other logical systems and vocabularies. On the model-theoretic side, the semantical bilateralism that draws an explicit distinction between the support of truth and the support of falsity that a statement can receive at an information state will be extended to a general semantical framework suitable for capturing very weak contradictory logics. As to the philosophy of logic, it is expected that the project obtains a clear understanding of what working with a negation inconsistent logic means against the background of a plurality of logical systems for theory formation and for the status logic has among the sciences.
The next steps include further experimental work on inference patterns that lead to or display contradictions. On the proof-theoretic side, it is expected that the methods that have been developed can be applied the other logical systems and vocabularies. On the model-theoretic side, the semantical bilateralism that draws an explicit distinction between the support of truth and the support of falsity that a statement can receive at an information state will be extended to a general semantical framework suitable for capturing very weak contradictory logics. As to the philosophy of logic, it is expected that the project obtains a clear understanding of what working with a negation inconsistent logic means against the background of a plurality of logical systems for theory formation and for the status logic has among the sciences.