Philosophical Institute of Czech Academy of Sciences PhDs in Logic, Prague May 2, 2018
Plurality of logics as philosophical problem There are many logical systems, yet it is not clear what this fact tells us about logic itself Are more logics equally legitimate? Can logical laws be binding in such a situation? Or is there just one logic and the various systems are not really logics in the strict sense? What is their role, then?
Concepts as constitued by rules One can see a concept as constituted by inference rules which specify a use of a corresponding expression Thus the concept of dog is constituted, among other rules, by the rule saying that every dog is a mammal The rules in general can remain implicit in our practice, e.g., when we approve inferences done according to them and criticize those violating it They can, on the other hand, be also rendered explicit
Necessity of implicit rules Concepts can be explicit to different degrees Yet some implicitness is present in our use of every concept Take the concept of dog again - is having lungs a necessary part of the concept of a dog? New applications of a concept in new contexts can reaveal what was only implicit so far
Dog and lungs Imagine we encounter animals which are like dogs in almost all respects They just lack lungs Will we claim that these animals are dogs? It appears that neither answer is better, we are just free to decide
Having a concept Yet, of course, in most cases the issue whether all dogs have lungs does not arise The principle that they have is relatively robust Some people may be more inclined to say that all dogs have lungs Yet this hardly means that these people have a different concept of dog or lungs If the differences between the two people were more numerous and essential, it could be meaningful to say that they have two different concepts of dog Yet such cases of people sharing just an expression but not the corresponding concept have to be mere exceptions - otherwise we could not share a language
Change of a concept By change of the constitutive rules a concept can change Such a change can either happen under our control or spontaneously As it is always to some degree open what the original rules were, it is not always clear whether a real change happened or how great it was Despite this fuziness, it certainly makes sense to maintain that the concept still is numerically identical, even if some of the constitutive rules change When the changes are too radical, it can make sense to say that the concept is not the same anymore - it has then not developed, it has been rather replaced
Making the inference rules explicit by means of logic According to logical expressivism, inference rules are rendered explicit by means of logic We have logical instruments in natural languages, namely their logical vocabulary With them, we have the corresponding concepts of negation, conjunction, conditonal etc. Furthermore, we also have the concept of making inference rules explicit - the concept of expressive rationality for short
Constitutive rules of logical concepts The logical concepts are also constituted by inference rules Not all of these rules are fully explicit, just as those of extra-logical concepts They can thus be spelled out in multiple ways - which is done by various formal logics This already opens a pathway to explanation of plurality of logics
Change of logical concepts As any concept can change, so also the logical ones Logical concepts can change without our notice, just as any other ones The question is whether we can also make them explicit and thus change them by means of a rational discussion The problem is that logic would have to make itself explicit
Making logic explicit We can make some of these rules explicit by using other logical rules It is like repairing the Neurathian ship - only a very limited part can be repaired at time and only slowly and cautiously What the various formal logics present is nevertheless only a possible form our expressive rationality could attain They thus point to possible modifications of our expressive rationality
Implementation of rules A change of some inference rules we agree on can prove difficult to realize Thus there still is a long path from the proposal to its implementation This holds much more for logical rules The change of practice of making inference rules explicit thus happens very slowly
Development of logic Logic as the instrument of our expressive rationality thus can change Rather than saying that different people or cultures have different logic, we can say that they share one logic which they develop The development attained new possibilities with the plurality of logical systems
Example of implicitness and potential to develop Just as the concepts of dog and lungs do not determine whether all dogs have lungs, so the concept of disjunction and negation do not determine, whether the law of excluded middle holds Of course, in most cases it holds, just as most dogs have lungs This again does not mean that people with different opinions about the law have different concepts of disjunction and negation We can discuss the reasons for and against acceptance of the law
Developing the logical systems The various logical systems can be seen as proposals of modification of expressive rationality Yet any such proposal can inspire other proposals Besides the actual rules of expressive rationality, the logical systems can also develop One logical system can inspire another one, a combination of virtues of two systems can become a goal for someone etc.
Enrichment of expressive rationality Not only are the logical systems capable of modifying our logical concepts, they can also bring new ones Thus one logic can develop a concept of conjunction into two different conjunctions This is the case in some substructural logics Also the generalized quantifiers can be seen as an enrichment of our expressive capacities
Logical dynamism My position is best called logical dynamism It reconciles logical monism with logical pluralism There is some truth to both of the doctrines
Monist side of logical dynamism A concept is constituted by rules which hold thanks to our normative attitudes The rules cannot be private Any concept, including the logical ones thus has to be shared We can speak of sharing the concept even if we do not agree about it completely
Identity of concepts Sometimes it is just a matter of terminology whether we say that two people have different concepts or whether they just have different opinions on the same concept Logical concepts enable the rational use of language in the first place Thus there is a strong preference to say that we share the logical concepts Thanks to that we also share one logic
Pluralist side of logical dynamism Having one logic, there are many ways we can modify it We can decide to endorse the law of the excluded middle in all contexts We can also decide that it is better to reckon with exceptions to this rule Thus the one expressivist rationality enables the plurality of logical systems as proposals of modification This pluralist line in my view is, nevertheless, very far from relativism - logic is something we share, inluding its dynamic character which makes it never complete Validity of a given logical law is not relative to a given context or speaker or community
Rivalry between logics There is indeed a rivalry between various logics We can compare the logical systems as to their assets and drawbacks Not only the motivation or scope of logics but also what we find about the systems in the course of investigation can be relevant Thus a given logic can be, e.g., relatively unnatural, yet have neat technical properties such as completeness, compactness etc.
The best logic? Now there seems to be little argument available to support the thesis that some specific logic is clearly the best There does not seem to be any hiearchy of the criteria according to which we should assess logics Should logic rather be simple, or expressive, is completeness necessary? etc. Reasonable pluralism can mean exactly this - there is no clearly best logic This does not mean that all logics are equal, some can be better than others
The best logic - maybe but.. It is entirely possible that one logic will prove to have by far the best properties It might even prove to be easily implemented by education But still it will never be the one true logic It is essential for our expressive rationality that it is dynamic and open to further change