# What justifications have been given for using particular systems of logical calculus?

I think that using any particular system of logical calculus should be properly justified.

This justification should be seen as particularly important and pressing in science and technology, and possibly elsewhere, such as in strategic and geopolitical studies. Since mathematics is fundamental at least to science and to technology, such a justification should be produced in mathematics as well.

I also think that this justication can only be achieved on the basis, somehow, of the way we think, i.e. what I would call our sense of logic. What do you think are the most effective justifications produced so far, if any, and why do you see them as effective?

• Maybe useful : Carlo Cellucci, Rethinking Logic : Logic in Relation to Mathematics Evolution and Method, Springer (2013). – Mauro ALLEGRANZA May 14 '18 at 9:54
• You might also want to look into the literature on logical pluralism, see: philpapers.org/browse/logical-pluralism – possibleWorld May 14 '18 at 15:29
• Agreement with "logical intuitions" is only one consideration in motivating use of logical systems (and it is widely acknowledged that those are, at face value, incoherent). Typically, it has to be balanced with good formal properties, ease of application, etc. A specific difficulty is that any justification of logic will be circular, for it has to use logic. Logical harmony is often cited as an attractive property for logics to have, Girard's "transcendental syntax" is another approach. – Conifold May 15 '18 at 3:28
• @Conifold - So many false notions are widely regarded as true that I don't have to be impressed by the fact that our logical intuitions be widely regarded as incoherent. It seems this view comes from the fact that we often catch other people saying incoherent things. I would agree that this is often the case. However, there's no indication whatsoever that this is due to an incoherent sense of logic. Rather, people will say logically incoherent things whenever they elect to ignore what their sense of logic suggests they should say. – Speakpigeon May 15 '18 at 12:18
• @Conifold - I also don't see that using our sense of logic to produce a justification for our use of a particular formal logic system would necessarily be a problem. – Speakpigeon May 15 '18 at 12:33

There seem to be two questions in one here. The first is, what justification can we give for using deductive reasoning generally? The second is, what justification is there for one particular system over another? One cannot really address the second without attempting to answer the first.

The first question is seeking to understand what is the epistemology of logic. How do we know logical truths are true, or that valid arguments are valid? There have been many different responses to this question. Broadly, they divide into internalist and externalist categories, and each in turn has several variants.

Some have defended the idea that our intuition or insight as rational beings provides a kind of privileged a priori knowledge of logic that is incontrovertible. One might challenge this on many grounds. The history of philosophers claiming that various things are intuitively and certainly true is not a happy one: often the claimed truths turn out to be false or even absurd. Our intuitions aren't really that reliable. Also, if logic is incontrovertible, why is there so much disagreement between competent logicians about which logic is the 'correct' one, or even concerning what logic is fundamentally about?

Some hold that logic is justified inferentially by the relations of logical consequence that we consider correct. This is often defended by appealing to the concepts of logical harmony and logical stability. This runs into a circularity objection, pointed out by Lewis Carroll.

Some contend that logic is justified purely syntactically by the proofs it is capable of generating. Or that proofs correspond to computations, and so logic is justified indirectly by our understanding of computation.

Another approach is to try to base logic on a concept of analyticity. The idea is that some sentences have no empirical content and are therefore true 'come what may', or true in virtue of the meaning of the terms they contain. This position was popular with the logical positivists in the 1920s and 1930s and still has some defenders today, though it took a fair beating from Quine, Putnam and others, and does not appear to be especially popular with philosophers.

Another possible attempt at justification would be to appeal to natural selection. If we were not good at logical reasoning, we would be selected against and be less like to survive and propagate. This runs into the objection that we cannot be sure that logic always provides a strongly positive selection bias. Also, we know that humans are quite spectacularly bad at reasoning with probabilities and uncertainties, among other things, so it seems dubious to place too many expectations on natural selection.

Another approach is called anti-exceptionalism and maintains that logic is similar to a scientific theory. It has no special properties of a priority or analyticity and is potentially revisable in the light of empirical discoveries. On this account, logic is justified in the same way scientific theories are: we subject it to criticism and attempt to solve problems with it. If we succeed we keep trying more problems, and if we fail we look for something better. The logic or logics we are left with are the ones that work best because they have survived critical testing.

Which of the various systems of logic is 'correct'? On the rational intuition account, presumably we must just consult our intuitions on the matter. For myself, I don't see how this helps. Are the defenders of the various different logics just wrong in their intuitions? Is it intuitively obvious that the principle of explosion should hold, or that it should not? Is it intuitively obvious that universal statements have existential import, or that they do not?

On an analytic account, we would have to come up with a theory of meaning for natural language and argue that one particular logic does the best job of correctly accounting for, or at least conforming to, that theory. Michael Dummett took this approach in arguing for intuitionistic logic. Quine and the later Wittgenstein argued that this approach does not work.

Anti-exceptionalism potentially allows for logical pluralism, so the question of which system is 'correct' need not arise.

There are a number of useful references in my answer to this question.

• Thanks for your extensive but synthetic answer and for the references. My question is explicitly about what people think are the most effective justifications. Your answer shows that many opinions and very different ones have been produced over the years but you don't actually get to say which of them you think would be the most effective and why. This suggests you think none of them is really convincing, if at all. – Speakpigeon May 16 '18 at 7:41
• The references you give also seem to be of philosophers rather than the actual practitioners of logic in mathematics, science and technology. I think it is these people who should really worry about this question and try to provide a justification! – Speakpigeon May 16 '18 at 7:42
• As far as I am concerned, I believe that the best justification should be somehow based on the adequacy of the formal logic you use to our sense of logic, and I mean literally "sense", like in our "visual sense" for example. And I fail to see what the problem would be with this approach. I doubt it's even possible to do any different. Still, I wanted to leave the question open... – Speakpigeon May 16 '18 at 7:43

I feel you've given the answer. The best or most useful logic is the one with which we normally think, (notionally and ideally) and this would be because we want to understand the world. No point in using a logic that does not allow us to do this (dialethism, paraconsistent logic etc).

I would vote for Aristotle's dialectic as the correct logic for philosophy,and have never come across a reason to abandon or modify it. The only problem is that it is regularly misused by philosophers and scientists alike but this is not the fault of the system. If we apply it correctly it does the job.

• I agree. Yet, you're not quite answering my question. I can only assume you are implicitly suggesting that no one could possibly produce a proper justification for modern logic. I certainly think so myself. Still, did Aristotle or anybody else ever provide any proper justification of syllogistic logic? And can you give examples of the application of syllogistic logic to complex logical formulae? – Speakpigeon May 15 '18 at 12:46
• I know little about modern logic or the reasons for using it. In philosophy the old-fashioned logic seems to work fine. To me the justification for the dialectic is that we use it instinctively and it works, Maybe I should get with the times. . – PeterJ May 16 '18 at 8:37

If you are dealing with logical propositions that are unambiguously either true or false, then natural deduction based on classical (or standard) logic is all you will ever need in a system of logic. It is by far the most widely used form of logic in mathematics, science, engineering, commerce and daily life. The reason for this success IMHO is that it uniquely models the ways we actually think about such propositions.

Yes, there are other forms of logic, but they seem to be applicable in only very narrowly defined areas of study. They are no substitute for natural deduction based on classical logic IMHO. (Just be aware that there many different ways to present these rules of logic, some more intuitive than others. You may have to shop around for the best presentation for you. It took me several years of self-study and "experimentation" with computer software to piece it all together.)

• Thanks for the link on natural deduction. I'll have to have a close look at that. Any best textbook ever on the subject you'd be aware of? – Speakpigeon May 16 '18 at 16:21
• @Speakpigeon You might try my homepage at dcproof.com – Dan Christensen May 16 '18 at 22:41
• After looking into Gentzen's method of natural deduction, and how it is used in the context of theorem provers, I interpret Gentzen as a straightforward mathematical generalisation of Aristotle, Aristotle, which as I see it, was essentially the realisation that we usually solve logical problems by making use of intuitively known logical truths, called in Gentzen, "rules of inference". In that respect, both terms, "natural" and "deduction", also seem to be a pointed reference to Aristotle. Any comment on that? – Speakpigeon Jan 15 at 21:17
• I also feel confident little could go wrong with natural deduction because of its grounding in the logical truths that we've discovered since Aristotle and that I think are accepted by most people as intuitively evident. I can't say the same thing of the method of logic based on truth tables and the arbitrary principle of the truth-functionality of logical operators. In fact, I'm confident it's wrong, though I couldn't justify this view outside the same kind of meta-logical considerations I've just given. Any comment on that? – Speakpigeon Jan 15 at 21:32
• @Speakpigeon You may be interested my blog posting at dcproof.com/IfPigsCanFly.html There, I derive the truth table for material implication using only the following rules of natural deduction: direct proof (=> intro), proof by contradiction (~ intro), join (& intro), split (& elim), detachment (=> elim), double negation (~~ elim). – Dan Christensen Jan 16 at 14:51