If an argument can be valid in one logical system, but invalid in another, are logical arguments "meaningful"?

Question

Typically, classical logic, or extensions of classical logic, are used in all scientific and mathematical contexts to justify conclusions. By doing this, aren't we implicitly assuming that all other systems of logic that provide contrary conclusions, are "wrong" in some way?

It seems much deeper than a matter of interpretation, because, for example, in classical logic, if the premises contain a contradiction, then any argument containing those premises is valid, and thus, it's possible to "prove" anything true with those premises. Whereas, in paraconsistent logical systems, contradictions don't necessarily imply everything.

So, if a contradiction (i.e. something being both true and false in the same respect at the same time) were discovered in the real world, would that imply every possible statement about the real world is true, or would it mean that there are still false statements, and we should have been using a paraconsistent logical system instead?

More generally, for any valid argument you could make using classical logic, I could imagine an alternative logical system in which that argument is invalid. So, then, how could you say, that an argument is really "valid" or "true" in any meaningful or universal sense? What makes us so confident that classical logic is the "correct" logical system to use?

On top of this, if one were to argue that one logical system was "better" than another, you would have to be using some system of logic to make such an argument, which implicitly assumes that the system of logic you were using was the "correct" one from the onset. How can I even talk about this subject without presupposing a system of logic?

Answer 1

Here's how I explain it when I teach logic.

Formal or mathematical logic uses mathematics to represent "good reasoning." These models are like maps: they can be extremely useful for certain purposes; but every useful map introduces simplifications, distortions, and omissions. A good map has simplifications, distortions, and omissions that make it more useful for its designated purpose. But there's no such thing as a universal, literally all-purpose map.

For example, sentence logic or propositional logic — the formal system you learn when you first study logic — assumes truth-functionality and bivalence. Truth-functionality means that the truth value of a compound sentence (like "either p or q") depends only on the truth value of the component sentences ("p" and "q"). Bivalence means that every sentence is either true or false, and no sentence can be both. There's no "partly true" or "I'm not sure." Truth-functionality and bivalence are extremely useful for representing operators like "not," "and," and "or." But they do weird things to "if-then," and are simply incompatible with "p because q," "I would prefer that p," or "all humans are mortal; and Hypatia is human; therefore Hypatia is mortal." They also lead to the the "fact" that a contradiction implies anything; this isn't so much a "fact" as a distortion created by the simplifications involved in assuming truth-functionality and bivalence.

Consider a subway map. The distances between station markers on the map don't correspond to distances between stations. When reading the map, we use conventions and prudence to avoid drawing incorrect inferences. In the same way, when using sentence logic, we should use conventions and prudence to avoid overinterpreting explosion or the weirdness of the material condition.

Other formal systems use different assumptions, in order to do a better job of representing some of the things that sentence logic can't really represent. But they have simplifications, distortions, and omissions of their own. For example, paraconsistent logic does weird things to "or". This means that there is no one universal formal system. A street map isn't a great way to represent the organization of a subway system; and neither is a good way to represent where different species of birds live in the city. For three different tasks — navigating by bike, navigating by subway, and avian ecology — we need different maps.

All of this is compatible with some kinds of realism about "good reasoning." If formal systems of logic are like maps, then actual good reasoning is like the city represented in the maps. The city is real, even if no one map can perfectly represent it in every aspect and we need to exercise "subjective" prudential judgment in order to correctly use any given map. In an analogous sense, you might say that reasoning can really be good or bad, even if no one formal system can perfectly represent it in every aspect and we need to exercise non-formal judgment in order to avoid overinterpreting the quirks of any particular formal system.

Answer 2

Bertrand Russell once said that all statements fall into two categories. Those that can be said (Proven) to be either true or false, and those that cannot. And then he said that the necessary corollary is that no statement in the first group (the ones that can be proven to be either true or false) can ever be associated with or have anything at all to do with reality. i.e., only those statements that are totally abstract, and have nothing whatsoever to say about the real universe, (like one plus one equals two), can ever be proven to be true or false. and any statement that purports to say anything about reality can never be proven to be absolutely true, or false.

Answer 3

No, "contrary conclusions" needn't be "wrong in some way". The simplest example is perhaps modal logics, where the statement "it is raining" may be true for some places while not others, or true for the same place at some times and not others. But that doesn't involve any deduction, per se.

In those cases, the choice of appropriate logic typically involves your universe of discourse. For example, a typical classical rule of inference would be,
  A==>B     A==>C
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      A==>B&C
Now, at first glance you'd perhaps think that might be tautologically true in all universes of discourse. But consider the "vending machine universe" where, say, you can buy a Coke for one dollar, which we'll write 1$==>Coke. And you can also buy a candy bar for one dollar, 1$==>Candy. But now it's wrong to infer that 1$==>Coke&Candy because the dollar gets used up. Slightly more formally, the process of proving the conclusion "uses up" (or "discharges" in the terminology of linear logic) the premises. Constructive logics (substructural logics in general) typically have somewhat different rules of inference than classical logic, often leading to different provable conclusions from the same premises. But that doesn't mean either is "right" or "wrong".

Answer 4

I think a good way to approach this is to say that different logics have different natural semantics. It is commonplace, for example, to say that intuitionistic logic can be interpreted as the logic of provability or verifiability (this is the BHK interpretation). So, when an intuitionist writes a proposition A we understand this to mean A is provable, and when he writes A or B this is understood to mean A is provable or B is provable. This gives rise to a logic that differs from classical, and so an argument can be classically valid but intuitionistically invalid. The two needn't conflict provided we keep the semantics separate. Classical logic is about truths and falsehoods, while intuitionism is about provability. Similarly, relevance logic can be understood as having the natural semantics of information passing. At least one form of dialethic logic can be understood as having the natural semantics of falsifiability. A deontic logic would have the natural semantics of obligation. Linear logic has the natural semantics of resource bounded interactions (among other things). Bayesian confirmation theory can be understood as a kind of logic of partial belief.

This is written from a point of view that classical logic is indeed the 'correct' logic of truths. There are real, substantive disputes between advocates of some logics, e.g. Michael Dummett with intuitionism, Stephen Read with relevance logic and Graham Priest with dialethic logic, in which they are claiming that their logics are the logics of truth and falsehood and that classical logic is not. Defenders of classical logic maintain that classical logic is about truths and these other logics are about something else.

You refer to the principle of explosion: that a contradiction entails all propositions in classical logic. This is unproblematic provided we remember that we are talking about truths. We must be careful not to shift the semantics to that of belief: having inconsistent beliefs does not entitle me to infer all beliefs. We cannot "discover" a contradiction in the real world because a contradiction is a proposition not a thing. If we use classical logic, then on discovery of some conflicting observations we would look for some way to distinguish them - some additional variable that we had overlooked. Another way to think of this is that as a matter of method, a scientist who performs an experiment twice and gets different results does not conclude that some contradiction is true, but assumes there is some unknown variable that needs to be identified and controlled. This assumption is in effect a form of realism.

As to what persuades us that classical logic is a good logic to use, a response in the spirit of Quine would be to say that classical logic justifies itself because of the contribution it makes to our scientific understanding of the world. If it didn't work we would discard it and try something else. Indeed there have been empirically motivated suggestions for other logics, such as quantum logic. Another line of reasoning might be to claim that classical logic corresponds to the concept of computability via the Curry-Howard correspondence. Various other approaches to the epistemology of logic exist.

Your last sentence about how we can talk about logic without presupposing it in effect asks, what is the logic of our meta-language? In practice this is often classical, though it is not impossible that we could use some other logic in our meta-language. Again, if it didn't work well, we would look for another.

Answer 5

If a computer program works on one operating system, but not another, is there any point to it?

Of course, computer programs are utilitarian (in the non-philosophical sense), and we have only the expectation that they function in the context of the system they were designed for. But isn't the same true of logical arguments?

It may be that you have expectations that logical systems should grant you insight into larger metaphysical truths, expectations you don't have for operating systems. But just as both Euclidian and non-Euclidan geometries reflect certain aspects of the universe as we observe it, so too do different systems of logic reflect different aspects of reality. What you might have to give up is your intuition that there is one right, universe logical system, rather than multiple systems that are useful in different contexts, and towards different ends.

Answer 6

There are a number of ways of answering this, based on what some of those terms mean: logical system, valid/invalid, and 'meaning'

• a logical system consists of an algebraic/symbolic part (the logical connectors, rules of inference, and their syntax) and a semantic part (what is intended by them). These are intended together to capture what we intuitively think of as 'logic', a formalization of what we vaguely call rational thought. So it is not (vaguely) unreasonable to think that there is more than one way of representing logical thought in symbols that may come to different conclusions. These may be even entirely opposite conclusions, but more likely there will be lots of overlap or even more likely that the thoughts that can be formalized and proved in one system is a subset of another. What ever it is these formalizations formalize, they are meaningful. If two systems contradict each other in some specifics, then the inference you should make is not that 'all of logic is meaningless' but rather that these two systems are formalizing two different intuitions that are meaningful in their own way (correspond to two slightly different intuitive situations).

• 'validity' is a technical term in logic, but let's treat it informally. If a statement is valid in one logic but invalid in another, it could be that, yes, the two logics are contradicting each other and that in the situation the two statements describe is rendered incomprehensible. But that doesn't mean that all other statements are meaningless. It could mean that the two logics, while mostly the same are just describing two different things. But to be concrete, what usually happens with logics, like between classical vs intuitionistic logic, is that while the law of excluded middle is 'true' in classical, as to intuitionistic it is not so much false as it is non-derivable. It is just that you can prove more things in classical logic.

• 'meaning' has more than one meaning. Often in logic, the technical meaning of a sentence is whether it is true or not, and that is all. The 'meaning' of "A->A" is true. But there is the non-technical, intuitive meaning of 'meaning' which is all the connections a concept makes in your head, like "A->A" is a good axiom to have and is sometimes a good equivalent for '-A v A' and sometimes not. So "A->A" is meaningful in many ways, though it may not be the same meaning depending on context.

Logic is just a way to formalize thought. If you have two logics that contradict each other, it doesn't mean that all logic is a sham and buildings and bridges are going to fall apart, it means that you possibly made a mistake in operation of one of your logics, or it may mean that the two formalizations capture different kinds of ideas, or it may mean that a statement is true in both but provable in one but not the other.