How can we know (with absolute certainty) that our logic is correct?

Question

How can we know (with absolute certainty) that our logic is correct?

Even statements like Descartes’ “I think therefore I am” relies on our logic: that I exist is a logical consequence of my experience. But how can we know that our logic isn’t a (very consistent) collective delusion (caused by natural selection or a malin génie)? Perhaps we experience ‘non-existence’, something that is completely nonsensical to us, but only because we rely on our logic.

Is there something about logic that makes it necessary to be true (apart from the formal definitions of the axioms)? Or can we only use logical reasoning after accepting certain axioms/ assumptions? And if these assumptions always need to be made, how can anything other than radical skepticism be defended? How is this issue addressed in philosophical literature?

Answer 1

Logical pluralism might be described as the belief that there is no "One True Logic." You might also be interested in the story of TONK:

Consider the case of a new binary connective, TONK, proposed in Prior (1960), whose (complete) meaning was to be given by the introduction rule: from ϕ infer (ϕ TONK ψ), and the elimination rule: from (ϕ TONK ψ) infer ψ. If logic rules are to be given merely in terms of some Introduction and some Elimination conditions, then TONK has just been given a proper Int-Elim definition. But as Prior notes, logic becomes much more “egalitarian” with this new connective… too egalitarian, perhaps!

Note: in addressing the very question of whether there is a "One True Logic," we will have to be using some form of reasoning. Does that mean that, if we could be fully self-aware of how we reason using said form, we would have clearly apprehended the fundamental logical system? I am tempted to say so. However, I doubt that first-order propositional logic, or standard modal logic, for example, are (alone) this fundamental system. While Conifold once provided me with a link to a well-argued essay whose thesis was that imperative logic is not a viable program, I think imperatives do play a common and standard role in logic: firstly, it is usually in prescriptive format that fundamental assumptions are listed in abstract arguments. We can, for example, say, "Assume that [something] is true," or, "Let it be true that [something]." Secondly, though, I see a very evident proof-of-concept that imperatives can be subjected to some kind of logical structure, in the existence of imperative programming. ---Anyway, I also see erotetic logic as fundamental. I'll have to find the citation, but I remember it being said that the only three moods(?) common to all languages are assertoric, prescriptive, and interrogative moods. It's also said, by some, that logic and language go together in some quintessential way.

Answer 2

I don't offer the "philosophical literature" answer, as the current Analytic focus of philosophy does not have a good answer for the questions you are asking.

Historically, philosophers and mathematicians believed in the "one true" logic or math. Kant used the intrinsic truth of Euclidian Geometry as his paradigmatic example of a truth we can know to be true, just from reasoning. However, within a century, non-Euclidean geometries were developed as a thought exercise, THEN they proved to actually apply to our world, under General Relativity!!!!

Modern math thinking is that math is basically infinite. One can postulate basically any set of mathematical premises, and derive axioms etc. from them. Hence no math is "true" or not, the question instead is whether we find it a useful tool or model to use to describe part of our world.

Once infinite pluralism was accepted in math, it was only a matter of time before it was be accepted in logic -- because logic is structurally the same as math -- a set of postulates and axioms, from which one can derive a system. Logic pluralism has not yet been fully accepted, but the last several decades have seen it become the dominant view among logicians. The delay appears to be the consequence of an evolutionarily driven psychological need -- we MUST trust our logic to be "true" else we dare not risk our lives on our own decision making!" But a psychological "wish it were so" need, does NOT constrain reality!

The question for logics, is once more their pragmatic usefulness for us to deal with the world.

We humans seem to be gifted with an intrinsic logic sense by evolution. This logic sense is flawed --as one can demonstrate by pulling tricks on children. Our logic sense is NOT a valid logic. But if we apply our evolutionary logic sense to itself, and make corrections to our logic intuition based on this self-critique, then the simplest logic that IS valid, appears to be Classical logic. This is the reason for the strong intuitive preference among most thinkers for classical logic.

However, Classical Logic has a variety of failings, when we apply it to our world. For one, it is absolutist -- it presumes certainty. It includes the Law of the Excluded Middle. But ALL our empirical knowledge, is UNcertain. For every proposal about our empirical world, we bin that proposal into FOUR categories, not the two of classical logic. These are "Well enough supported to accept as a working hypothesis", "Well enough refuted to reject as a working hypothesis", "Incoherent or otherwise non-evaluable in principle", and "currently uncertain as to how to respond to the hypothesis". Note none of these categories is EITHER true or false, hence all four violate the Law of the Excluded Middle.

For a second failing, it requires objects in our world to satisfy the "identity" criteria. But all objects in our world, with the possible exception of elementary particles, are bundle objects. The "Ship of Theseus" thought problem was intended to show how there cannot be a THE ship of Theseus, but it is generalizable to any object like a ship. Add or remove molecules, boards, plating, crew, etc, from the bundle and you may still have A ship, but the ship is no longer identical moment to moment, hence classical logic cannot validly be applied to it.

Classical logic is still USEFUL when applied to objects in our world, but it can fail at unexpected times. An example is one of the aspects of our bundle that is selfhood. All of us compartmentalize, as we generally do not have 100% coherence across all aspects of our knowledge and lives. If selfhood were simple and unitary -- compartmentalizing would not be possible -- but selfhood appears to have sub-modules that allow us to compartmentalize with ease. A self that can compartmentalize, can lie to itself. Lying to oneself -- that I believe both A, and not-A at the same time, is a violation of the Law of Non Contradiction. Yet it is a feature of our world.

Note also, that even translating thought problems into logic form -- requires a level of precision of definition that NO language can provide. See section 1-IV of Quines Two Dogmas of Empiricism: https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf Quine was arguing that analyticity depends on an uncertain set of definitions that we infer from usage -- IE synthetically, hence there is no real analytic-synthetic distinction. I believe Quine overstepped his argument -- he instead showed that analyticity is not distinguishable from synthetic BY ITS OWN STANDARDS -- but if one instead evaluated it by pragmatic "synthetic" standards, there still might be utility in making an analytic-synthetic distinction. But despite this caveat -- the uncertainties he noted in all language use, prohibit 100% confidence in translating ANY language claim into a logic statement.

Logics then, are chooseable, and are chosen by us based on their pragmatic utility, and none are 100% valid in our world.

Note my repeated references to "pragmatic". This is pragmatic answer, which is currently a very minority viewpoint in philosophy.

How does a pragmatic approach deal with radical skepticism? All humans alive, in late infancy, develop a world model. And a theory of self, and a theory of other minds. All of these models are speculative -- they are pragmatic empirical inferences. They are not logically derived deductions, and one cannot be certain of any of them. This, of course is also true of all empirical knowledge. There may be a few infants who instead of pragmatic inference, instead pursue analytic certainty, and postulate radical skepticism. Those infants would be among the sad set who die in late infancy, characterized by death due to "failure to thrive".

Revisiting this choice, and embracing radical skepticism at any later point in life, would lead to the same outcome. There is a pragmatic Darwinian process at work in our world, that helps limit the adherence to radical skepticism. If this is not sufficient to convince you, the only thing a pragmatist can do at that point is doff our hat in sadness at the consequence.

Answer 3

For reasoning in the abstract, absolute certainty can be achieved in Logic same as in maths. Results of reasoning can be objectively verified.

However, in general, it is impossible to know that any reasoning about the real world is correct. That has several reasons. One reason is the frame problem, we never know if we have used all relevant information to a problem. In the real world it is also commonly difficult to know if all information used is true (also because we lack infinite time to double check), and whether our models of reality are suitable or not. There are other effects like chaos, which prevents us from predicting the weather far into the future.

A pragmatic view however is that absolute certainty is never required for real world problems. Acting while knowing our reasoning might be wrong is still possible if we made big efforts to validate our reasoning. That's why space missions are possible, or building skyscrapers, or running global businesses.

Even if we always know that we cannot be absolutely sure and some calculation might be wrong, from experience we can reliably predict well enough to not fall back into radical skepticism.

Answer 4

How can we know (with absolute certainty) that our logic is correct?

Certainty applies to beliefs, so I will assume that "absolute certainty" is code word for knowledge.

We usually believe our logic to be correct. However, most people do not really think about logic per se. Rather, they tend to consider only arguments and reasonings, and tend to express agreement or disagreement according to whether they agree or not with the conclusion. In other words, most people couldn't care less about logic per se. If not for Ancient Greek philosophers and in particular Aristotle, we might still be unaware of it today, using it but never considering it in itself.

The question is further complicated by the long history of formal logic. Most people confuse formal logic with logic itself. However, it is clear that most people make routine logical inferences even though they have no training in formal logic. Plato noticed that people without education could give logical answers. One of the first recorded ad absurdum argument was a century before Aristotle in a poem deriding Homer's idea that the gods had the same appearance as human beings--cows and horses would say that the gods had the same appearance as respectively cows and horses, and it is not possible for the gods to have the same appearance as humans, cows and horses.

Formal logic is at best a model of our logic, and as such may be correct or incorrect of it. However, it is trivial to admit that we do not know whether any formal logic is correct in this sense and I will assume that this is not the question.

So I take the question to be whether we know that the innate logic of our deductive reasoning is correct.

We can treat logic in this sense as a cognitive sense, somewhat similar to vision, memory, etc. As such, we can say that we are essentially dependent on our senses to the point that we do not have any alternative. If we were to distrust our vision, we would have nothing to replace it, at least given the current state of our technology. And so it is for our logic. We may like to fancy that a better logic is possible, but even if there was one, we could only apply it vicariously through our own logic, much in the same way as we play games. The difficulty would be in accepting any conclusion derived using this different logic that would be false according to our own logic. I doubt anyone would accept that. We would probably be unable to decide whether it is really better.

We usually trust our vision but we now understand that it provides an image of the world in the same sense that a map is an image of the territory while not being the territory itself. Is our vision correct of the world? Definitely not, but what we need is to be able to use it to survive, prosper and reproduce. It is in fact so good that we have been able to use it, together with our other senses, to produce scientific theories that greatly improve our ability to predict some events in the future.

However, logic is completely unlike any one of our other senses, although each of them is pretty unlike any other. The point however is that logic does not provide us with an image of the world. Rather, it allows us to build models of it on the basis of the data we get from our other senses. Scientific theories provide a good example of the process by which we start with one model and update it to take into account any new data coming in, sometimes revising almost completely our model. This is what our logic allows us to do. This is what everybody does. Scientists work in a professional and systematic way where the process itself is scrutinised and improved. Outside science, we cannot usually afford the time necessary to carry any systematic investigation to build models. Rather, our brain does it without us necessarily being aware that it is doing it.

Another use of logic is in allowing us to convince other people through verbal arguments. This effectively allows humans to pool their cognitive powers in a way beyond what other animal species can do. Our current civilisation is a demonstration of the effectiveness of this process. All sciences and professional activities rely also on this particular use of our logic.

The result on the face of it is quite good. Human beings have survived in a difficult environment for at least 500,000 years. That we make mistakes it apparent in pretty much everything we do. However, the effectiveness of our logic is that we can modify our models to take into account any new data that contradict them. Scientists do it, business people do it, engineers do it, and the brain of every human being does it even when the subject is not aware of the process. Even dogmatic religious people do it. If they didn't, they would be dead.

So the notion that our logic has to be correct or incorrect is a misconception. Our vision is not strictly speaking correct, but we can use it to survive in the real world, or at least so it seems. It should be noticed that every one of us has plenty of opportunities to disagree with the logic employed by everyone, and I don't know of anyone who seriously disagree with the validity of the modus ponens (Vann McGee's contention to the contrary was a fallacious argument). Mathematicians, who may be regarded as the most systematic users of our logic, rely essentially on the modus ponens, 2,400 years after it was first proposed as a logical truth by Ancient Greece logicians. Formal logic is correct or incorrect, at least to the extent that it purports to provide a model of our logic. But our logic is either useful or useless, effective or not effective. In all appearances, it is very useful and very effective.

It should be noticed that if we tried to verify that our logic is consistent, we would have to use it to carry out this verification because we have no alternative method to do the job, and we would presumably fail to see any problem even if there was one.

Further, we can take our environment to have provided a very comprehensive test-bench that effectively verified that it worked during at least the last 500,000 years. If it didn't work in the natural environment of the Earth, we would not have survived and the human species would have quickly disappeared. The fact that we are here and very successful proves that our logic works, that it is useful, that it is very effective.

It is perfect? That we don't know. However, it should be noticed that humans have come to apply their logic well beyond our natural environment, in particular to new technologies and to sciences, in particular to General Relativity and Quantum Physics, even though these as extremely puzzling aspects of the real world. General Relativity and Quantum Physics both have practical applications and it cannot be doubted that scientists got at least something right in this context.

So, no, we don't know that our logic is correct, but we don't need to. Instead, we need to be able to use it to survive, prosper and reproduce, and we certainly do that every day of our lives.