What is logic, and can logic itself be true or false?

Question

In my understanding, logic is the process by which we can determine whether a conclusion is true or false, starting from a bunch of premises.

What kinds of logic are there, then, and what would it mean to claim that logic is false (or true). What would it mean? Would it even make sense to state that logic is true or false? If so, in what way, and under what conditions?

Note, I am not only asking whether a "piece" of logic can be true or false (would that be the same as asking whether the piece of logic holds or not)? I am also asking whether the logic itself can be true or false.

Does my question make sense, from any perspective, or does it not make sense?

Thanks.

Answer 1

It doesn't really make sense to say that logic can be true or false. A system of logic can have properties such as being sound and complete. A logic is semantically sound if it only proves sentences that are logically true under the relevant semantics, and it is complete if all its logical truths are provable. If a logic is demonstrably unsound, this would tend to make it useless, since it would be capable of proving things that were false, or proving false conclusions from true premises. A logic can also have the property of being consistent, meaning typically that no contradictions can be derived from the logic itself.

A particular application of logic occurs when an argument is constructed. This consists of a set of premises and a conclusion, with the intended claim that the conclusion follows from the premises. But again, we don't say that an argument is true or false. An argument is said to be valid, or more precisely deductively valid, when the premises guarantee that the conclusion holds, and invalid otherwise. This does not mean that the premises have to be true, only that if the premises are true, then the conclusion must also be true. A valid argument with true premises is said to be a sound argument.

There are different kinds of logic. This is partly because a particular approach to logic can be formulated in different ways. Also, a particular approach to logic can have many extensions added to it, in order to make it more expressive. Also, there are just different approaches to doing logic. The most commonly used kind of logic is called classical logic, but there are others, such as intuitionistic logic, minimal logic, linear logic, the logic of paradox, probability logic, a whole family of relevance logics, and many more.

Even though there are many different logics, we still don't usually say of any that they are true or false. There are people called logical monists, who hold that only one kind of logic can be fundamentally correct, and they may advocate for a particular kind. Other people are logical pluralists and maintain that we can accept a plurality of logics. Explaining how a multiplicity of logics can coexist is a complex question in the philosophy of logic.

Answer 2

Logics might be understood as systems of the truth of propositions. And, it's tempting to apply what we know about logical statements, that they can be true or false generally, to the system of statements, but doing so would be the fallacy of composition.

Where as we are interested in the truth-conditions of the individual statement, for instance, is it true Socrates is in the kitchen?, generally in logical systems, we are more interested in finding if there is a contradiction. So, if it's true that Socrates is in the kitchen, and it's true Socrates is outside the house, what should we make of that?

In psychology, the thinking is that when we have contradictions, we experience a mild form of anxiety called cognitive dissonance which helps guide our intuitions about truth. This practice is so important in understanding and using systems of logics, there's an entire theory of truth devoted to it, the coherence theory of truth:

Coherence theories of truth characterize truth as a property of whole systems of propositions that can be ascribed to individual propositions only derivatively according to their coherence with the whole. While modern coherence theorists hold that there are many possible systems to which the determination of truth may be based upon coherence, others, particularly those with strong religious beliefs, hold that the truth only applies to a single absolute system.

Logicians explore what it means to have different sorts of systems. The classic Laws of Thought, for instance, include the Law of the Excluded Middle, where a statement is either true or it's opposite is true, but perhaps it makes sense to talk of something that is both partially true and partially false. This is perfectly acceptable in fuzzy logic, where degree of truth is part of the logic. Different logics have different rules.

Answer 3

One way to interpret (alleged) topic neutrality in logic is to think of pure logic sentences/propositions as schematic sentences. Depending on the degree of schematism to a pure sentence's name, one might suppose that its (the sentence's) evaluation as true or false (or whatever) is unavailable unless the schematics are interpreted in such a way that they can be true or false (or whatever).

For example, one might represent a noncontradiction axiom as:

∀A∀~A: ~(A & ~A) (something like "for any pair of incompatible A, it is not (true) that A and ~A").

But you might be of a mind to be wondering whether we ought to impose this ∀-scheme on possible inconsistency unless we "checked" every pair of possibly inconsistent sentences (or facts, or whatever) to see whether the universal rule really holds. A strongly a priori (or, more clearly-put, a more proactive) model of "knowledge of logic" would allow that we can know the universal scheme to be true, indeed by knowing that all its instances (satisfactions) are true, but this because we sort of "project" all possible cases of the question (the question behind the schematics) into our reflective intellect and so we can then "see" how all possible cases are resolved/resolvable in advance. (When logical necessity and identity are tight-knit enough to questions of possible consistency, the immediacy of this "seeing" is acute to an ultimate extent, for then the identity of mere possibility in itself is made to turn on consistency as such.)

Or maybe logic isn't so topic-neutral after all, but is about things that are particular in their own way (or even nominalism is true, and everything is particular by the by anyway, but to mention that further would be to digress...). "No self-contradictory conjunction is true," or even, "Necessarily, no self-contradictory conjunction is true," might be an "instantly (ideally!) recognizable" logical fact. Or, "Every proposition is true or false," or, "There are Continuum-many truth-values,", or, "Truth is first and foremost an object named by true sentences, not as much a direct property of those sentences by themselves,", or, "The proposition, 'A,' is true if and only if A," or so on and on: some or all or none of these, and/or plenty of others besides, would themselves be true, or false, or both, or neither, or "only" one or the other, or "categorically false (resp. true)", even.

Now, if you're familiar with Lewis Carroll's musings on modus ponens, you might notice in the back of your thoughts a representation of logical rules not as assertions about metaphysical architecture so much as imperatives of inference and reasoning (or thought). John Stuart Mill, IIRC, traced the standard noncontradiction axiom to (I assume, based on his description) a "mental sensation" that we now often refer to as "cognitive dissonance." At any rate, one can construe the consistency requirement less as a matter of the truth-content of a belief and more a matter of the formal grounding of the belief. This is an especially perspicuous option if self-conflicting conjunctions "literally" turn into voids in our thoughts, i.e. if they are (sufficiently) "meaningless."

So, even so, for logical axioms that are elementary imperatives of understanding more than (or instead of) assertions or assertion schemes about abstract possibilities, we are at least referring to seemingly hypothetical imperatives, or moments in general instrumental reasoning: "If your end is inferring further truths from given truths, then..." Alas, the philosophy-pirates have smuggled truth-aptitude back into the capital city of the land of logic, for the time being...

Answer 4

Logic, in one view, are those rules by which we can go from one truth to another. It's about valid modes of reasoning. It's what Aristotle called prior analytics.

Logic, in another view, is about the ultimate foundation of our world. Logos in an older world view, Hegels absolute spirit. And more commonly now, in our secularised world, the logical foundations of our world. This is what, for example, Wittgenstein theorised in his logical idealism.

A logic worthy of its name would be necessarily true but insomuch logics are created artifacts of the human mind they can be wrong or false.

Answer 5

What is logic, and can logic itself be true or false?

What kinds of logic are there, then, and what would it mean to claim that logic is false (or true). What would it mean? Would it even make sense to state that logic is true or false? If so, in what way, and under what conditions?

The assertion that logic is false has enormous consequences for the claimant.

To claim that logic itself is false would mean, among other things, that the Laws of Thought are false. On that assumption, all three laws collapse into one. The following “Law of Non-identity” becomes true: Nothing is identical to itself (A is not-A). “Excluded Middle” becomes “Included Middle “: everything must be and not be (A and not-A). “Non-contradiction” becomes “Contradiction”: two contradictory statements can be true at the same time (A and not-A). All three of these say the same thing.

Given the current rules of the syllogism, there are 256 syllogisms, given 4 figures and 64 moods. Of the 256, exactly 15 or exactly 24 are valid, depending on one’s assumptions. If logic is false, then all 256 are valid, because any term can mean anything. The reasoning process becomes pointless.

To say that logic is true is to say that its rules produce conclusions that are observably true. To say that logic is false is to say that its rules produce conclusions that are true and false randomly. Such is the danger of disavowing logic; the denier strikes themselves blind.

Answer 6

Logic is a mental faculty and as such it is neither true nor false. The right question is whether logic is useful or not to us. It is apparent that logic allows any community of human beings to develop a natural language. It is also clear that natural languages are instrumental in the development of any civilisation, and therefore also our own. So, is logic useful? Yes, it is and this is in evidence around us in the technology humans developed, in the scientific discoveries, but also in the political organisations of societies of millions and indeed hundreds of millions of people.

What may be true or false, however, is any system of formal logic. A system of formal logic is meant to represent and model our logical faculty. As such, it is either correct or incorrect, and specific statements in that systems will be true or false. For example, it is true or false that Aristotle's syllogisms are logically valid. Here "logically valid" can only mean valid according to our logical faculty.

It is of course trivial to verify for oneself whether the following syllogism is logically valid:

All sailors are courageous;

Some Greeks are sailors;

So, some Greeks are courageous.

Note that this requires no expertise in formal logic. All you need is to understand what the syllogism means and this should be enough for you to decide that if the premises are true, then the conclusion is true.

Asserting that a syllogism is logically valid, or asserting that some other logical statement is a logical truth is just one species of theoretic statements that may be true or false of the logic of human deductive reasoning.

Aristotle's syllogistic is just one possible system of formal logic but there is only one logic of human deductive reasoning, so each system will be either true or false, and they won't possibly be all true.

They may even be all false. Personally, I believe they are all false. The fact that many people are still working on different systems is also evidence that they themselves believe that no existing system is true of the logic human deductive reasoning.

Answer 7

Logic is the formal expression of the rules of reason. Our reason uses Logic to find what is true or false. More on truth here.

Many philosophers sustained the position that Logic is tautological, like Russell. But you don't need Russell for that: any rule validating logic must necessarily be... a logical rule. You see? Logic is valid in a circular fashion. There are propositions to sustain Logic on external rules, but that's a different and quite controversial philosophical subject.

Considering that, the answer to the second question is yes. Given that Logic is tautological, Logic itself is true.

Answer 8

Recently, I read an Amazon review by one Robert W. Sawyer which fully expresses my own deliberations in its last paragraph,

https://www.amazon.com/Defending-Axioms-Philosophical-Foundations-Theory-ebook/dp/B005UMHV9Q/ref=mp_s_a_1_2?crid=CZANA6X9JIG0&keywords=penelope+maddy&qid=1660303590&sprefix=penelope+maddy%2Caps%2C121&sr=8-2

If one takes formalism in mathematics seriously, then words like "true" and "false" are nothing more than meaningless constants subject to interpretation. What becomes important, then, is meaningfulness.

What is the specification of syntax by structural induction other than a description of what is and what is not meaningful relative to an alphabet of symbols? So, a syntactic transformation in a logical calculus is a rule relating one meaningful formula to a subsequent meaningful formula.

Clearly, the notion of "truth" as the sole semantic valuation governing such transformations establishes the plurality of valuations as a pointed set. But, it only expresses truthfulness relative to what a language user believes. One reason one may now speak of a plurality of logics is because the very idea of a reduction to logic is dubious to some.

The word "pragmatics" had been popularized by Rudolph Carnap in response to criticism that his work devolved into meaningless syntax. So, instead of a dichotomy between syntax and semantics, one has a trichotomy between syntax, semantics, and pragmatics. Pragmatics is intended to refer to how syntax is meaningful to language users.

Relative to such an account, semantics is motivated by the fact that a logical calculus assumes a pointed set of valuations. Indeed, one may understand Tarski's notion of satisfaction as the means of assuring that formulas in a derivation with free variables can be understood as "true" relative to this presupposition of a logical calculus.

Appropriately, semantics would not be motivated by what language users "believe" as is commonly assumed in rhetorical argumentation.

If one recognizes that these logical deliberations from the last two centuries really cannot differentiate truth from belief, doxastic logic becomes a very interesting study,

https://en.m.wikipedia.org/wiki/Doxastic_logic

Good luck learning about logic. It is a very intriguing subject. If you look around at people just trying to survive in this world, it is clear that logic is not meaningful to many.