# What defines if an inference is correct or not, (regarding different logics)?

For me, it makes sense to say for example:

1. From a follows b.
2. Not b.
3. Therefore not a.

I can't explain it, but it's "logical". Now I've read that there are different kind's of logic out there and this here for example might be the "classical logic". What about Buddhist logic for example? What defines, that my inference from (1, 2) to (3) is valid? I was assuming, that when not b, there could never be a because otherwise it would have lead to b which leads to a contradiction with (2) but are there other logics, where that inference could be invalid?

• What defines it is the interpretation. In intuitionistic logic, for example, "from a follows b" is interpreted as "any proof of a can be turned into proof of b" and you have to show, for this inference to be valid, that any proof of not-b can be turned into proof of not-a. That is possible. But equally "logical" (classically) "from not-b follows not-a; a; therefore, b" is not valid intuitionistically, see Math SE. Oct 14, 2023 at 0:10
• Your so called ‘inference correct or not’ is technically defined as validity or soundness loosely speaking in all useful logics with harmony which can be easily googled, usually they all have a soundness theorem to ensure their inference is ‘correct’ under their respective logic calculus semantics… Oct 14, 2023 at 7:06

There are, as you say, many different logics. And sometimes they differ as to whether a particular form of inference is valid. The form you have chosen there is called modus tollens and is valid in nearly all commonly used logics. There have been some claimed counterexamples to this form, but none that have found general acceptance. As far as I can see, this form would fail only in a paraconsistent logic that tolerates true contradictions.

A better example might be something like this:

1. Not everything has the property F.
2. Therefore, there is something that lacks the property F.

This argument form is valid in classical logic, but not in intuitionistic logic. The difference here may be attributed to a difference in what the intuitionist understands by a claim that something exists. Classical logic permits non-constructive existence claims: something of a particular description exists, but I can't show you what it is. Intuitionistic logic is more fussy and does not permit a claim of existence unless it is accompanied by a constructive warrant.

Another example is this:

1. P is true.
2. Therefore, P or Q is true.

This form is valid in classical logic, but is not accepted in some relevance logics, because the Q in the conclusion is irrelevant to the premise.

It is common to understand classical logic as accepting an argument as valid if it has no counterexample, i.e. there is no possible case where the premises are all true and the conclusion false. This may also be expressed as saying that in a classically valid argument, there is a guarantee of truth preservation from premises to conclusion.

For many other logics, it is possible to understand them as preserving some other property. Intuitionistic logic preserves constructive provability or warranted assertability. Probability logic preserves high probability (approximately). Relevance logic may be thought of as preserving the integrity of information in a channel. Linear logic may be thought of as preserving the integrity of resource-bound interaction.

More generally, it makes sense when examining a logic to ask about its semantics. What do the symbols in this logic mean, and how are they intended to be used? The answer to this question usually provides the answer to whether particular rules of inference are valid or not. The rules are validated against their use.

• Thank you! So indeed, different logics can have different fallicies? I'm wondering, why that's not mentioned in so many books about logic like "The art of reasoning, An introduction to Logic by David Kelley", I suppose that many logic books take the common classical logic (or the "Aristotle logic") for granted? Is it, because the inference rules/ assumptions (non-contradiction, excluded middle, (idk what classical logic else contains) are so close to our everyday experience?
– iwab
Oct 14, 2023 at 1:30
• Buddhist logic for example, doesn't necessarily need law of identity a=a, because it's said that nothing have an inner nature that percist. For most Westerners, a river is always the same, while Buddhism says it changes every second - just as everything else is constantly changing. I wonder, if those types of logic could also be used for the scientific method.
– iwab
Oct 14, 2023 at 1:35
• So to wrap it up from general to specific: Reasoning > Logic > inference. So reasoning is the broad cognition, evaluating information, making decisions and so on..., then we make an inference based on the available information using an specific logic?
– iwab
Oct 14, 2023 at 1:47
• I would be inclined to agree that most introductory books take classical logic for granted. It is definitely the most commonly used. Oct 14, 2023 at 2:13