Can we have a logic different from classical logic without breaking or modifying any logic rule?

Question

Can we have a logic different from classical logic without breaking or modifying any logic rule?

Let's assume that classical logic is defines as a logic with these rules:

Law of excluded middle and double negation elimination
Law of noncontradiction, and the principle of explosion
Monotonicity of entailment and idempotency of entailment
Commutativity of conjunction
De Morgan duality: every logical operator is dual to another

Can we have a different logic system without changing any of these rules and modifying some other properties within the classical logic system, by properties I mean basically anything, even things that aren't properties. I was making the assumption that properties are the only other thing besides logic rules, but maybe I am mistaken.

Answer 1

Non-classical logics can be divided into three categories. There are those that may be called sublogics, because the set of their theorems is a proper subset of the theorems of classical logic. There are extensions of classical logic, which have the property that the set of their theorems is a proper superset of those of classical logic. And there are contra-classical logics, whose theorems differ from those of classical logic.

An example of the first type is intuitionistic logic. It differs from classical logic by lacking the law of excluded middle and double negation elimination. There are also substructural logics that lack structural rules such as contraction or weakening. With these, some of the rules and properties of classical logic are dropped or restricted.

The second category is trickier to describe, because we have to be careful with the term 'extension'. In many cases, we cannot extend a logic just by adding axioms or rules. Some logics have the property that they are Post complete meaning that they have no consistent proper extension. This is true of classical propositional logic, for example. But we can create new logics by introducing additional logical operators or connectives and using classical logic as the underlying logic. This is true with the family of modal logics, for example, or temporal logic, or the various conditional logics.

The third category includes connexive logic and abelian logic. These have different rules from classical logic.

So the answer to your question, is: yes we can have non-classical logics that don't break the classical rules, but only by extending classical logic.

Answer 2

Yes, you can add new rules.

There are only 3 ways to change something:

• Modify a part ("in place" i.e. transform it)
• Delete a part
• Add a new part

Note that changing interactions of parts is not an independent / elemental way. Its because to change interaction between two parts you have to modify the parts themselves, and this is already included above.

Your question ruled out modifying and deleting any part (rule) of a system. So you are left with only adding new parts (rules).

A new rule can be as trivial as:

"All other rules in this system work in reverse on wednesdays"

There, you have a new logical system. Fully valid (consistent within).

Now, to tell that your system is of any worth, that is, it tells true whats true and false whats false when you run data in it, you have to check the truthiness with reality (data / evidence/ observation).

Answer 3

It looks like you are aware of some non-classical logics and your conditions are designed to rule out the ones you know. For example your first condition is designed to rule out multivalued logics and Intuitionistic logic, and your second rule is intended to rule out Relevance logic.

I don't believe you have ruled out modal logics, or other types of logics that use modal operators such as temporal logics, deontic logics, or intensional logics. You don't seem to have ruled out free logic which allows variables to range over things that don't exist.

There are probably more. You might want to go to the Stanford Encyclopedia of Philosophy and search for "logic".