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

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.

• Sure you can only retain, say the set of {and, or} connectives, as functional complete without the need of other connectives. Also in 1st order case you may modify your system's property to not include equality relation symbol whose presence is the default in your classic case. You may even relax the requirement on the domain to go completely free logic way with many more philosophical applications and implications... Jan 7 at 5:11
• What you listed are the laws of Boolean algebra, so your logic would have to be Boolean. There are Boolean algebras other than classical 0-1 algebra, trivially, the four valued one whose values are bit pairs of 0s and 1s with bitwise operations. Obviously, it modifies other properties, namely, the number of truth values. Its interpretation as "logic" is also questionable. Jan 7 at 9:14
• I don't follow. You define classical logic in terms of these rules, and then you ask if you can have a non-classical logic which validates these rules. This does not make any sense to me, can you clarify what you are asking? (This also does not make any sense to me: "by properties I mean basically anything, even things that aren't properties".) Jan 7 at 15:00
• Essentially, I am asking if logical systems are just a collection of logical laws, or there are things outside of logical laws that define them. Jan 7 at 15:21

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.

• OP specifies without changing any of these rules in above spec... Jan 7 at 5:18
• I'm running rules and properties together, since I'm not sure what you mean by distinguishing them. If your question is really whether the rules and properties you have listed are sufficient to define classical logic, then no, we should include transitivity of entailment, bivalence, existential assumptions, and some other stuff as well. You cannot change classical logic without changing/adding/removing/restricting some of its rules and properties. Jan 7 at 9:23
• Is modal logic an extension of classical logic? What about fuzzy logic? Is it too, like in the game universe, an expansion pack? Jan 7 at 11:56
• I would say that a logic is an extension of classical logic if it includes all theorems of classical logic as theorems. In that sense, propositional modal logic is an extension of classical propositional logic. Having said that, there is nothing stopping you from having an intuitionistic modal logic in which the axioms of modal logic are added to an underlying intuitionistic logic. Similarly with fuzzy logic, there are fuzzy systems based on different logics. Gödel-Dummett logic is an extension of intuitionistic logic. Łukasiewicz logic is a kind of many-valued logic. Jan 7 at 12:42
• I would prefer to talk about properties and say that some of the rules are properties. Much of what you listed are properties rather than rules. The law of excluded middle and the law of noncontradiction are theorems of classical logic, not rules. Jan 7 at 14:42

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

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).

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".