This leads it to reject certain rules of inference found in classical logic that are not compatible with this assumption.
This way, various fundamental philosophical concepts, like possibility, necessity, obligation, permission, and time, are treated in a logically precise manner by formally expressing the inferential roles they play in relation to each other.
[1] Treatises on modern logic often treat these different systems as a list of separate topics without providing a clear classification of them.
[13][1][4][9] These new symbols are then integrated into the logical mechanism by specifying which new rules of inference apply to them, like that possibility follows from necessity.
In this sense, they are not mere extensions of it but are often formulated as rival systems that offer a different account of the laws of logic.
[13][15] Expressed in a more technical language, the distinction between extended and deviant logics is sometimes drawn in a slightly different manner.
[1] Pluralists, on the other hand, hold that a variety of different logical systems can all be correct at the same time.
[13][20][4] So not everyone agrees that all the formal systems discussed in this article actually constitute logics, when understood in a strict sense.
[5][14] Classical logic was initially created in order to analyze mathematical arguments and was applied to various other fields only afterward.
[5] For this reason, it neglects many topics of philosophical importance not relevant to mathematics, like the difference between necessity and possibility, between obligation and permission, or between past, present, and future.
[13][5] In first-order logic, the propositions themselves are made up of subpropositional parts, like predicates, singular terms, and quantifiers.
[27][28][30] It is usually advantageous to have the strongest system possible in order to be able to draw many different inferences.
But this brings with it the problem that some of these additional inferences may contradict basic modal intuitions in specific cases.
[34][14][35] Of central importance in ethics are the concepts of obligation and permission, i.e. which actions the agent has to do or is allowed to do.
[41] While similar approaches are often seen in physics, logicians usually prefer an autonomous treatment of time in terms of operators.
This is also closer to natural languages, which mostly use grammar, e.g. by conjugating verbs, to express the pastness or futurity of events.
[12] For example, Peano arithmetic and Zermelo-Fraenkel set theory need an infinite number of axioms to be expressed in first-order logic.
[9][26][6] This is often interpreted as meaning that higher-order logic brings with it a form of Platonism, i.e. the view that universal properties and relations exist in addition to individuals.
[50] These modifications of classical logic are motivated by the idea that truth depends on verification through a proof.
[18] On this interpretation, the law of excluded middle would involve the assumption that every mathematical problem has a solution in the form of a proof.
Free logic avoids these problems by allowing formulas with non-denoting singular terms.
[51] In free logic, often an existence-predicate is used to indicate whether a singular term denotes an object in the domain or not.
[56] Formal semantics of classical logic can define the truth of their expressions in terms of their denotation.
Some versions introduce a second, outer domain for non-existing objects, which is then used to determine the corresponding truth values.
[56][53] Neutral semantics, on the other hand, hold that atomic formulas containing empty terms are neither true nor false.
[56][53] This is often understood as a three-valued logic, i.e. that a third truth value besides true and false is introduced for these cases.
Instead, it is usually formulated with the goal of avoiding certain unintuitive applications of the material conditional found in classical logic.
[67][14][68] For example, the material conditional "if all lemons are red then there is a sandstorm inside the Sydney Opera House" is true even though the two propositions are not relevant to each other.
The fact that this usage of material conditionals is highly unintuitive is also reflected in informal logic, which categorizes such inferences as fallacies of relevance.
[67][14][68] A difficulty faced for this issue is that relevance usually belongs to the content of the propositions while logic only deals with formal aspects.