It is a formal science that investigates how conclusions follow from premises in a topic-neutral manner, i.e. independent of the specific subject matter discussed.
Another question concerns the nature of premises and conclusions, i.e. whether to understand them as thoughts, propositions, or sentences, and how they are composed of simpler constituents.
Together, premises and a conclusion constitute an inference, which can be either deductive and ampliative depending on whether it is necessarily truth-preserving or introduces new and possibly false information.
The criteria of valid inference and logical truth can themselves be specified in different ways: based on syntactic or semantic considerations.
[2] One problem for this characterization is that logic is not an empirical discipline studying the regularities found in actual human thinking: this subject belongs to psychology.
[2] This reflects the practical significance of logic as a tool to improve one's reasoning by drawing good inferences and becoming aware of possible mistakes.
[14] For example, it could be argued that first-order logic has individuals as its subject matter, due to its usage of singular terms and quantifiers, and is therefore not completely topic-neutral.
[4] It has been argued that one central requirement is that the marks and how they are manipulated can be interpreted in such a way as to reflect the basic intuitions about valid arguments.
[5] This is usually achieved by expressing them through a formal symbolism to make these features explicit and independent of the ambiguities and irregularities of natural language.
[5] This means that for a valid argument, it is not possible to produce true premises with a false conclusion by substituting their constituents with elements belonging to similar categories while keeping the logical constants in place.
This distinction lies at the very heart of the syntactic approach due to its role in the definition of valid inference or logical truth.
[2] On the other hand, it ignores the relationship between language and world, since it defines truth based on the interpretation that takes place only between symbols and set-theoretic objects.
For a long time in history, Aristotelian syllogistics was treated as the canon of logic and there were very few substantial improvements to it for over two thousand years until the works of George Boole, Bernard Bolzano, Franz Brentano, Gottlob Frege, and others.
[3] These developments were often driven by a need to increase the expressive flexibility of logic and to adapt it to specific areas of usage.
[27] Formal logic is concerned with the validity of inferences or arguments based only on their form, i.e. independent of their specific content and the context in which they are used.
[12] One recurrent problem concerns the word "is" in the English language, which has a variety of meanings depending on the context, such as identity, existence, predication, class-inclusion, or location.
But this does not solve many of the problems that the usage of natural language brings with it, like ambiguities, vague expressions, or implicitly assuming premises instead of explicitly stating them.
[1][8] An influential interpretation of modal operators, due to Saul Kripke, understands them as quantifiers over possible worlds.
[1][8] One problem for this type of characterization is that they seem to be circular since possible worlds are themselves defined in modal terms, i.e. as ways how things could have been.
[8] Even when restricted to alethic modal logic, there are again different types of possibility and necessity that can be meant by these terms.
The law fails for cases where no such proof is possible, which exist in every sufficiently strong formal system, according to Gödel's incompleteness theorems.
[2][6] A central motivation for this type of modification is that free logic can be used to analyze discourse with empty singular terms, like in the expression "Santa Claus does not exist".
Some versions claim that a proposition is true if believing it is useful, if it is the ideal result of an endless inquiry, or if it meets the standards of warranted assertibility.
[55] This is sometimes expressed by stating that analytical truths are tautologies, whose denial would imply a contradiction, while it is possible for synthetic propositions to be true or false.
The proposition "some bachelors are happy", on the other hand, is synthetically true since it depends on empirical factors not included in the meaning of its terms.
This approach has been rejected by various philosophers since it has proven difficult to specify clear identity criteria for these types of entities.
[5][66][67] According to Alfred Tarski, deductive inference has three central features: (1) it is formal, i.e. it depends only on the form of the premises and the conclusion; (2) it is a priori, i.e. no sense experience is needed to determine whether it obtains; (3) it is modal, i.e. that it holds by necessity for the given propositions, independent of any other circumstances.
[1] One approach to this issue is to characterize the norms of logic as generalizations of the inferential practices found in natural language or the sciences.
Free logic, for example, allows empty singular terms, which do not denote any object in the domain and therefore carry no ontological commitments.
Logic, on the other hand, has the objective of discovering the laws of correct reasoning, independently of whether actual human thinking often falls short of this ideal.