Logical consequence in the context of "Logical reasoning"

Reason is the capacity of consciously applying logic by drawing valid conclusions from new or existing information, with the aim of seeking truth. It is associated with such characteristically human activities as philosophy, religion, science, language, and mathematics, and is normally considered to be a distinguishing ability possessed by humans. Reason is sometimes referred to as rationality, although the latter is more about its application.

Reasoning involves using more-or-less rational processes of thinking and cognition to extrapolate from one's existing knowledge to generate new knowledge, and involves the use of one's intellect. The field of logic studies the ways in which humans can use formal reasoning to produce logically valid arguments and true conclusions. Reasoning may be subdivided into forms of logical reasoning, such as deductive reasoning, inductive reasoning, and abductive reasoning.

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ${\displaystyle \land }$ (and) or ${\displaystyle \to }$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts.

In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems.

An argument is one or more premises—sentences, statements, or propositions—directed towards arriving at a logical conclusion. The purpose of an argument is to give reasons for one's thinking and understanding via justification, explanation, or persuasion. As a series of logical steps, arguments are intended to determine or show the degree of truth or acceptability of a logical conclusion.

The process of crafting or delivering arguments, argumentation, can be studied from three main perspectives: through the logical, the dialectical and the rhetorical perspective.

Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other than its logical constants). Thus, logical truths such as "if p, then p" can be considered tautologies. Logical truths are thought to be the simplest case of statements which are analytically true (or in other words, true by definition). All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.

Logical truths are generally considered to be necessarily true. This is to say that they are such that no situation could arise in which they could fail to be true. The view that logical statements are necessarily true is sometimes treated as equivalent to saying that logical truths are true in all possible worlds. However, the question of which statements are necessarily true remains the subject of continued debate.

A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion.

An argument is meaningful for its conclusion only when all of its premises are true. If one or more premises are false, the argument says nothing about whether the conclusion is true or false. For instance, a false premise on its own does not justify rejecting an argument's conclusion; to assume otherwise is a logical fallacy called denying the antecedent. One way to prove that a proposition is false is to formulate a sound argument with a conclusion that negates that proposition.

Eliminative materialism (also called eliminativism) is a materialist position in the philosophy of mind that expresses the idea that the majority of mental states in folk psychology do not exist. Some supporters of eliminativism argue that no coherent neural basis will be found for many everyday psychological concepts such as belief or desire, since they are poorly defined. The argument is that psychological concepts of behavior and experience should be judged by how well they reduce to the biological level. Other versions entail the nonexistence of conscious mental states such as pain and visual perceptions.

Eliminativism about a class of entities is the view that the class of entities does not exist. For example, materialism tends to be eliminativist about the soul; modern chemists are eliminativist about phlogiston; modern biologists are eliminativist about élan vital; and modern physicists are eliminativist about luminiferous ether. Eliminative materialism is the relatively new (1960s–70s) idea that certain classes of mental entities that common sense takes for granted, such as beliefs, desires, and the subjective sensation of pain, do not exist. The most common versions are eliminativism about propositional attitudes, as expressed by Paul and Patricia Churchland, and eliminativism about qualia (subjective interpretations about particular instances of subjective experience), as expressed by Daniel Dennett, Georges Rey, and Jacy Reese Anthis.