An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
In logic, the law of non-contradiction (LNC) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "the house is white" and "the house is not white" are mutually exclusive. Formally, this is expressed as the tautology ¬(p ∧ ¬p). For example it is tautologous to say "the house is not both white and not white" since this results from putting "the house is white" in that formula, yielding "not ", then rewriting this in natural English. The law is not to be confused with the law of excluded middle which states that at least one of two propositions like "the house is white" and "the house is not white" holds.
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below.
Objectivism is a philosophical system named and developed by Russian-American writer and philosopher Ayn Rand. She described it as "the concept of man as a heroic being, with his own happiness as the moral purpose of his life, with productive achievement as his noblest activity, and reason as his only absolute".
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.
Relativism is a family of philosophical views which deny claims to objectivity within a particular domain and assert that valuations in that domain are relative to the perspective of an observer or the context in which they are assessed. There are many different forms of relativism, with a great deal of variation in scope and differing degrees of controversy among them. Moral relativism encompasses the differences in moral judgments among people and cultures. Epistemic relativism holds that there are no absolute principles regarding normative belief, justification, or rationality, and that there are only relative ones. Alethic relativism is the doctrine that there are no absolute truths, i.e., that truth is always relative to some particular frame of reference, such as a language or a culture, while linguistic relativism asserts that a language's structures influence a speaker's perceptions. Some forms of relativism also bear a resemblance to philosophical skepticism. Descriptive relativism seeks to describe the differences among cultures and people without evaluation, while normative relativism evaluates the word truthfulness of views within a given framework.
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
In mathematics and logic, an axiomatic system is any collection of primitive notions and axioms to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.
In philosophical epistemology, there are two types of coherentism: the coherence theory of truth, and the coherence theory of justification.
The Critique of Pure Reason is a book by the German philosopher Immanuel Kant, in which the author seeks to determine the limits and scope of metaphysics. Also referred to as Kant's "First Critique", it was followed by his Critique of Practical Reason (1788) and Critique of Judgment (1790). In the preface to the first edition, Kant explains that by a "critique of pure reason" he means a critique "of the faculty of reason in general, in respect of all knowledge after which it may strive independently of all experience" and that he aims to decide on "the possibility or impossibility of metaphysics".
The is–ought problem, as articulated by the Scottish philosopher and historian David Hume, arises when one makes claims about what ought to be that are based solely on statements about what is. Hume found that there seems to be a significant difference between descriptive statements and prescriptive statements, and that it is not obvious how one can coherently transition from descriptive statements to prescriptive ones.
A truism is a claim that is so obvious or self-evident as to be hardly worth mentioning, except as a reminder or as a rhetorical or literary device, and is the opposite of a falsism.
Dialetheism is the view that there are statements that are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true contradictions", dialetheia, or nondualisms.
Ethical intuitionism is a view or family of views in moral epistemology. It is foundationalism applied to moral knowledge, the thesis that some moral truths can be known non-inferentially. Such an epistemological view is by definition committed to the existence of knowledge of moral truths; therefore, ethical intuitionism implies cognitivism.
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic.
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
A self-refuting idea or self-defeating idea is an idea or statement whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are called self-refuting by their detractors, and such accusations are therefore almost always controversial, with defenders stating that the idea is being misunderstood or that the argument is invalid. For these reasons, none of the ideas below are unambiguously or incontrovertibly self-refuting. These ideas are often used as axioms, which are definitions taken to be true, and cannot be used to test themselves, for doing so would lead to only two consequences: consistency or exception (self-contradiction).
Evidence for a proposition is what supports the proposition. It is usually understood as an indication that the proposition is true. The exact definition and role of evidence vary across different fields. In epistemology, evidence is what justifies beliefs or what makes it rational to hold a certain doxastic attitude. For example, a perceptual experience of a tree may serve as evidence to justify the belief that there is a tree. In this role, evidence is usually understood as a private mental state. In phenomenology, evidence is limited to intuitive knowledge, often associated with the controversial assumption that it provides indubitable access to truth.
