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.
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law / principleof the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur or "no third [possibility] is given". In classical logic, the law is a tautology.
In logic, the semantic principleof bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic.
Truth or verity is the property of being in accord with fact or reality. In everyday language, it is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences.
In their most common sense, the terms thought and thinking refer to cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, reasoning, concept formation, problem solving, and deliberation. But other mental processes, like considering an idea, memory, or imagination, are also often included. These processes can happen internally independent of the sensory organs, unlike perception. But when understood in the widest sense, any mental event may be understood as a form of thinking, including perception and unconscious mental processes. In a slightly different sense, the term thought refers not to the mental processes themselves but to mental states or systems of ideas brought about by these processes.
A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
A fallacy is the use of invalid or otherwise faulty reasoning in the construction of an argument that may appear to be well-reasoned if unnoticed. The term was introduced in the Western intellectual tradition by the Aristotelian De Sophisticis Elenchis.
Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is sound if it is valid and all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning.
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. For instance the sentence "The sky is blue" denotes the proposition that the sky is blue. However, crucially, propositions are not themselves linguistic expressions. For instance, the English sentence "Snow is white" denotes the same proposition as the German sentence "Schnee ist weiß" even though the two sentences are not the same. Similarly, propositions can also be characterized as the objects of belief and other propositional attitudes. For instance if one believes that the sky is blue, what one believes is the proposition that the sky is blue. A proposition can also be thought of as a kind of idea: Collins Dictionary has a definition for proposition as "a statement or an idea that people can consider or discuss whether it is true."
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 term logic, the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate On Interpretation and its distinction between two oppositions: contradiction and contrariety. However, Aristotle did not draw any diagram; this was done several centuries later by Apuleius and Boethius.
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word infer means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular evidence to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction.
Intuitionistic type theory is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types.
Relevance is the concept of one topic being connected to another topic in a way that makes it useful to consider the second topic when considering the first. The concept of relevance is studied in many different fields, including cognitive sciences, logic, and library and information science. Most fundamentally, however, it is studied in epistemology. Different theories of knowledge have different implications for what is considered relevant and these fundamental views have implications for all other fields as well.
Ethical subjectivism is the meta-ethical view which claims that:
- Ethical sentences express propositions.
- Some such propositions are true.
- The truth or falsity of such propositions is ineliminably dependent on the attitudes of people.
The Topics is the name given to one of Aristotle's six works on logic collectively known as the Organon. In Andronicus of Rhodes' arrangement it is the fifth of these six works.
According to the redundancy theory of truth, asserting that a statement is true is completely equivalent to asserting the statement itself. For example, asserting the sentence "'Snow is white' is true" is equivalent to asserting the sentence "Snow is white". The philosophical redundancy theory of truth is a deflationary theory of truth.
The psychology of reasoning is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory.
Heuristics is the process by which humans use mental shortcuts to arrive at decisions. Heuristics are simple strategies that humans, animals, organizations, and even machines use to quickly form judgments, make decisions, and find solutions to complex problems. Often this involves focusing on the most relevant aspects of a problem or situation to formulate a solution. While heuristic processes are used to find the answers and solutions that are most likely to work or be correct, they are not always right or the most accurate. Judgments and decisions based on heuristics are simply good enough to satisfy a pressing need in situations of uncertainty, where information is incomplete. In that sense they can differ from answers given by logic and probability.
