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 wikipedia/en/Logical_connective like ∧(wikipedia/en/Logical_conjunction) or →figure1: [bin/assets/69dd90e3dc20bc4ee18427e50ac6cc03_MD5.svg]. 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.
Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have the strongest form of support: if their premises are true then their conclusion must also be true. This is not the case for ampliative arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into inductive and abductive arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens. Abductive arguments are inferences to the best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer. Arguments that fall short of the standards of correct reasoning often embody fallacies. Systems of logic are theoretical frameworks for assessing the correctness of arguments.
Logic has been studied since antiquity. Early approaches include Aristotelian logic, Stoic logic, Nyaya, and Mohism. Aristotelian logic focuses on reasoning in the form of syllogisms. It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as Gottlob Frege. Today, the most commonly used system is classical logic. It consists of propositional logic and first-order logic. Propositional logic only considers logical relations between full propositions. First-order logic also takes the internal parts of propositions into account, like predicates and quantifiers. Extended logics accept the basic intuitions behind classical logic and apply it to other fields, such as metaphysics, ethics, and epistemology. Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic.
The History of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic (or term logic) as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
Christian and Islamic philosophers such as Boethius (died 524), Avicenna (died 1037), Thomas Aquinas (died 1274) and William of Ockham (died 1347) further developed Aristotle’s logic in the Middle Ages, reaching a high point in the mid-fourteenth century, with Jean Buridan. The period between the fourteenth century and the beginning of the nineteenth century saw largely decline and neglect, and at least one historian of logic regards this time as barren. Empirical methods ruled the day, as evidenced by Sir Francis Bacon’s Novum Organon of 1620.
Logic revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formal discipline which took as its exemplar the exact method of proof used in mathematics, a hearkening back to the Greek tradition. The development of the modern “symbolic” or “mathematical” logic during this period by the likes of Boole, Frege, Russell, and Peano is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.
Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.
The four laws of logic, also known as the laws of thought, are the fundamental principles underlying all logical reasoning: the Law of Identity, Law of Non-Contradiction, Law of Excluded Middle, and Law of Sufficient Reason. These laws provide a framework for evaluating arguments and determining their validity.
Law of Identity: This law states that something is what it is. In simpler terms, “A is A”. It means that a thing is identical to itself. For example, a cat is a cat, not a dog or a chair.
Law of Non-Contradiction: This law states that a statement cannot be both true and false at the same time. In other words, “A cannot be both B and not B”. For example, a coin cannot be both heads and tails at the same time (in the same context).
Law of Excluded Middle: This law states that a statement is either true or false; there is no middle ground. For example, a statement like “It is raining” must be either true or false, there isn’t a third option. It can be represented as “either A or not A”.
Law of Sufficient Reason: This law suggests that everything has a reason or cause for its existence. While not universally accepted as a core logical law like the first three, it is often included as a foundational principle in reasoning, suggesting that things don’t just happen randomly. It implies that if something exists, there must be a reason for it.
These laws, particularly the first three, are considered foundational to formal logic and are used to construct and evaluate arguments.
Law of Identity
The Law of Identity, a fundamental principle in logic and philosophy, states that every object is identical to itself. In simpler terms, “A is A” or “A thing is what it is”. It’s one of the three classical laws of thought, alongside the law of non-contradiction and the law of the excluded middle.
Formal Representation: In formal logic, the law is often expressed as “∀x (x = x)”, where “∀x” means “for all x” (every object) and ”=” denotes identity.
Basic Principle: It asserts that an object possesses a specific nature and that this nature is consistent with itself.
Implications:
Distinguishability: The law implies that everything is distinct from everything else, as each thing has its own unique identity.
Knowability: Because reality has a definite nature (identity), it is knowable.
Non-Contradiction: The law also underlies the law of non-contradiction, as a contradiction implies that something is both itself and not itself, which is impossible according to the law of identity.
Beyond Logic: The law of identity has implications beyond formal logic, influencing how we understand concepts like self-identity, personal identity, and even the nature of reality itself.
Critiques: Some philosophers, like Hegel, have critiqued the law of identity, arguing that it oversimplifies reality by treating things as static and unchanging, neglecting the dynamic and relational aspects of existence.
Law of Excluded Middle
The Law of the Excluded Middle is a fundamental principle in classical logic. It states that for any proposition, either that proposition is true, or its negation is true; there is no third option or middle ground. Essentially, every well-formed logical statement must be either true or false.
Core Idea: The law asserts that there is no “middle” state between a statement being true and its negation being true.
Formal Statement: For any proposition P, either P is true, or the negation of P (¬P) is true.
Example: If the statement “The sky is blue” is considered, the Law of the Excluded Middle dictates that either the statement “The sky is blue” is true, or the statement “The sky is not blue” is true. There is no third possibility within this framework.
Classical Logic: The Law of the Excluded Middle is a cornerstone of classical logic.
Contrasted with Intuitionistic Logic: In contrast to classical logic, some logical systems, like intuitionistic logic, do not universally accept the Law of the Excluded Middle.
Relevance to Proofs: The law is used in proofs by contradiction (reductio ad absurdum), where proving the negation of a statement leads to a contradiction, thus proving the original statement.
Law of Non-Contradiction
The Law of Non-Contradiction, a fundamental principle of logic, states that a statement and its negation cannot both be true at the same time and in the same sense. Essentially, something cannot both be and not be simultaneously. This principle is considered a cornerstone of reasoning and is widely accepted as a self-evident truth.
Core Idea: The law of non-contradiction asserts that if a statement is true, then its opposite (negation) must be false.
Formal Representation: In logic, this is often expressed as ¬(p ∧ ¬p), where ‘p’ represents a statement and ‘¬p’ represents its negation. This formula means “not (p and not p)”, signifying that both p and its negation cannot be true simultaneously.
Example: If a statement is “The cat is on the mat,” then its negation, “The cat is not on the mat,” cannot be true at the same time and in the same way, wikipedia/en/Law_of_noncontradiction.
Importance: The law is crucial for avoiding logical fallacies and ensuring coherent reasoning. According to Philosophy Stack Exchange, if contradictions were allowed, anything could be proven, rendering logic meaningless.
Philosophical Debate: While widely accepted, some philosophers, like Hegel, have proposed alternative logical systems that challenge the law’s universality, according to Quora.
Applications: The law of non-contradiction is fundamental in various fields, including mathematics, computer science, and everyday reasoning.
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