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Logical validity is a property of deductive arguments where, if the premises are true, the conclusion must also be true. It focuses on the form of the argument rather than the actual truth of the premises or conclusion.
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Deductive reasoning is a logical process where conclusions are drawn from a set of premises that are assumed to be true, ensuring the conclusion must also be true if the premises are correct. This method is often used in mathematics and formal logic, providing certainty and clarity in arguments by moving from general principles to specific instances.
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Syllogism is a form of deductive reasoning where a conclusion is drawn from two given or assumed propositions (premises), each sharing a common term with the conclusion. It is a foundational element in formal logic, providing a structured way to derive logical conclusions from general statements.
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Formal logic is a system of reasoning that uses structured and symbolic representation to deduce the validity of arguments. It provides a framework for distinguishing between valid and invalid reasoning through rules and principles that are universally applicable across different contexts.
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Truth value is a fundamental concept in logic and mathematics that determines the truth or falsity of a proposition or statement. It is typically represented as either 'true' or 'false' in classical logic, but can have more complex representations in other logical systems, such as many-valued or fuzzy logic.
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Soundness refers to the property of an argument where if the premises are true, the conclusion must also be true, ensuring both validity and truthfulness. It is a crucial concept in logic and reasoning, providing a standard for evaluating the reliability of deductive arguments.
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Inference is the cognitive process of drawing conclusions from available information, often filling in gaps where data is incomplete. It is fundamental in reasoning, allowing us to make predictions, understand implicit meanings, and form judgments based on evidence and prior knowledge.
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Propositional logic is a branch of logic that deals with propositions, which can be either true or false, and uses logical connectives to form complex statements. It is fundamental in mathematical logic and computer science for reasoning about truth values in a formal, structured manner.
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Predicate logic extends propositional logic by including quantifiers and predicates, allowing for more expressive statements about objects and their properties. It forms the foundation of formal reasoning in mathematics and computer science, enabling the representation and manipulation of complex logical expressions.
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Logical consequence is a fundamental concept in logic that describes the relationship between premises and conclusion, where if the premises are true, the conclusion must also be true. It is central to understanding the validity of arguments and is used to determine whether a set of statements logically entails another statement.
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Truth preservation refers to the property of logical systems where the truth of premises guarantees the truth of the conclusion. It is a fundamental feature of valid deductive arguments, ensuring that if the initial statements are true, the resulting statement must also be true.
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The Completeness Theorem, established by Kurt Gödel, states that every logically valid formula in first-order logic is provable, ensuring that the axioms and inference rules are sufficient to derive all truths expressible in the system. This theorem is fundamental in mathematical logic as it bridges the gap between semantic truth and syntactic provability, highlighting the power and limitations of formal systems.
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Logical meaning refers to the interpretation of statements or propositions in terms of their truth values and the relationships between them within a formal logical system. It is crucial for understanding the validity of arguments and for ensuring clarity and precision in reasoning processes.
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Gödel's Completeness Theorem states that every logically valid formula in first-order logic can be proven within a formal system, meaning that if a formula is true in every model, there is a finite proof of it using the axioms and rules of inference of the system. This theorem establishes a fundamental link between semantic truth and syntactic provability, highlighting the power and limitations of formal systems in capturing logical truths.
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Categorical logic is a branch of logic that deals with the use of categorical propositions and syllogisms to deduce conclusions from premises. It is foundational to understanding classical logic and provides a framework for analyzing the validity of arguments based on the logical relationships between categories or classes of objects.
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Validity and soundness are critical concepts in logic and philosophy, determining the strength and reliability of arguments. An argument is valid if the conclusion logically follows from the premises, and it is sound if it is both valid and its premises are true.
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Entailment refers to a logical relationship where one statement necessarily follows from another. In a valid entailment, if the premises are true, the conclusion must also be true, making it a fundamental concept in logic and reasoning.
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The universal quantifier, denoted as ∀, is a logical symbol used in predicate logic to assert that a given property or predicate holds for all elements within a particular domain. It is foundational in mathematical logic and formal reasoning, allowing for the expression of statements like 'for all x, P(x)' which signifies that the predicate P is true for every element x in the domain considered.
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The paradoxes of implication highlight the counterintuitive outcomes that can arise from the logical implication, where a false statement can imply any statement, and a true statement is implied by any statement. These paradoxes challenge our understanding of logical consequence and necessitate a deeper examination of the principles underlying logical systems.
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Term logic, originating from Aristotle's syllogistic logic, is a formal system that focuses on the logical relationships between terms or categories, such as 'all', 'some', and 'none'. It serves as the foundation for classical logic, emphasizing the structure of arguments through the arrangement of these terms to deduce conclusions from premises.
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