Logic
Logic is the systematic study of the principles of valid inference and correct reasoning, serving as a non-empirical science akin to mathematics that evaluates arguments through their structure rather than content.[1] Originating in ancient Greece, logic was pioneered by Aristotle in the 4th century BCE through his development of syllogistic reasoning, a deductive method analyzing categorical propositions in works collectively known as the Organon, which laid the foundation for evaluating the validity of arguments based on premises and conclusions.[2] This Aristotelian framework dominated Western thought for over two millennia, influencing medieval scholasticism and early modern philosophy until the 19th century, when advancements in symbolic notation transformed the field.[3] Key figures like George Boole introduced algebraic approaches to logic in his 1847 work The Mathematical Analysis of Logic, while Gottlob Frege's 1879 Begriffsschrift established modern quantificational logic, enabling precise formalization of mathematical proofs and paving the way for Bertrand Russell and Alfred North Whitehead's Principia Mathematica (1910–1913), which sought to ground mathematics in logic.[4] In the 20th century, Kurt Gödel's incompleteness theorems (1931) revealed fundamental limits to formal systems, profoundly impacting mathematical logic.[4] Contemporary logic encompasses diverse branches, including formal logic, which uses symbolic languages to assess deductive validity, and informal logic, which examines everyday argumentation and fallacies.[1] Within formal logic, propositional logic deals with truth-functional connectives like conjunction and negation, while predicate logic (or first-order logic) incorporates quantifiers to handle relations and variables, forming the basis for automated theorem proving.[5] Specialized areas such as modal logic explore necessity and possibility, temporal logic addresses time-dependent statements, and intuitionistic logic rejects the law of excluded middle, aligning with constructive mathematics.[6] Beyond theory, logic underpins critical disciplines: in philosophy, it clarifies concepts like truth and knowledge; in mathematics, it supports set theory and proof theory; in computer science, it drives programming languages, artificial intelligence, and circuit design; and in linguistics, it models natural language semantics.[7] These applications highlight logic's enduring role in advancing human understanding and technological innovation.[7]Definition
Formal Logic
Formal logic is a branch of logic that examines the validity of inferences based on their structural form rather than their specific content, employing symbolic languages to represent statements and formal rules to derive conclusions from premises. This approach abstracts away from the particular meanings of words or propositions, focusing instead on patterns of reasoning that guarantee truth preservation. By using symbols—such as variables for objects and predicates for properties—formal logic enables the precise analysis of arguments, ensuring that conclusions follow necessarily if the premises are true.[8] Key characteristics of formal logic include its emphasis on precision, deductivity, and the elimination of ambiguity. Precision arises from a strictly defined syntax that specifies how symbols combine to form valid expressions, preventing misinterpretation. Deductivity refers to the use of inference rules, such as modus ponens, which allow step-by-step derivations within a proof system, ensuring that every conclusion is logically entailed by the premises. Ambiguity is avoided through the interplay of syntax and semantics: syntax governs the form of expressions, while semantics assigns interpretations to those forms, clarifying truth conditions across possible worlds or models. These features make formal logic a rigorous tool for evaluating argument validity independently of empirical content.[8] In formal systems, syntax involves the recursive construction of well-formed formulas (wffs), starting from basic atomic formulas (e.g., predicate applied to terms) and building compound expressions according to precise rules. Semantics, in turn, provides interpretations—mappings of symbols to domains and relations—and models, which are structures where a formula holds true if it is satisfied under the interpretation for all relevant assignments. For instance, a classic syllogism can be symbolized to highlight its deductive structure: premises stating that all members of one category possess a property and that all in that category belong to another lead formally to the conclusion that the first category shares the second property, derivable via rules without regard to the categories' content. Formal logic thus contrasts with informal logic, its counterpart in analyzing everyday discourse, by prioritizing symbolic rigor over contextual nuances.[8][9]Informal Logic
Informal logic is the branch of logic whose task is to develop non-formal standards, criteria, and procedures for the analysis, interpretation, evaluation, critique, and construction of argumentation in everyday discourse.[10] It centers on the study of everyday reasoning in natural language, with a particular emphasis on detecting weaknesses in arguments related to relevance (whether premises bear on the conclusion) and acceptability (whether premises are plausible or justified).[10] This approach addresses arguments as they appear in ordinary communication, including public debates, editorials, and casual discussions, rather than idealized or symbolic forms.[10] Key techniques in informal logic include argument reconstruction, which entails clarifying the structure of an argument by identifying its explicit components and uncovering any unstated elements. A central part of this process is the identification of implicit premises—unstated propositions required to connect stated premises to the conclusion, essential for fully understanding and critiquing the argument's logic.[11] Evaluation relies on criteria such as relevance, acceptability, and sufficiency (whether the premises provide enough support for the conclusion), applied contextually to determine an argument's overall strength.[10] Informal logic differs from rhetoric in its focus on truth-seeking through normative standards for rational argumentation, rather than on effective persuasion or audience influence.[10] While rhetoric prioritizes communicative strategies to sway opinions, informal logic promotes critical scrutiny to advance understanding and resolve disputes on evidential grounds.[10] Practical examples of informal analysis include diagramming arguments to map their components visually, such as James Freeman's model, which adapts Stephen Toulmin's layout of claims, data, warrants, and backings for natural language evaluation.[10] Another approach involves assessing dialectical exchanges, as in pragma-dialectics, where arguments are examined within structured discussions to ensure adherence to rules for orderly resolution of differences of opinion. For instance, in analyzing a debate on policy, one might reconstruct implicit assumptions about societal values and evaluate their sufficiency against counterarguments.Basic Concepts
Propositions and Truth Values
In logic, a proposition is the abstract content or meaning expressed by a declarative sentence, which can be evaluated as either true or false but not both.[12] This distinguishes propositions from sentences themselves, which are concrete linguistic forms varying by language or phrasing, whereas propositions capture the invariant semantic content that bears a truth value.[13] For instance, the English sentence "The sky is blue" and its French equivalent "Le ciel est bleu" express the same proposition, which is true under conditions where the sky appears blue due to atmospheric scattering of light.[14] Central to classical logic is the principle of bivalence, which asserts that every meaningful proposition is exactly true or exactly false, excluding any third value, indeterminacy, or gap in truth assignment.[15] This principle underpins the semantic framework of classical systems, ensuring that truth evaluations are exhaustive and mutually exclusive for all propositions.[16] Truth values thus function as semantic assignments, reflecting whether the proposition corresponds to reality: an atomic proposition like "Paris is the capital of France" receives the value true because it accurately states a geographical fact, while "Paris is the capital of Germany" is false.[17] Propositions serve as the foundational elements in logical reasoning, where their truth values enable the construction of arguments by evaluating premises and conclusions.[18]Arguments and Inference
In logic, an argument is defined as a set of statements, known as premises, intended to provide reasons for accepting another statement, called the conclusion, through a process of inference.[19] The premises are propositions that offer support or evidence, while the conclusion is the claim that follows from them.[20] This structure allows for the evaluation of reasoning by examining whether the premises adequately justify the conclusion.[21] Arguments can be explicit, where all premises and the conclusion are fully stated, or implicit, where some elements are omitted under the assumption that they are understood by the audience.[19] A classic example of an explicit argument is the syllogism: "All men are mortal; Socrates is a man; therefore, Socrates is mortal," in which the premises explicitly lead to the conclusion.[20] Implicit arguments often take the form of enthymemes, which are arguments with one or more suppressed premises that the audience is expected to supply based on shared knowledge.[19] For instance, the enthymeme "Socrates is a man, therefore he is mortal" implicitly relies on the premise that all men are mortal.[22] Inference refers to the reasoning process by which a conclusion is drawn from given premises, aiming to extend or apply the information provided.[20] Within arguments, inferences connect premises to conclusions, often with the goal of preserving truth: if the premises are true, the conclusion should follow as true./05:_What_is_Logic/5.01:_Core_Concepts) This truth-preserving aspect underscores the reliability of the inference in logical discourse.[23] Propositions in arguments carry truth values—true or false—that influence the overall assessment of the inference's strength.[21]Validity, Soundness, and Logical Truth
In logic, an argument is valid if, in every possible interpretation or scenario, the truth of all its premises guarantees the truth of its conclusion, irrespective of whether the premises themselves are actually true in the real world.[8] This semantic notion of validity, formalized model-theoretically by Alfred Tarski, emphasizes preservation of truth across all models where the premises hold, ensuring no counterexample exists where premises are true but the conclusion false.[24] For instance, the argument "All humans are mortal; Socrates is human; therefore, Socrates is mortal" is valid because its structure ensures the conclusion follows necessarily from the premises, though the actual truth of the premises depends on empirical facts.[8] Soundness builds upon validity by requiring not only that the argument's form preserves truth but also that all premises are factually true in the given context, thereby guaranteeing the conclusion's truth.[24] In proof-theoretic terms, a deductive system is sound if every provable argument is semantically valid, meaning derivations from true premises yield true conclusions without error.[8] Thus, the aforementioned Socrates argument is sound only if "All humans are mortal" and "Socrates is human" are indeed true, distinguishing soundness from mere validity by incorporating empirical verification of premises.[24] Logical truth pertains to statements that are necessarily true due to their logical form alone, holding in all possible interpretations or models, such as the tautology "If P, then P" or "Either it is raining or it is not raining."[25] These are often called tautologies in propositional logic or theorems derivable without premises in formal systems, reflecting their a priori necessity as articulated by philosophers like Aristotle and Leibniz.[25] Unlike factual truths, which are contingent and empirically verifiable (e.g., "Water boils at 100°C at sea level"), logical truths depend solely on syntactic structure and semantic rules, independent of worldly content or observation.[25] This distinction underscores that logical truths are formal necessities, not discoverable through experience but through analysis of form.[25] These concepts—validity, soundness, and logical truth—form the foundation for evaluating arguments in deductive reasoning, ensuring reliable inference from premises to conclusions.[8]Types of Reasoning
Deductive Reasoning
Deductive reasoning is a form of inference in which the truth of the conclusion is guaranteed by the truth of its premises, meaning that if the premises are true, the conclusion must necessarily be true.[26] This non-ampliative process ensures that the conclusion does not introduce new information beyond what is already entailed by the premises, distinguishing it from forms of reasoning that extend knowledge probabilistically.[27] Key characteristics of deductive reasoning include its certainty, monotonicity, and analytic nature. Certainty arises because the inference preserves truth: a valid deductive argument cannot lead from true premises to a false conclusion.[27] Monotonicity refers to the property that adding further premises to a valid argument cannot invalidate the conclusion; the entailment remains intact or strengthens.[28] The analytic nature means that the conclusion is logically contained within the premises, deriving its truth solely from their meanings and logical relations rather than empirical observation.[29] Classic examples of deductive reasoning include categorical syllogisms and hypothetical reasoning. A categorical syllogism, such as "All A are B; all B are C; therefore, all A are C," demonstrates how universal premises lead to a necessary conclusion about categories.[30] Hypothetical reasoning involves conditional statements, where premises establish a necessary connection, such as deriving an outcome from an antecedent and its condition, ensuring the conclusion follows inescapably.[26] Deductive reasoning forms the foundation for proofs in formal logical systems, where arguments are constructed and verified to establish entailments rigorously.[8] In contrast to ampliative reasoning, which allows for conclusions that go beyond the premises with some uncertainty, deductive methods provide conclusive certainty when premises hold.[27]Ampliative Reasoning
Ampliative reasoning refers to forms of inference in which the conclusion extends beyond the information strictly contained in the premises, introducing new content or generalizations that are not deductively entailed but are supported to varying degrees of probability or plausibility.[27] Unlike deductive reasoning, which preserves truth from premises to conclusion with certainty, ampliative inference allows for the expansion of knowledge while acknowledging uncertainty, making it essential for scientific discovery, everyday decision-making, and hypothesis formation.[31] This type of reasoning, often contrasted with explicative or analytic inference, amplifies the scope of beliefs by drawing conclusions that add substantive information not explicitly present in the initial data.[32] Inductive reasoning, a primary form of ampliative inference, involves generalizing from specific observations to broader principles or predictions, where the conclusion goes beyond the observed instances but gains strength from the size and relevance of the sample.[27] For example, repeatedly observing white swans in various locations might lead to the generalization that all swans are white, though this remains probabilistic and vulnerable to counterexamples like black swans discovered later.[33] The justification for such inferences traces back to David Hume, who highlighted the "problem of induction" by questioning how past regularities can reliably project to unobserved cases without circular assumptions.[33] In practice, the strength of an inductive argument depends on factors such as sample size, diversity of evidence, and absence of bias, enabling applications in fields like statistics and empirical science.[27] Abductive reasoning, another key ampliative process, consists of inferring the most plausible hypothesis that explains given evidence, often termed "inference to the best explanation."[31] Introduced by Charles Sanders Peirce in the late 19th century, it posits that when multiple hypotheses could account for data—such as unusual symptoms suggesting a specific disease—the one offering the simplest, most comprehensive explanation is preferred.[31] A classic example is inferring that a kitchen mess results from a late-night snack rather than a burglary, based on contextual clues like open snack packages.[31] In scientific contexts, abductive steps have driven discoveries, such as hypothesizing Neptune's existence to explain irregularities in Uranus's orbit.[31] Unlike induction's focus on patterns, abduction emphasizes explanatory power, though it too involves uncertainty since alternative explanations may emerge.[31] The evaluation of ampliative reasoning relies on measures of evidential support, such as probabilistic confirmation and Bayesian updating, which assess how evidence increases the likelihood of a hypothesis relative to alternatives.[27] Confirmation theory, developed by philosophers like Rudolf Carnap, quantifies support through likelihood ratios, where evidence confirms a hypothesis if it is more probable under that hypothesis than under rivals.[27] Bayesian approaches conceptualize this via prior beliefs updated by new evidence to yield posterior probabilities, as in Bayes' theorem, which formally balances initial plausibility with evidential fit without guaranteeing truth.[27] These methods provide a framework for weighing inductive generalizations or abductive hypotheses, though challenges like the choice of priors persist.[34] Fallacies, such as hasty generalization in induction or overlooking rival explanations in abduction, can undermine these inferences.[27]Fallacies and Errors
Fallacies and errors in logic refer to flawed patterns of reasoning that undermine the validity of deductive arguments or the strength of ampliative ones, leading to conclusions that do not logically follow from the premises.[35] These errors are broadly classified into formal fallacies, which arise from structural defects in the logical form regardless of content, and informal fallacies, which stem from issues in the argument's content, context, or relevance.[36] Such flaws can occur across deductive and ampliative reasoning, compromising the reliability of inferences in both.[37] Formal fallacies involve invalid logical structures that fail to preserve truth from premises to conclusion, detectable through analysis of the argument's form.[35] A classic example is denying the antecedent, where one argues: "If P, then Q; not P; therefore, not Q." This is invalid because the absence of P does not preclude Q from occurring through other means.[38] Other formal fallacies include affirming the consequent ("If P, then Q; Q; therefore, P"), which similarly overlooks alternative causes for Q.[39] These errors highlight the importance of ensuring that the logical form guarantees the conclusion's truth when premises are true.[36] Informal fallacies, by contrast, depend on the specific content or context of the argument rather than its abstract structure, often involving irrelevance, ambiguity, or insufficient evidence.[40] The ad hominem fallacy occurs when an arguer attacks the character, motives, or circumstances of the opponent instead of addressing the argument itself, such as dismissing a policy proposal by claiming the proponent is untrustworthy due to personal flaws.[41] Another common type is the slippery slope fallacy, where a minor action is claimed to inevitably lead to a chain of extreme, undesirable consequences without supporting evidence for the causal links, for instance, arguing that legalizing a substance will lead to societal collapse.[42] Hasty generalization represents an inductive error by drawing a broad conclusion from an unrepresentative or insufficient sample, such as concluding that all members of a group share a trait based on one atypical example.[43] Detecting and avoiding fallacies plays a central role in critical thinking and debate by promoting rigorous evaluation of arguments.[44] For formal fallacies, one can scrutinize the argument's structure against valid forms, while informal fallacies require assessing relevance, evidence quality, and potential biases in the content.[37] Avoidance involves constructing arguments with clear premises, sufficient support, and direct relevance to the conclusion, thereby enhancing the persuasiveness and integrity of discourse in philosophy, science, and everyday reasoning.[44]Core Formal Systems
Propositional Logic
Propositional logic, also known as sentential logic, is a branch of logic that deals with the structure of compound statements formed from simpler atomic statements using truth-functional connectives, focusing on their validity without regard to internal content.[45] It provides the foundational framework for analyzing arguments based on how the truth values of components determine the truth value of the whole.[46] Atomic propositions, denoted by uppercase letters such as P, [Q](/page/Q), or [R](/page/R), represent basic declarative statements that are either true or false, without further decomposition in this system.[45] Compound propositions are constructed by applying connectives to atomic or other compound propositions. The standard connectives include negation (\neg P), which reverses the truth value of P; conjunction (P \land [Q](/page/Q)), true only if both P and [Q](/page/Q) are true; disjunction (P \lor [Q](/page/Q)), true if at least one of P or [Q](/page/Q) is true; implication (P \to [Q](/page/Q)), false only if P is true and [Q](/page/Q) is false; and biconditional (P \leftrightarrow [Q](/page/Q)), true if P and [Q](/page/Q) have the same truth value.[45][47] The semantics of these connectives are defined by truth tables, which enumerate all possible truth value assignments to the atomic propositions and compute the resulting truth value of the compound proposition. The following table presents the truth tables for the connectives, where T denotes true and F denotes false:| P | Q | \neg P | P \land Q | P \lor Q | P \to Q | P \leftrightarrow Q |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
| P | Q | \neg P | \neg P \lor Q | P \to Q | (P \to Q) \leftrightarrow (\neg P \lor Q) |
|---|---|---|---|---|---|
| T | T | F | T | T | T |
| T | F | F | F | F | T |
| F | T | T | T | T | T |
| F | F | T | T | T | T |