Fact-checked by Grok 2 weeks ago

Tautology

A tautology is a concept with primary applications in logic, rhetoric, and philosophy, denoting either a statement that is invariably true due to its logical structure or a redundant repetition of meaning through synonymous expressions.^[1]^[2] In propositional logic, a tautology is defined as a compound formula that evaluates to true for every possible assignment of truth values to its atomic components, making it a fundamental principle of valid inference.^[1] For instance, the proposition p \lor \neg p (read as "p or not p") is a tautology, embodying the law of the excluded middle, which holds true irrespective of whether p is true or false.^[1] This property is verified through truth tables, where all rows yield a true outcome under the formula's main connective.^[3] In rhetorical analysis, tautology refers to the restatement of an idea using alternative words, phrases, or sentences, often viewed as a stylistic fault when it introduces redundancy but sometimes employed deliberately for emphasis.^[2] An example appears in Abraham Lincoln's Second Inaugural Address: "with malice toward none, with charity for all," which reiterates the notion of benevolence through parallel phrasing.^[2] This usage stems from classical traditions, where tautology highlights repetition to reinforce concepts, though it risks weakening discourse if overused.^[4] Philosophically, tautologies extend logical notions to encompass analytically true statements that derive their truth solely from the meanings of their terms, without empirical content, such as "all animals are animals."^[5] These are distinguished from synthetic truths, which require external verification, and play a role in discussions of logical necessity and analyticity in philosophy of language.^[5] Overall, the term underscores the interplay between form, meaning, and expression across these disciplines, influencing fields from formal reasoning to literary criticism.

Logic and Philosophy

Definition in propositional logic

In classical propositional logic, propositions are basic statements that can be either true or false, and compound propositions are formed using logical connectives such as conjunction (\land, "and"), disjunction (\lor, "or"), negation (\lnot, "not"), and implication (\to, "implies"). These connectives allow the construction of complex formulas from atomic propositions, enabling the analysis of their truth values under different interpretations.^[6] A tautology is a compound proposition that is always true, regardless of the truth values assigned to its atomic propositions, making it logically equivalent to the constant true value.^[7] Formally, a formula \phi is a tautology if its truth value is true for every possible truth assignment to the propositional variables it contains; that is, for all valuations v, v(\phi) = \top.^[8] The concept of such always-true statements traces back to Aristotle's formulation of the law of the excluded middle, which asserts that a proposition or its negation must hold, while the specific term "tautology" was introduced in modern logic by Ludwig Wittgenstein in his 1921 Tractatus Logico-Philosophicus.^[9]^[10]

Examples and truth tables

Truth tables provide a systematic method to evaluate the truth value of a propositional formula under all possible assignments of truth values to its atomic propositions, thereby verifying whether it is a tautology by checking if the final column contains only true values. To construct a truth table, first identify the atomic propositions (e.g., p and q) and list all $2^n possible combinations of their truth values (T for true, F for false) in the initial columns, where n is the number of distinct atoms; then, compute the truth value for each compound subformula column by column using the definitions of the connectives, such as disjunction (p \lor q is true unless both are false) or negation (\neg p flips the value of p). Interpretation involves examining the final column: if it is entirely T, the formula is a tautology; if mixed T and F, it is contingent; and if entirely F, a contradiction.^[1] A classic example of a tautology is the law of excluded middle, expressed as p \lor \neg p, which states that a proposition is either true or false, with no middle ground. The truth table for this formula, involving one atomic proposition p, is as follows:

p	\neg p	p \lor \neg p
T	F	T
F	T	T

The final column is always T, confirming it as a tautology regardless of p's value.^[1] Another fundamental tautology is the law of non-contradiction, given by \neg (p \land \neg p), asserting that a proposition cannot be both true and false simultaneously. Its truth table is:

p	\neg p	p \land \neg p	\neg (p \land \neg p)
T	F	F	T
F	T	F	T

Here, the conjunction p \land \neg p is always F, so its negation is always T, verifying the tautology.^[1] The material implication can also be understood through the tautology (p \to q) \leftrightarrow (\neg p \lor q), which equates the conditional to the disjunction of the negation of the antecedent and the consequent. The complete truth table for this equivalence, with two atomic propositions, demonstrates identical truth values in the final two columns:

p	q	p \to q	\neg p	\neg p \lor q	(p \to q) \leftrightarrow (\neg p \lor q)
T	T	T	F	T	T
T	F	F	F	F	T
F	T	T	T	T	T
F	F	T	T	T	T

The biconditional column is entirely T, establishing the logical equivalence and thus the tautological nature of the formula.^[1] A common pitfall in identifying tautologies is confusing them with contingent formulas, which are true only under some assignments but not all; for instance, p \to p yields all T in its truth table and is thus a tautology, whereas p \land q has mixed values (T only when both are T) and is contingent, depending on specific truth assignments rather than holding universally.^[8]

Role in deductive reasoning

In deductive reasoning, tautologies function as inherently valid arguments because their conclusions follow necessarily from the premises under every possible truth assignment. For instance, any tautological formula implies a true statement, ensuring that the inference preserves truth regardless of the specific content of the propositions involved. This property makes tautologies foundational to logical deduction, as they represent cases where the conclusion is guaranteed by the logical structure alone, without reliance on empirical facts.^[6] A key application of tautologies lies in testing the validity of arguments: an argument is deemed valid if the implication formed by its premises as the antecedent and the conclusion as the consequent is itself a tautology. This method confirms that no scenario exists where the premises are true but the conclusion false, thereby upholding the soundness of deductive inferences in propositional logic. Such verification relies on the exhaustive nature of truth assignments, allowing logicians to systematically check for universal truth preservation.^[11] Tautologies also underpin the concept of logical equivalence between formulas, where two propositions are considered equivalent if their biconditional (i.e., \phi \leftrightarrow \psi) qualifies as a tautology, meaning they share the same truth value across all interpretations. This equivalence relation facilitates simplification and transformation in deductive proofs, enabling the replacement of one formula with another without altering the argument's validity.^[6] Philosophically, tautologies are regarded as analytic truths in the philosophy of language, embodying necessary conditions of thought that hold independently of contingent experience, as articulated in Wittgenstein's Tractatus Logico-Philosophicus. There, Wittgenstein posits that the propositions of logic are tautologies, senseless yet unconditionally true, serving to delineate the boundaries of meaningful discourse rather than describing the world.^[12]^[10] Despite their utility, tautologies have inherent limitations: they convey no substantive new information about the world, operating as vacuously true statements that merely reiterate logical necessities without advancing empirical knowledge. This vacuity underscores their role as tools for validation rather than sources of discovery in deductive reasoning.^[11]

Linguistics and Rhetoric

Definition as redundancy

In linguistics and rhetoric, a tautology is defined as the needless repetition of an idea, statement, or word through the use of synonymous or near-synonymous terms, where the redundant elements fail to convey additional meaning or information.^[13] This form of redundancy occurs when the implied sense of a phrase is restated explicitly without necessity, rendering the expression inefficient. The concept emphasizes stylistic overlap rather than substantive expansion, distinguishing it from deliberate repetition for emphasis in poetry or oratory. The term "tautology" derives from the ancient Greek tautologos, meaning "saying the same thing," a compound of tautos ("the same") and logos ("word" or "reason").^[14] Introduced into English in the late 16th century via Late Latin tautologia, it initially critiqued verbose expression in rhetorical analysis.^[13] In contrast to logical tautologies—statements invariably true due to their propositional structure, irrespective of content—linguistic tautologies pertain solely to semantic redundancy in natural language, viewed as a communicative inefficiency rather than a formal truth.^[13] Tautologies frequently appear in spoken English, where they may arise spontaneously to underscore assertions or bridge conversational gaps, and in advertising or casual writing, where brevity is often sacrificed for perceived reinforcement.^[15] However, they are widely regarded as a stylistic flaw, particularly in formal discourse, because they dilute precision and prolong expression without value.^[16] Linguists note their prevalence in everyday communication, yet recommend avoidance to maintain clarity and conciseness.^[17]

Common examples in English

Common tautological expressions in English frequently illustrate redundancy by combining terms that inherently convey the same meaning, often for emphasis or habit. One prevalent example is free gift, where "gift" already implies something given without cost, rendering "free" unnecessary. This phrase is commonly used in advertising to highlight promotional offers, despite its inherent repetition. Similarly, advance warning repeats the idea of prior notice, as a warning is typically given in advance. These constructions demonstrate how everyday language can embed superfluous elements without altering the core message. A significant category of tautologies arises from acronyms, where the expanded form is redundantly appended to the abbreviation itself, a phenomenon termed RAS syndrome (redundant acronym syndrome syndrome). For instance, PIN number expands to "personal identification number number," repeating "number" because PIN already denotes a numeric code. Likewise, ATM machine stands for "automated teller machine machine," duplicating "machine" as the acronym refers to a device. Such expressions have become so ingrained in spoken and written English that speakers often use them unconsciously, prioritizing fluency over precision. Regional variations in English can yield distinct tautologies, particularly in place names influenced by historical languages. In British English, River Avon exemplifies this, as "Avon" derives from a Celtic word meaning "river," resulting in a literal "river river." This pattern appears in several UK river names, reflecting the layering of Anglo-Saxon descriptors onto pre-existing Celtic terms. In contrast, American English features similar redundancies in borrowed names, such as La Brea Tar Pits ("the tar tar pits" from Spanish), though these are less tied to everyday speech. Tautologies also appear in idioms and proverbs, where they serve cultural functions like expressing resignation or inevitability. The phrase it is what it is is a prime example, tautologically stating the obvious to convey acceptance of an unchangeable situation, often in response to disappointment. This expression has permeated modern English idioms, functioning as a shorthand for philosophical detachment despite its repetitive structure. Over time, many tautologies evolve from perceived errors to accepted norms through widespread usage and lexicalization. For example, RAS syndrome phrases like PIN number have entered dictionaries without correction, as their redundancy no longer impedes communication and adds rhythmic familiarity to language. This acceptance highlights English's adaptability, where initial stylistic critiques give way to conventional adoption in both formal and informal contexts.

Relation to pleonasm and other devices

In linguistics and rhetoric, pleonasm refers to the use of superfluous words or phrases that can be omitted without altering the meaning of a sentence, often resulting in redundancy for emphasis or stylistic effect.^[18] For instance, constructions like "I myself saw it" add an unnecessary pronoun for intensification, but the core message remains intact without it.^[18] In contrast, a tautology involves a stricter form of redundancy where two or more elements express logically equivalent or synonymous ideas, rendering one entirely dispensable due to inherent overlap in meaning.^[19] Tautology can be viewed as a subset of pleonasm, specifically where the repetition occurs through near-synonyms or logically identical concepts, such as "free gift," which reiterates the absence of cost in both terms.^[20] While pleonasm encompasses broader superfluousness, including non-synonymous additions like adjectives that merely amplify without equivalence, tautology demands semantic equivalence, making it a more precise violation of conciseness.^[19] This distinction highlights tautology's focus on logical redundancy over general wordiness.^[20] Tautology relates to other rhetorical devices through shared themes of repetition but differs in intent and scope; for example, redundancy serves as a broader umbrella term that includes both pleonasm and tautology as forms of unnecessary information.^[18] Unlike intentional repetition in devices like anaphora, which repeats words or phrases at the beginning of successive clauses for rhythmic emphasis or persuasion—as in Martin Luther King Jr.'s "I have a dream" sequence—tautology typically lacks such deliberate structure and is often inadvertent.^[2] Similarly, tautophony involves the repetition of sounds rather than meanings, such as alliteration or assonance, focusing on phonological rather than semantic overlap.^[21] In stylistic contexts, tautologies are generally advised against in prose, where they are seen as errors that violate clarity and efficiency, potentially weakening arguments by diluting impact.^[15] However, in poetry, they may be employed acceptably for emphasis, rhythm, or to evoke emotional resonance, as seen in William Wordsworth's "The Thorn," where repetitive phrasing underscores themes of doubt and perception.^[22] This selective use balances artistic license against the risk of perceived sloppiness.^[23] From a linguistic perspective, tautologies and related pleonastic forms are analyzed in pragmatics and semantics as violations of Grice's maxim of quantity, which requires contributions to be as informative as needed without excess.^[18] Such violations can signal implicatures, like emphasis or hedging, but in semantics, they highlight how synonymous expressions fail to add propositional content, often critiqued for breaching conversational efficiency.^[24]

Mathematics and Formal Systems

Tautologies in mathematical logic

In first-order predicate logic, a tautology is a sentence that holds true in every possible model, meaning it is semantically valid across all interpretations of its non-logical symbols and all non-empty domains.^[25] This contrasts with propositional logic by incorporating quantifiers and relations over structures, where a model consists of a domain and assignments to predicates, functions, and constants that satisfy the sentence universally.^[26] A classic quantified example is the law of excluded middle expressed as \forall x (P(x) \lor \neg P(x)), which is true in any model because, for every element in the domain, the predicate P either applies or does not, independent of the specific interpretation of P.^[27] Another key example arises from universal instantiation: the schema \forall x \phi(x) \to \phi(t), where t is any term, is a tautology, as it follows from the semantics of the universal quantifier—in any model with domain D, if \phi holds for all elements of D, it holds for the element denoted by t.^[27] Model-theoretically, this validity stems from the universal quantifier ranging over the entire domain, ensuring instantiation preserves truth without requiring additional assumptions about the structure. Gödel's completeness theorem establishes that every tautology in first-order logic corresponds to a provable theorem, meaning any sentence true in all models is derivable from the logical axioms using the rules of inference.^[25] Specifically, for the empty theory, semantic validity (\emptyset \models \phi) is equivalent to syntactic provability (\emptyset \vdash \phi), bridging the gap between truth in structures and formal deduction.^[26] Unlike propositional tautologies, which can be verified exhaustively via finite truth tables, first-order tautologies require consideration of potentially infinite models, complicating direct reduction. Herbrand's theorem addresses this by showing that the validity of a first-order sentence can be reduced to the propositional validity of its Herbrand expansion—a process that eliminates quantifiers through substitutions of terms, yielding a propositional formula whose tautology status determines the original's validity.^[28] For instance, in a universal theory, provability equates to the existence of a finite disjunction of instances that is propositionally tautological. In modern extensions like modal logic, tautologies incorporate necessity operators; for example, \Box (p \to p) is a tautology, as any propositional tautology holds necessarily across all accessible worlds in a Kripke model.^[29] This builds on classical tautologies by ensuring that truths independent of truth assignments remain true under modal constraints.

Applications in proof theory

In Hilbert-style proof systems for classical propositional logic, the axioms are typically chosen as schemes that instantiate to tautologies, ensuring that every theorem derived is semantically valid. A foundational axiom is A \to (B \to A), which is a tautology verifiable by truth table analysis, as it holds regardless of the truth values assigned to A and B.^[6] Other common axioms include (A \to (B \to C)) \to ((A \to B) \to (A \to C)) and (\neg A \to A) \to A, all of which are tautological and form the basis for deriving further theorems without relying on semantics during proof construction.^[6] These systems, pioneered by Frege and refined by Hilbert and Bernays, emphasize a minimal set of inference rules applied to such axioms to achieve completeness, meaning every propositional tautology is provable.^[6] The primary inference rules in Hilbert-style systems—modus ponens and uniform substitution—preserve the tautological status of derived formulas. Modus ponens states that from \phi and \phi \to \psi, one may infer \psi; since the axioms are tautologies and this rule corresponds to the valid implication ((\phi \land (\phi \to \psi)) \to \psi), it ensures that all derivable formulas remain tautologies.^[6] Uniform substitution replaces propositional variables with arbitrary formulas while maintaining structural integrity, thereby preserving validity because tautologies are closed under such substitutions in classical logic.^[30] These rules enable the syntactic derivation of all tautologies, with soundness guaranteed by the fact that no non-tautology can be proven.^[6] A key meta-theorem illustrating the power of these systems is the deduction theorem, which states that if \Gamma, \phi \vdash \psi, then \Gamma \vdash \phi \to \psi. This is proven by induction on the length of the derivation of \psi from \Gamma \cup \{\phi\}, handling cases for axioms, modus ponens, and assumptions.^[6] For example, to prove the tautology (A \to (A \to B)) \to (A \to B), assume A \to (A \to B) and derive A \to B: from the assumption and the axiom A \to ((A \to B) \to B) (an instance of X \to (Y \to X) with X = A \to B, Y = A), apply modus ponens twice to obtain B; then, by the deduction theorem applied to the assumption of A, infer A \to B; finally, apply the deduction theorem again to the outer assumption, yielding the desired tautology.^[31] Tautologies play a central role in automated theorem proving, where checking whether a propositional formula is a tautology reduces to verifying the unsatisfiability of its negation using SAT solvers. These solvers employ resolution, a refutation procedure that derives the empty clause from an unsatisfiable set of clauses, thereby confirming tautologies via clausal form conversion and conflict-driven clause learning.^[32] For instance, modern SAT solvers like MiniSat or Glucose generate resolution proofs for propositional unsatisfiability, which can be extracted and verified to certify tautologies in theorem provers such as Isabelle/HOL.^[33] This integration enables efficient automation for large-scale verification tasks, though proof sizes can grow exponentially in the worst case.^[32] Despite these advances in propositional logic, where tautology checking is decidable via truth tables or SAT solving, limitations arise in higher-order logics, where the validity problem—determining if a formula is a tautology in all models—is undecidable. Alonzo Church proved in 1936 that no algorithm exists to decide validity in first-order logic, a result extending to higher-order systems due to their greater expressive power.^[34] The Church-Turing thesis reinforces this by positing that any effective procedure corresponds to a Turing machine computation, and since no such machine can solve the halting problem (reducible to validity), tautology checking remains inherently non-algorithmic in these logics.^[35]

Distinctions from identities

In mathematics, an identity is an equation that holds true for all values of the variables involved within a specified algebraic structure, such as a group or ring. For instance, in group theory, the equation x + 0 = x is an identity because the additive identity element 0 leaves every element x unchanged under the group operation, regardless of the specific group.^[36]^[37] A key distinction between tautologies and identities lies in their scope and nature: tautologies are truth-functional statements in propositional logic that are true under every possible assignment of truth values to their variables, independent of any particular mathematical domain, whereas identities are domain-specific equalities that rely on the axioms and operations of a given algebraic system. For example, the trigonometric identity \sin^2 \theta + \cos^2 \theta = 1 holds for all real \theta due to the properties of the real numbers and trigonometric functions, but it is not a logical tautology because it depends on the interpretation of sine and cosine rather than pure truth values. This contrasts with a tautology like p \lor \neg p, which is true solely by the semantics of logical connectives.^[36]/02%3A_Logic/2.05%3A_Logical_Equivalences) Despite these differences, overlaps exist where logical tautologies underpin universal algebraic laws through semantic interpretation in formal systems. In Boolean algebra, for example, propositional tautologies directly correspond to algebraic identities; the absorption law A \land (A \lor B) = A is both a tautology in logic (as p \land (p \lor q) \equiv p) and an identity in the Boolean ring, holding for all elements due to the structure's two-valued semantics.^[38]^[39] Philosophically, W.V.O. Quine critiqued the analytic-synthetic distinction in his 1951 essay "Two Dogmas of Empiricism," arguing that the boundary between logical tautologies and other analytic statements like mathematical identities is unclear and untenable, as both rely on intertwined notions of synonymy and meaning that cannot be sharply delineated from empirical content.^[40]

Other Contexts

Degeneracy of the genetic code

In molecular biology and genetics, the term "degeneracy" refers to the redundancy inherent in the genetic code, where multiple nucleotide triplets (codons) encode the same amino acid, ensuring that the translation process remains robust despite variations in the DNA sequence. This degeneracy means that out of the 64 possible codons, only 20 standard amino acids and 3 stop signals are specified, with most amino acids assigned 2 to 6 synonymous codons; for instance, leucine is encoded by the six codons CUU, CUC, CUA, CUG, UUA, and UUG.^[41] The concept of this coding redundancy was explored by Francis Crick in his 1968 paper on the origin of the genetic code, where he described how an initially simple coding system expanded to incorporate more amino acids through synonymous assignments, eventually "freezing" into its current form to avoid disruptive changes across evolving organisms.^[42] Crick's earlier wobble hypothesis (1966) provided the mechanistic explanation for this redundancy, proposing that flexible base-pairing at the third codon position allows a single transfer RNA (tRNA) molecule to recognize multiple synonymous codons, thereby reducing the number of required tRNAs from 61 to about 40.^[41] This degeneracy has significant implications for genetic stability, as it buffers against the effects of point mutations: a substitution in the third codon position often results in a synonymous change that does not alter the amino acid incorporated into the protein, thereby minimizing deleterious outcomes.^[43] The wobble mechanism further enhances this resilience by tolerating certain mismatches during translation, which helps mitigate errors from mutations or replication inaccuracies.^[41] In contemporary understanding, while the genetic code's degeneracy shares conceptual parallels with logical tautologies through its invariant output despite varied inputs, it is viewed primarily as adaptive functional redundancy rather than mere repetition, with codon usage biases influencing translation speed, accuracy, and protein folding.^[43] Evolutionary debates persist regarding its origins, with proposals ranging from stereochemical affinities between codons and amino acids to optimization for error minimization, though no single hypothesis fully accounts for its structure.^[44] Related to this are silent mutations, which exemplify degenerate alterations by changing a codon without affecting the resulting amino acid sequence, often preserving protein function entirely.^[45]

Everyday and legal usage

In everyday language, "tautology" refers to the needless repetition of an idea, statement, or word, often using different phrasing that adds no new information or clarity.^[13] This colloquial usage dismisses statements as obvious or circular, such as "war is war," which restates the subject without advancing understanding, or "it is what it is," which merely affirms a situation's self-evidence.^[46] Such expressions are common in casual discourse to emphasize redundancy, though they can frustrate listeners seeking substantive insight; for instance, phrases like "8 a.m. in the morning" repeat the temporal indicator unnecessarily since "a.m." already denotes morning hours.^[46] In legal contexts, tautology manifests in redundant clauses within contracts and statutes, where phrases like "null and void" reiterate invalidity without enhancing precision, a holdover from traditional drafting aimed at emphasis despite modern critiques for inefficiency.^[13] Courts apply the "rule against tautology" in statutory interpretation to presume that legislatures avoid superfluous words, ensuring every provision has independent meaning and preventing interpretations that render parts mere surplusage.^[47] For example, in R. v. Fox (2023), the Ontario Court of Appeal interpreted criminal code provisions on child exploitation by attributing distinct roles to terms like "recklessness" and "belief" to avoid redundancy.^[47] Similarly, Canada v. Canada North Group Inc. (2021) reinforced this principle in the Supreme Court of Canada, holding that statutory language must be harmonized without overlap.^[47] In media and politics, "tautology" critiques arguments or policies as circular or vacuous, such as accusations of "change for change's sake," which promises transformation without specifying benefits.^[24] Political speeches often employ tautological devices for rhetorical emphasis, including identical repetitions like "Do you miss me yet?" to build audience connection, or pleonastic listings such as "wonderful friends, conservatives, and fellow citizens" where terms overlap in implying shared identity.^[24] These flout conversational norms of brevity to imply deeper solidarity or urgency, as seen in analyses of speeches at events like the Conservative Political Action Conference.^[24] Culturally, tautology intersects with debate and literature as a form of fallacy akin to petitio principii (begging the question), where circular reasoning masquerades as proof, such as claiming "the policy works because it succeeds," which assumes its conclusion.^[13] In literature, authors like Mark Twain highlighted tautologies satirically, noting repetitions in political rhetoric to underscore emptiness.^[13] Over time, the term evolved from its 16th-century English debut denoting rhetorical repetition—derived from Greek tautologia ("saying the same")—to a 20th-century logical sense introduced by Ludwig Wittgenstein, before settling into modern colloquial use for any obvious redundancy.^[14] This shift reflects broader linguistic adaptation from strict formalism to informal critique of unhelpful verbiage.^[14]

References

[1]
Truth Tables, Tautologies, and Logical Equivalences
A tautology is a formula which is "always true" --- that is, it is true for every assignment of truth values to its simple components. You can think of a ...
[2]
Glossary of Rhetorical Terms | University of Kentucky College of Arts ...
Tautology: repetition of an idea in a different word, phrase, or sentence. *With malice toward none, with charity for all. Lincoln, Second Inaugural. Zeugma: ...
[3]
[PDF] 7.5 Tautology, Contradiction, Contingency, and Logical Equivalence
Definition: A compound statement is a tautology if there is a T beneath its main connective in every row of its truth table. Examples. Either it is raining ...
[4]
RHETORICAL TERMS | Dickinson College Commentaries
tautology: the repetition of the same idea in different ways. Etymology: from Greek tauto, 'the same', and logos, 'word, idea'. Example: 'The ... mouse ...
[5]
Formal Logic - Philosophy 160 (002) - UMSL
A sentence that is explicitly logically true is sometimes called a tautology. E.g. 'all animals are animals' Logical falsity.<|control11|><|separator|>
[6]
Propositional Logic - Stanford Encyclopedia of Philosophy
May 18, 2023 · Propositional logic is the study of the meanings of, and the inferential relationships that hold among, sentences based on the role that a specific class of ...
[7]
tautology - Introduction to Propositional Logic
A proposition that is always true called a tautology. There are also propositions that are always false such as (P P). Such a proposition is called a ...
[8]
[PDF] 2. Propositional Equivalences 2.1. Tautology/Contradiction ...
A tautology is a proposition that is always true. Example 2.1.1. p ∨ ¬p. Definition 2.1.2. A contradiction is a proposition that is always false.Missing: philosophy | Show results with:philosophy
[9]
[PDF] The Principle of Excluded Middle in Aristotle - e d o c . h u
because it is an instance of the law of excluded middle and therefore a tautology, even though ... Aristotle treated the laws of non-contradiction and excluded ...<|control11|><|separator|>
[10]
Wittgenstein's Logical Atomism - Stanford Encyclopedia of Philosophy
Nov 22, 2004 · According to this conception, a tautology, being true, is, in contrast to a nonsensical string, in the running for possessing a non-zero ...Names and Objects · Linguistic Atomism · Metaphysical Atomism · Bibliography
[11]
Propositional Logic | Internet Encyclopedia of Philosophy
Definition: a wff is a tautology if and only if it is true for all possible truth-value assignments to the statement letters making it up. Tautologies are also ...
[12]
Tractatus Logico-Philosophicus (English) - The Ludwig Wittgenstein Project
Summary of each segment:
[13]
TAUTOLOGY Definition & Meaning - Merriam-Webster
Oct 17, 2025 · The meaning of TAUTOLOGY is needless repetition of an idea, statement, or word. How to use tautology in a sentence.Missing: linguistics Oxford
[14]
Tautology - Etymology, Origin & Meaning
Tautology, from Greek tautologos ("repeating what has been said"), means the repetition of the same idea or words in different terms within the same ...
[15]
Tautology | Meaning, Definition & Examples - Scribbr
Aug 4, 2023 · A logical tautology is a proposition or statement that is always true because it excludes no logical possibility. Logical tautologies don't ...
[16]
What Is Tautology? Definition and Examples | No Film School
In everyday speech and writing, tautologies are often considered a flaw because they are redundant and don't add new information. But like many “mistakes ...
[17]
Are you using these 6 common tautologies? - Lingoda
Oct 9, 2024 · A tautology is when you use different words to repeat the same idea. For example, the phrase, “It was adequate enough,” is a tautology.What's A Tautology? · Using Tautologies In English · Examples Of Common...
[18]
Tautology - Rhetorical Devices - Ultius
The word “tautology” is derived from the Greek tautologos meaning “the same” and “word or idea.” Its first known use was in 1574.
[19]
Tautology | PDF | Linguistics | Semantics - Scribd
Why Do People Use Tautology? Reason : Explanation Emphasis: Adds emotional weight or clarity. Style/Rhythm: Especially in speeches, poems, or songs ...Missing: psychological | Show results with:psychological
[20]
[PDF] Pleonasm and hypercharacterization - Christian Lehmann
Feb 23, 2006 · The concepts of pleonasm and tautology have been current in rhetoric, linguistics and philosophy since antiquity. At the beginning, we can ...
[21]
The Role of Pleonasms and Tautology in the Text - ResearchGate
Oct 5, 2025 · Tautology is a particular type of pleonasm which requires the similar notion or meaning in different words within one expression.
[22]
Tautology and Pleonasm in Political Interviews: A Semantic study
Jun 1, 2024 · The current study aims at analysing two political interviews in the light of pleonasm and tautology concepts and find out the differences between them ...
[23]
Tautology (Grammar, Rhetoric, and Logic) - ThoughtCo
Mar 9, 2019 · In rhetoric and logic, a tautology is a statement that is true by its form alone. Examples of tautology include phrases like 'pair of twins' and ...
[24]
The Use of Tautology in “The Thorn” by William Wordsworth
Abstract: A tautology is to say the same thing twice where part of the sentence is redundant. The words are used and then expanded further in a way as if it ...
[25]
Definition and Examples of Tautology in Literature - Literary Devices
Tautology, as a literary device, is a repetitive use of phrases or words which have similar meanings used for emphasis or poetic license.
[26]
A pragmatic Study of Tautology in Trump's Political Speech
Maxim of quantity: saying only informative information, no more and no less. ... pleonasm tautology as unity. includes strength. 28. In the use of " We will ...
[27]
The completeness and compactness theorems of first-order logic
Apr 10, 2009 · The famous Gödel completeness theorem in logic (not to be confused with the even more famous Gödel incompleteness theorem) roughly states the following:
[28]
completeness theorem in nLab
### Summary of Gödel's Completeness Theorem for First-Order Logic
[29]
[PDF] CS311H: Discrete Mathematics First Order Logic, Rules of Inference
(P(x) → Q(x)) is satisfiable. ▷ An FOL formula F is valid if, for all interpretations, F ... ▷ This idea is formally called universal instantiation: ∀x.P ...
[30]
[PDF] On Herbrand's Theorem - UCSD Math
If, in addition, C does not involve equality, C will be tautologically valid. Thus, Herbrand's theorem reduces provability in first-order logic to gen- eration ...
[31]
[PDF] Basic Concepts in Modal Logic - Stanford University
(b) Show that (ϕ → ⊥) ↔. ¬ϕ is a tautology. Remark: These examples and exercises show that our definition of a tau- tology allows us to prove that certain ...
[32]
[PDF] CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction ...
The Hilbert proof systems put major emphasis on logical axioms, keeping the rules of inference to minimum, often in propositional case, admitting only Modus ...
[33]
[PDF] A Primer for Logic and Proof - Appalachian State University
Applying the deduction theorem, we obtain A → (B → C) `L B → (A → C). A second application of the deduction theorem yields `L (A → (B → C)) → (B → (A → C)),.
[34]
[PDF] Verifying Refutations with Extended Resolution
We present a new proof format for checking unsatisfiability proofs produced by SAT solvers that use techniques such as extended resolution and blocked clause ...Missing: proving | Show results with:proving
[35]
[PDF] Efficiently Checking Propositional Resolution Proofs in Isabelle/HOL
Both SAT solvers generate resolution-style proofs for (instances of) propositional tautologies. These proofs are verified by the theorem prover. The presented ...
[36]
[PDF] Undecidability of First-Order Logic - Computer Science
Church's proof uses the fact that it is undecidable whether two expressions in λ-calculus are equivalent, and then reduces this to the decision problem. On the ...
[37]
The Church-Turing Thesis - Stanford Encyclopedia of Philosophy
Jan 8, 1997 · The Church-Turing thesis concerns the concept of an effective or systematic or mechanical method, as used in logic, mathematics and computer science.
[38]
Algebraic Identities - Two & Three Variable, Factorization - Cuemath
An algebra identity means that the left-hand side of the equation is identically equal to the right-hand side, for all values of the variables. Here we shall ...
[39]
Identity Property - (College Algebra) - Vocab, Definition, Explanations
The identity property is a fundamental axiom in the field of abstract algebra, where it is used to define and analyze algebraic structures such as groups ...
[40]
Boolean algebra, Part 1 | Review ICS 241
Oct 23, 2015 · Each equivalence or tautology in propositional logic corresponds to an identity or law in Boolean algebra. Important identities and laws are ...
[41]
Algebraic Laws for Logical Expressions
DeMorgan's Laws are algebraic laws (or equivalences) that allow us to rewrite any logical expression so that NOT is only applied to propositional variables.
[42]
The Analytic/Synthetic Distinction
Aug 14, 2003 · “Analytic” sentences, such as “Pediatricians are doctors,” have historically been characterized as ones that are true by virtue of the meanings of their words ...
[43]
https://pmc.ncbi.nlm.nih.gov/articles/PMC3293468/
[44]
https://royalsocietypublishing.org/doi/10.1098/rsfs.2019.0038
[45]
Origin and evolution of the genetic code: the universal enigma - PMC
Crick FH. The origin of the genetic code. J Mol Biol. 1968;38(3):367–79. doi: 10.1016/0022-2836(68)90392-6. [DOI] [PubMed] [Google Scholar]; 7. Ycas M. The ...
[46]
On the origin of degeneracy in the genetic code | Interface Focus
Oct 18, 2019 · Here, we provide a new theory that explains the origin of the degeneracy based only on symmetry principles.Missing: review | Show results with:review
[47]
Genetic code degeneracy is established by the decoding center of ...
The degeneracy of the genetic code confers a wide array of properties to coding sequences. Yet, its origin is still unclear. A structural analysis has shown ...
[48]
TAUTOLOGY Definition & Meaning | Dictionary.com
Tautology definition: needless repetition of ... Word History and Origins. Origin of tautology. C16: from Late Latin tautologia, from Greek, from tautologos.
[49]
Statutory Interpretation - Rule Against Tautology - Isthatlegal
Oct 17, 2023 · The rule against tautology presumes that legislatures avoid superfluous words, and that every word in a statute has a specific role, avoiding ...