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Tautology


Definition
Tautology Notation
Tautology vs Equivalence
Logic Laws as Tautologies
Tautologies that are not laws
Tautology vs Contradiction
Formula Summary
Common Pitfalls
Tautology FAQ



Introduction

Tautologies represent a fundamental concept in propositional logic, embodying logical certainty. A tautology is a formula that evaluates to true under all possible interpretations of its variables. The classic example is P ∨ ¬P (the law of excluded middle), which states that either a proposition or its negation must be true – an inescapable truth.

Unlike regular propositions whose truth depends on specific circumstances, tautologies are guaranteed to be true, providing absolute certainty in logical systems. This property makes tautologies essential in logical reasoning, serving as the foundation for valid arguments, logical equivalences, and formal proofs.

This page explores tautologies comprehensively, examining their properties, relationship to contradictions, and their crucial role in establishing the framework of logical deduction and mathematical reasoning.



Definition

Five formulas, and the question is which of them can never come out false.
Some of these say something about the world — whether PP and QQ happen to be true. Others might be true no matter what. Before naming that property, list them.
candidateformulain words
AP ∨ ¬PP or not P
BP ∧ ¬PP and not P
CP → Qif P then Q
DP ∧ (P ∨ Q)P, and also P or Q
EP → (Q → P)if P, then (if Q then P)
Checked in every row of a truth table, only A and E are true throughout.
To test a formula, list every truth assignment of its variables in a truth table and evaluate the formula in each row. With nn variables there are 2n2^n rows: two for P∨¬PP∨¬P, four for a formula in PP and QQ, eight for one in PP, QQ and RR. The formula is a tautology exactly when its column holds T in all of these rows.
PQABCDE
TTTFTTT
TFTFFTT
FTTFTFT
FFTFTFT
every row T?yesnononoyes
Pitfall: Checking only some rows
Calling P→(P∧Q)P \to (P \land Q) a tautology after the rows P=TP = T, Q=TQ = T and P=FP = F misses the row P=TP = T, Q=FQ = F, where P∧QP \land Q is F and the formula is F. It is not a tautology.
Each rejected formula fails in a particular row, and the near-miss E fails in none.
B is false in every row — it can never be true. C fails in the single row where PP is true and QQ is false. D fails whenever PP is false, because then "PP and anything" is false. E looks as if it should depend on QQ, but it cannot fail: when PP is true the inner Q→PQ \to P is true, and when PP is false the whole implication is true automatically.
The rule: true under every assignment of truth values, whatever the components say.
A formula with this property is true because of its shape, not because of facts about PP or QQ. The most familiar example is A:
Example
                                                P∨¬PP∨¬P
(The law of the excluded middle: "P or not P" is always true.)
AB
P ∨ ¬P covers every case

Let circle A stand for the cases where P is true. P ∨ ¬P shades A together with everything outside it, so no case is left unshaded: the formula is true under every assignment, which is what makes it a tautology. Check other formulas on the Venn truth-table explorer.

A tautology always fills the whole diagram; a contradiction always leaves it empty.
The definition
A tautology is a proposition that is always true, regardless of the truth values of its components. In propositional logic, a tautology is a formula that evaluates to true under all possible truth assignments.
Of the five candidates, A and E are tautologies.
Being a tautology is a property of a formula: it either has it or does not.
"Tautology" classifies formulas, the way "even" classifies numbers. Every formula falls into exactly one of three groups: true in every row (a tautology, like A and E), false in every row (a contradiction, like B), or true in some rows and false in others (like C and D).
A rejected formula is usually still useful: it is true somewhere.
C and D are not tautologies, but each has rows where it is true — they are satisfiable, and most useful statements are of this kind. Only B is hopeless, false everywhere. And any rejected formula can be turned into a tautology by joining it with its own negation: C∨¬CC \lor \lnot C is true in every row.
The rest of the page covers tautology notation, how tautologies relate to logical equivalence, the laws of logic read as tautologies, other important tautologies, and how tautologies and contradictions mirror each other.

Tautology Notation

Notation

Tautology Notation

The constant that is true by definition, and the two turnstiles that certify truth by different means — one by proof, one by inspection of every case. Telling those two apart is the oldest distinction in formal logic.
The connectives and letters come from propositional logic notation, the meta-level ≡\equiv from equivalence notation; the mirror constant ⊥\bot belongs to the contradiction page.
⊤\top
top; verum; the true constant
A constant, not a variable: ⊤\top denotes truth itself — a formula true under every assignment, needing no letters. It stands for the whole class of tautologies the way a single number stands for a value.
CasesAs a constant it can sit inside formulas — P∨⊤P \vee \top is ⊤\top, P∧⊤P \wedge \top is PP; the laws page states these as the identity and domination rules.
Same glyph elsewhereThe same glyph is the perpendicularity mark's upside-down twin, and in other fields marks the top of an ordering; in logic it is always the true constant.
⊢P\vdash P
turnstile P; P is provable
The single turnstile is a syntactic certificate: ⊢P\vdash P claims a derivation of PP exists in the proof system — obtained by pushing symbols according to rules, without consulting truth values at all.
CasesWith premises on the left it becomes an argument: P,P→Q⊢QP, P \to Q \vdash Q reads "from these, QQ is derivable"; bare on the left, ⊢P\vdash P marks PP as a theorem of the system.
Do not confuseA statement inside the language. The turnstile is metalanguage — it talks about formulas from outside; writing ⊢\vdash between two connectives produces no formula the formation rules recognize.
⊨P\vDash P
double turnstile P; P is valid
The double turnstile is a semantic certificate: ⊨P\vDash P claims PP comes out true under every assignment — established by checking cases, as a truth table does, not by deriving anything.
CasesThe two turnstiles answer different questions about the same formula — ⊢P\vdash P asks "can it be proved?", ⊨P\vDash P asks "is it always true?"; that a system's answers agree in both directions is exactly what soundness and completeness assert.
Do not confuseThe single turnstile. One extra bar swaps proof for truth — and the pair also appears with premises, where Γ⊨P\Gamma \vDash P means every model of Γ\Gamma makes PP true, the semantic cousin of Γ⊢P\Gamma \vdash P.
P≡⊤P \equiv \top
P is equivalent to the true constant
The third spelling: rather than name PP a tautology, assert that it agrees with ⊤\top in every case — the equivalence relation applied to a constant.
CasesHandy where a proof already runs in equivalences — a chain of law-substitutions ending in ⊤\top has proved a tautology, which is the algebraic verification method in Are all equivalences tautologies? below.
Do not confuseAn assignment. P≡⊤P \equiv \top does not set PP true — it reports that PP was already true in every case; the ≡\equiv is a verdict, not an instruction.
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Notation · Cheat sheet
Propositional Logic Notation at a Glance
The marks from the notation cards across the propositional-logic pages: how each is read and what it means. Every group links to the full card on its page.
Syntax
P,Q,RP, Q, R · P1,P2P_1, P_2
proposition letters
Capital letters name atomic propositions; subscripts extend the supply.
¬  ∧  ∨  →  ↔\neg \; \wedge \; \vee \; \to \; \leftrightarrow
not; and; or; implies; if and only if
The five connectives: negation, conjunction, disjunction, implication, biconditional.
φ,ψ\varphi, \psi
phi and psi
Placeholders for whole formulas in the formation rules.
(φ∧ψ)(\varphi \wedge \psi)
parenthesised conjunction
Every binary connective brings its own parentheses; no precedence at this level.
Implication
p→qp \to q
if p then q; p implies q
The conditional connective: antecedent pp, consequent qq.
p⇒qp \Rightarrow q
p implies q, in the meta sense
The double shaft claims the implication holds in every case; older texts write p⊃qp \supset q.
p→qp \to q in words
the English phrasings of one arrow
"if pp, then qq"; "pp is sufficient for qq"; "qq is necessary for pp"; "pp only if qq".
Equivalences
P≡QP \equiv Q
P is logically equivalent to Q
A claim about formulas: PP and QQ agree in every case.
P⇔QP \Leftrightarrow Q
P if and only if Q, in the meta sense
The double-shafted spelling of the same claim.
P=QP = Q
P equals Q
Boolean algebra’s spelling, where formulas denote values in {0,1}\{0, 1\}.
TautologyThis page
⊤\top
top; verum; the true constant
Truth itself: a formula true under every assignment.
⊢P\vdash P
turnstile P; P is provable
The single turnstile: a derivation of PP exists in the proof system.
⊨P\vDash P
double turnstile P; P is valid
The double turnstile: PP is true under every assignment.
P≡⊤P \equiv \top
P is equivalent to the true constant
The third way to say that PP is a tautology.
Contradiction
⊥\bot
bottom; falsum; the false constant
Falsehood itself: a formula false under every assignment.
⊢¬P\vdash \neg P
the negation of P is provable
The proof-side way to declare PP a contradiction.
P≡⊥P \equiv \bot
P is equivalent to the false constant
The equivalence spelling of "contradiction".
Same marks, different job
→\to vs ⇒\RightarrowA connective inside a formula; a claim about formulas.
↔\leftrightarrow vs ≡\equivThe biconditional connective; logical equivalence in the metalanguage.
⊢\vdash vs ⊨\vDashProvable by the rules; true in every case.
⊃\supsetImplication in older logic texts; superset in set theory.
⊥\botThe false constant here; perpendicular in geometry.
Logic symbols tablePropositional logic
Propositional Logic Notation at a Glance·/logic/propositional-logicLearn Math Class

Tautology vs Equivalence

Are all equivalences tautologies?

As the definition of equivalence states, two propositions AA and BB are logically equivalent if they always have the same truth value in every possible scenario. This means that their truth tables are identical. This is a bilateral relationship meaning both expressions evaluate to true together or false together in all cases. It does not matter if they are true or false, the key here is that they are the same (equal).
Example
                                                (P→Q)≡(¬P∨Q)(P→Q)≡(¬P∨Q)
(Material implication rewritten in disjunctive form.)
As long as the equivalence is valid, its biconditional (P→Q)↔(¬P∨Q)(P→Q)↔(¬P∨Q) evaluates to true in every valuation and falls under definition of tautology.
So the final answer to that question is YES , as long as equivalence is valid- its biconditional form (A↔BA↔B) is a tautology.

Are all tautologies logical equivalences?

According to definition, a tautology is about a single proposition that is always true, while equivalence is a relationship between two propositions that always have the same truth value.
A tautology must always evaluate to true, but it does not necessarily have to express a relationship between two statements. In fact, there are numerous tautologies that are unilateral expressions that evaluate to true in all possible valuations but don't express a relationship of equality between two propositions.
Examples:
Law of Excluded Middle:
P∨¬PP∨¬P
Always true, but doesn't equate two propositions.

Self-implication:
P→PP→P
Tautological but doesn't equate different propositions.
More examples:

P∨(P→Q)P ∨ (P → Q)

(P→Q)∨(Q→P)(P → Q) ∨ (Q → P)

P→(Q→(P∧Q))P → (Q → (P ∧ Q))

All those are tautologies that do not fall under the definition of equivalence.

This distinction means that while every valid equivalence A≡BA≡B makes A↔BA↔B a tautology, not each tautology is equivalence because some tautologies do not express a two-sided relationship between two propositions.
This means that the biconditionals A↔BA↔B of all logical equivalences form a subset of the set of all tautologies.




Tautologies (e.g., P → P, P ∨ ¬P) Equivalences (e.g., P ↔ Q, when logically equivalent) P ↔ Q where P and Q are equivalent but not named laws Non-equivalence tautologies (e.g., ((P → Q) → P) → P, (¬P → P) → P)

Logic Laws as Tautologies

As discussed in previous section, a valid equivalence A≡BA≡B means its biconditional A↔BA↔B is a tautology.
Since all propositional logic laws apart from inference rules (such as resolution) and non-equivalences (such as the inverse law) are either equivalences or directly tautological statements, we conclude that all these laws yield tautologies.

Propositional Logic Laws as Equivalences

Fact: Named for saying nothing
Wittgenstein brought tautology, an old word for needless repetition, into logic in his Tractatus of 1921, where he held that every law of logic is one. A formula true in every row rules out no case, so it tells nothing about the world, and that is why it can never fail.
Many fundamental laws of propositional logic are expressed as equivalences, meaning both sides of the equation always yield the same truth value. Since the equivalence must always hold, the biconditional joining its two sides is a tautology.
Examples:
De Morgan’s Laws:

¬(P∨Q)≡(¬P∧¬Q)¬(P∨Q)≡(¬P∧¬Q)

¬(P∧Q)≡(¬P∨¬Q)¬(P∧Q)≡(¬P∨¬Q)

Commutative Laws:

P∨Q≡Q∨PP∨Q≡Q∨P

P∧Q≡Q∧PP∧Q≡Q∧P

The biconditionals of these laws are always true, making them tautologies. See more examples of basic propositional logic laws here.

Propositional Logic Laws That Are Directly Tautologies

Some laws are one-sided expressions that are true in all cases without establishing an equivalence. These are still tautologies.

Law of the Excluded Middle:

P∨¬P≡TP∨¬P≡T
This states that every proposition is either true or false, making it a tautology.

Law of Non-Contradiction:

¬(P∧¬P)≡T¬(P∧¬P)≡T
This states that no proposition can be both true and false simultaneously, which is always true.

Pitfall: Reading a tautological conditional as an equivalence
Turning the tautology ((P→Q)∧P)→Q((P \to Q) \land P) \to Q into (P→Q)∧P≡Q(P \to Q) \land P \equiv Q fails at P=FP = F, Q=TQ = T: the left side is F and QQ is T. The conditional holds in one direction only.
Since all propositional logic laws apart from inference rules and non-equivalences either express equivalences whose biconditionals are tautologies or standalone tautologies, we conclude that all these laws yield tautologies.

Visit corresponding page to learn more about propositional logic laws.

Use this tool to evaluate truth tables.


Tautologies that are not laws

While all equivalence and tautology laws of propositional logic yield tautologies, not all tautologies qualify as laws. A law in logic is a fundamental principle that defines how logical operations behave, often used in formal proofs and reasoning systems.
However, some tautologies are simply valid logical statements that always evaluate to true without establishing a fundamental logical principle.
These non-law tautologies may still be useful in proofs, derivations, or formal logic, but they do not define core logical rules like De Morgan’s Laws or the Law of the Excluded Middle. Instead, they are often the result of specific logical constructions, implications, or transformations that hold universally without being considered foundational.

nameexpressionexplanation
Self-Implication
P → P
If P then P - a proposition always implies itself
Self-Equivalence
P ↔ P
P if and only if P - a proposition is always equivalent to itself
Modus Ponens Form
((P → Q) ∧ P) → Q
If P implies Q and P is true, then Q must be true
Hypothetical Syllogism
((P → Q) ∧ (Q → R)) → (P → R)
If P implies Q and Q implies R, then P implies R
Weakening
P → (Q → P)
If P is true, then it remains true regardless of Q
Proof by Contradiction Form
(¬P → P) → P
If assuming not-P leads to P, then P must be true
Peirce's Law
((P → Q) → P) → P
Classical logical principle not provable in intuitionistic logic
Disjunctive Weakening
P ∨ (P → Q)
Either P is true or if P then Q - always true
Implication Disjunction
(P → Q) ∨ (Q → P)
Either P implies Q or Q implies P - always true
Conjunction Implication
(P → Q) → ((P ∧ R) → Q)
If P implies Q, then P and R together also imply Q
Conjunction Introduction Form
P → (Q → (P ∧ Q))
If P and Q are both true, their conjunction is true
Modus Ponens Alternative Form
(P ∧ (P → Q)) → Q
If P is true and P implies Q, then Q is true
Modus Tollens Form
(¬Q ∧ (P → Q)) → ¬P
If Q is false and P implies Q, then P must be false
Self-Contradiction Implication
(P → ¬P) → ¬P
If P implies its own negation, then P must be false
Implication Negation Equivalence
¬(P → Q) ↔ (P ∧ ¬Q)
An implication is false only when its antecedent is true and consequent is false
Contradiction Implication
(P → (Q ∧ ¬Q)) → ¬P
If P implies a contradiction, then P must be false
Reductio ad Absurdum
((P → Q) ∧ (P → ¬Q)) → ¬P
If P implies both Q and not-Q, then P must be false
Negation Introduction
(P → Q) → ((P → ¬Q) → ¬P)
If P implies contradictory results, P must be false
Material Implication Alternative
(P → Q) → (¬P ∨ Q)
If P implies Q, then either P is false or Q is true
Disjunctive Implication
(P ∨ Q) → (¬P → Q)
If either P or Q is true, then not-P implies Q
Constructive Dilemma
((P → Q) ∧ (R → S) ∧ (P ∨ R)) → (Q ∨ S)
Complex syllogism with disjunctive conclusion
Biconditional Expansion
(P ↔ Q) ↔ ((P → Q) ∧ (Q → P))
P if and only if Q means P implies Q and Q implies P

Use this tool to generate truth tables dynamically and evaluate those expressions.

Tautology vs Contradiction

While a tautology affirms itself under every possible interpretation, its logical counterpart — the contradiction — fails under all of them. These two aren’t just opposites in outcome; they’re structurally bound by negation.
The negation of a tautology is always a contradiction, and the negation of a contradiction is always a tautology.
Pitfall: Treating not a tautology as a contradiction
Calling P→QP \to Q a contradiction because it is not a tautology, and so taking ¬(P→Q)\neg(P \to Q) for a tautology, fails at P=TP = T, Q=TQ = T: there P→QP \to Q is T and ¬(P→Q)\neg(P \to Q) is F. Both formulas are contingencies.
This is not just an abstract symmetry — it's a fundamental mechanism in logical systems.
This reciprocal relationship plays out in reasoning strategies: to show something is necessarily true (a tautology), one might show that its negation leads to a contradiction — a technique known as reductio ad absurdum. The tension between the two is constructive: from the impossibility of one, we often infer the necessity of the other.
Take for example the classic tautology 'P ∨ ¬P'. Its truth table confirms that it's always true, regardless of whether 'P' is true or false. Now consider the negation: '¬(P ∨ ¬P)'. No matter what truth value you assign to 'P', this formula is always false — a contradiction. In practice, this duality underpins proof techniques, logic solvers, and model checkers. If a system can show that assuming '¬φ' leads to a contradiction, it immediately concludes that 'φ' must be a tautology — turning logical failure into proof.

negation (¬) negation (¬) Tautology Contradiction
</table>
Negation swaps the two extremes, so showing a formula is always true is the same task as showing its negation is never true.

Formula Summary

This summary sets out the laws of logic that the page reads as tautologies, in two families, the way the section on logic laws as tautologies divides them. Each family has a line linking back to that section, and each law a short note on what it says. The buttons above the cards show one family at a time.

The first family holds laws written as equivalences, the two De Morgan laws and the two commutative laws, whose biconditionals are true in every row. The second holds the laws that are tautologies on their own: the law of the excluded middle and the law of non-contradiction.
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Laws written as equivalences4 formulas
A valid equivalence A≡BA \equiv B means its biconditional A↔BA \leftrightarrow B is a tautology, as logic laws as tautologies explains.
¬(P∨Q)≡(¬P∧¬Q)\neg(P \lor Q) \equiv (\neg P \land \neg Q)
The negation of a disjunction is the conjunction of the negations, by De Morgan's laws.
¬(P∧Q)≡(¬P∨¬Q)\neg(P \land Q) \equiv (\neg P \lor \neg Q)
The negation of a conjunction is the disjunction of the negations, the second of De Morgan's laws.
P∨Q≡Q∨PP \lor Q \equiv Q \lor P
The order of the parts of a disjunction does not matter, by the commutative laws.
P∧Q≡Q∧PP \land Q \equiv Q \land P
The same holds for a conjunction, the second of the commutative laws.
Laws that are tautologies directly2 formulas
One-sided laws, true in every case without equating two formulas, from the same section.
P∨¬P≡TP \lor \neg P \equiv T
Every proposition is either true or false, the law of the excluded middle.
¬(P∧¬P)≡T\neg(P \land \neg P) \equiv T
No proposition is both true and false at once, the law of non-contradiction.
Logic laws as tautologies·All logic formulasLearn Math Class
Joining the two sides of a valid equivalence with a biconditional always gives a tautology, so every law here can be confirmed by checking that one formula is true in every row. Inference rules such as resolution are left out of both families, since they are not stated as equivalences.

Common Pitfalls

Three slips account for many wrong answers on the topics of this page. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.
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1Checking only some rows
A formula is evaluated in a few rows, comes out T in each of them, and is called a tautology. A tautology must be T in every row of its truth table, so the rows that were not checked can still hold an F.
Wrong turn
Error here
For P→(P∧Q)P \to (P \land Q), the row P=TP = T, Q=TQ = T gives T→TT \to T, which is T. When P=FP = F the formula is F→(P∧Q)F \to (P \land Q), which is T for either value of QQ. Every row tried gives T, so the formula is called a tautology.
Right path
The four rows for (P,Q)(P, Q) are (T,T)(T, T), (T,F)(T, F), (F,T)(F, T), (F,F)(F, F), and the formula takes the values T, F, T, T. In the row P=TP = T, Q=FQ = F, P∧QP \land Q is F and the formula is T→FT \to F, which is F. It is a contingency, not a tautology.
Why it happensOnce a few rows give T, the formula starts to look always true, and the remaining rows seem like a formality. The rows that are skipped are often the ones with mixed values, which are where formulas of this kind fail.
How to checkCount the rows first, 2n2^n for nn variables, and evaluate every one. A shortcut for conditionals is to look only for a row with the part before the arrow T and the part after it F; if no such row exists, the conditional is a tautology.
2Reading a tautological conditional as an equivalence
A conditional A→BA \to B that is a tautology is treated as if A≡BA \equiv B held. Only an equivalence makes the biconditional A↔BA \leftrightarrow B a tautology; a tautological conditional, such as the form of an inference rule, guarantees one direction only.
Start
((P→Q)∧P)→Q((P \to Q) \land P) \to Q, the form of modus ponens, is T in all four rows.
Wrong turn
Error here
So (P→Q)∧P≡Q(P \to Q) \land P \equiv Q, and ((P→Q)∧P)↔Q((P \to Q) \land P) \leftrightarrow Q is taken to be a tautology too.
Right path
In the row P=FP = F, Q=TQ = T, P→QP \to Q is T and (P→Q)∧P(P \to Q) \land P is T∧FT \land F, which is F, while QQ is T. The biconditional is F↔TF \leftrightarrow T, which is F, so (P→Q)∧P≢Q(P \to Q) \land P \not\equiv Q. The conditional is a tautology; the biconditional is not.
Why it happensBoth laws and inference rules are listed as tautologies, so their forms are read in the same way. The arrow →\to and the double arrow ↔\leftrightarrow also look alike, and the second direction is assumed without being tested.
How to checkTest the reverse conditional B→AB \to A on its own. Here Q→((P→Q)∧P)Q \to ((P \to Q) \land P) is F at P=FP = F, Q=TQ = T, so the reverse direction fails and the two sides are not equivalent.
3Treating not a tautology as a contradiction
The rule that the negation of a tautology is a contradiction is read as if every formula were one or the other. A formula that is not a tautology is then called a contradiction, and its negation is called a tautology.
Start
P→QP \to Q is F at P=TP = T, Q=FQ = F, so it is not a tautology.
Wrong turn
Error here
It is called a contradiction, and so ¬(P→Q)\neg(P \to Q) is taken to be a tautology.
Right path
For (P,Q)(P, Q) equal to (T,T)(T, T), (T,F)(T, F), (F,T)(F, T), (F,F)(F, F), the formula P→QP \to Q takes the values T, F, T, T, and ¬(P→Q)\neg(P \to Q) takes F, T, F, F. Each column holds both T and F, so both formulas are contingencies. The negation of a contingency is again a contingency.
Why it happensTautology and contradiction are shown side by side as opposites, and the negation rule links only these two. The third category, contingency, is not part of the picture and is easy to forget.
How to checkBefore negating, find one row with T and one row with F. If both exist, the formula and its negation are both contingencies, and neither is a tautology.
Common pitfalls with tautologies·/logic/propositional-logic/semantics/tautologyLearn Math Class
Each slip draws a conclusion the truth table does not support. A tautology must be T in every row, a tautological conditional A→BA \to B is not an equivalence unless B→AB \to A holds too, and a formula that is not a tautology may still be true in some rows, so its negation need not be a tautology. Evaluating every row, and testing the reverse conditional, catches each slip.

Tautology FAQ

What does the symbol ⊤ mean in logic?

+
⊤ (read 'top' or 'verum') is the constant for truth itself: a formula that is true under every assignment, containing no variables. Its mirror image ⊥ denotes the always-false constant. Inside formulas it simplifies by the identity and domination laws: P ∧ ⊤ reduces to P, while P ∨ ⊤ reduces to ⊤. Writing P ≡ ⊤ is another way of saying P is a tautology.Read more →

What is the difference between ⊢ and ⊨ in logic?

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The single turnstile ⊢ is syntactic: ⊢ P claims a derivation of P exists in the proof system, obtained by applying rules without consulting truth values. The double turnstile ⊨ is semantic: ⊨ P claims P is true under every assignment, as a truth table shows. That the two agree — everything provable is valid and everything valid is provable — is what soundness and completeness assert.Read more →

What is Peirce's law?

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Peirce's law is the tautology ((P → Q) → P) → P: if P follows from the mere assumption that P implies Q, then P holds outright. A truth table confirms it is true in all four rows. Its interest is philosophical — the law is valid in classical logic but not provable in intuitionistic logic, so it serves as a test case separating the two systems. Check it in the truth table generator.Read more →

If A → B is a tautology, are A and B equivalent?

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No, a tautological conditional guarantees one direction only. The modus ponens form ((P→Q)∧P)→Q((P \to Q) \land P) \to Q is true in all four rows, yet at P=FP = F, Q=TQ = T the left side (P→Q)∧P(P \to Q) \land P is F while QQ is T. Equivalence needs the biconditional A↔BA \leftrightarrow B to be a tautology, which means A→BA \to B and B→AB \to A both have to be.Read more →

Is every formula either a tautology or a contradiction?

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No, most formulas are neither. A formula that is true in some rows and false in others is a contingency, like P→QP \to Q, which takes the values T, F, T, T. Its negation ¬(P→Q)\neg(P \to Q) takes F, T, F, F and is a contingency too: negation turns a tautology into a contradiction and back, but a contingency stays a contingency.Read more →

Why is it called a tautology?

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Because it says nothing about the world. Wittgenstein brought tautology, an old word for needless repetition, into logic in his Tractatus of 1921, where he held that every law of logic is one. A formula true in every row rules out no case, so it carries no information about which case holds, and that is exactly why it can never fail.Read more →


PropertyTautologyContradiction
DefinitionAlways true under every interpretationAlways false under every interpretation
Symbolic Trait⊨ φ (φ is valid)⊨ ¬φ (¬φ is valid)
Truth TableAll rows are TAll rows are F
NegationBecomes a contradictionBecomes a tautology
Role in ProofsRepresents logical necessityUsed to derive absurdity in indirect proofs
Example'P ∨ ¬P''P ∧ ¬P'
Practical UseValidates general rules, laws of logicSignals inconsistency, contradiction in assumptions
Evaluation OutcomeTrue in all modelsFalse in all models
Logical StatusUniversally affirmedUniversally rejected