# Validity (logic)

**validityvalidinvalidlogically validlogical validityvalidlyvalid argumentvalidatedeductively validevaluated**

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.wikipedia

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### Logic

**logicianlogicallogics**

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

Logic (from the ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.

### Logical consequence

**entailsentailmentfollows from**

An argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid.

A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises.

### Logical form

**argument formschemaargument structure**

A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.

Attention is given to argument and sentence form, because form is what makes an argument valid or cogent.

### Premise

**connectedhypotheseslogical premises**

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

The proof of a conclusion depends on both the truth of the premises and the validity of the argument.

### Tautology (logic)

**tautologytautologiestautological**

In propositional logic, they are tautologies. |True preserving only: || Tautology ( \top ) * Biconditional (XNOR, ) * Implication ( \rightarrow ) * Converse implication ( \leftarrow )

In propositional logic, there is no distinction between a tautology and a logically valid formula.

### Argument

**argumentslogical argumentproof**

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. An argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid.

Validity

### Corresponding conditional

The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction.

An argument is valid if and only if its corresponding conditional is a logical truth.

### Syllogism

**syllogisticcategorical syllogismsyllogisms**

An example of a valid argument is given by the following well-known syllogism:

Boole's goals were "to go under, over, and beyond" Aristotle's logic by: (1) providing it with mathematical foundations involving equations, (2) extending the class of problems it could treat, as solving equations was added to assessing validity, and (3) expanding the range of applications it could handle, such as expanding propositions of only two terms to those having arbitrarily many.

### Logical conjunction

**conjunctionANDlogical AND**

|True and false preserving: || Proposition * Logical conjunction (AND, \and ) * Logical disjunction (OR, \or )

As a rule of inference, conjunction Introduction is a classically valid, simple argument form.

### Logical truth

**necessarily truenecessary truthlogical necessity**

The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction.

Validity

### Logical connective

**logical operatorconnectivesconnective**

! Preservation properties !! Logical connective sentences

Truth-preserving: The compound all those argument are tautologies is a tautology itself. E.g., ∨, ∧, ⊤, →, ↔, ⊂ (see validity).

### Soundness

**soundunsoundlogically sound**

The problem with the argument is that it is not sound.

Validity

### Mathematical fallacy

**Howlersfallacyinvalid proof**

Mathematical fallacy

Such an argument, however true the conclusion, is mathematically invalid and is commonly known as a howler.

### Well-formed formula

**formulaformulaswell-formed**

A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.

### If and only if

**iffif, and only ifmaterial equivalence**

A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.

### Interpretation (logic)

**interpretationinterpretationsinterpreted**

A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid. A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language.

### Contradiction

**contradictorycontradictionscontradict**

The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction.

### Formal language

**formal language theoryformal languageslanguage**

A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language.

### Model theory

**modelmodelsmodel-theoretic**

Model theory analyzes formulae with respect to particular classes of interpretation in suitable mathematical structures.

### Truth value

**truth-valuelogical valuetruth values**

In truth-preserving validity, the interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true'.

### Proposition

**propositionspropositionalclaim**

|True and false preserving: || Proposition * Logical conjunction (AND, \and ) * Logical disjunction (OR, \or )

### Logical disjunction

**ORdisjunctionlogical or**

|True and false preserving: || Proposition * Logical conjunction (AND, \and ) * Logical disjunction (OR, \or )

### Logical biconditional

**biconditionalA if and only if BBiconditional (if and only if, xnor)**

|True preserving only: || Tautology ( \top ) * Biconditional (XNOR, ) * Implication ( \rightarrow ) * Converse implication ( \leftarrow )

### Material conditional

**implicationconditionalmaterial implication**

|True preserving only: || Tautology ( \top ) * Biconditional (XNOR, ) * Implication ( \rightarrow ) * Converse implication ( \leftarrow )

### Converse implication

**A if BCIMP Converse implicationConverse implication ( \leftarrow )**

|True preserving only: || Tautology ( \top ) * Biconditional (XNOR, ) * Implication ( \rightarrow ) * Converse implication ( \leftarrow )