Validity in logic

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. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument’s conclusion. 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.

The concept of interpretation, which is central to this explanation, can be intuitively understood as a generalization of the variable assignment in propositional logicunderstand: Only by the assignment of the proposition variables of a propositional formula can the formula as a whole attribute a truth value. In more complex logics also assignments must be made to the formal components of a formula, which determine the truth value of the overall formula. In predicate logic, for example, the definition of a universe and an assignment of predicate symbols to predicates (on this universe) and of function symbols to functions (on this universe) takes place. Only by referring to a set of objects in a considered world can it be ascertained whether a formula can be fulfilled and whether it may always be fulfilled, that is, universally valid.

The following table lists some closely related terms and synonyms. The columns and are in an equivalence relationship, e.g. B. is just then universally valid, if   is unsatisfiable.

Synonyms condition
universal tautological (in propositional logic) All interpretations meet the formula. unattainable
satisfiable consistent, consistent There is an interpretation that satisfies the formula. falsifiable
falsifiable refutable There is an interpretation that disproves the formula. satisfiable
unattainable inconsistent, contradictory No interpretation fulfills the formula. universal

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. Under such conditions it would be self-contradictory to affirm the premises and deny the conclusion. The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a logical consequence of its premises.

An argument that is not valid is said to be “invalid”.

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

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

What makes this a valid argument is not that it has true premises and a true conclusion, but the logical necessity of the conclusion, given the two premises. The argument would be just as valid were the premises and conclusion false. The following argument is of the same logical form but with false premises and a false conclusion, and it is equally valid:

All cups are green.
Socrates is a cup.
Therefore, Socrates is green.

No matter how the universe might be constructed, it could never be the case[why?] that these arguments should turn out to have simultaneously true premises but a false conclusion. The above arguments may be contrasted with the following invalid one:

All men are immortal.
Socrates is a man.
Therefore, Socrates is mortal.

In this case, the conclusion contradicts the deductive logic of the preceding premises, rather than deriving from it. Therefore, the argument is logically ‘invalid’, even though the conclusion could be considered ‘true’ in general terms. The premise ‘All men are immortal’ would likewise be deemed false outside of the framework of classical logic. However, within that system ‘true’ and ‘false’ essentially function more like mathematical states such as binary 1s and 0s than the philosophical concepts normally associated with those terms.

A standard view is that whether an argument is valid is a matter of the argument’s logical form. Many techniques are employed by logicians to represent an argument’s logical form. A simple example, applied to two of the above illustrations, is the following: Let the letters ‘P’, ‘Q’, and ‘S’ stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:

All P are Q.
S is a P.
Therefore, S is a Q.

Similarly, the second argument becomes:

All P are not Q.
S is a P.
Therefore, S is a Q.
An argument is termed formally valid if it has structural self-consistency, i.e. if when the operands between premises are all true, the derived conclusion is always also true. In the third example, the initial premises cannot logically result in the conclusion and is therefore categorized as an invalid argument.

Valid formula
A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies.

These arguments are valid because they both have the form of a disjunctive syllogism, which is a valid argument scheme:

p o q
No p
Therefore, q
To determine the validity of a specific argument, then, it is enough to determine the validity of its argument scheme, and this can be achieved by semantic means or by syntactic means.

Semantic method
In the semantic method, an argument scheme is said to be valid when it is impossible for the premises to be true and the conclusion false. To determine if this is the case, the truth of the premises is assumed, and by applying the definitions of truth, one tries to deduce the truth from the conclusion. Or also, the premises are supposed to be true and the conclusion false, and by applying the definitions of truth, an attempt is made to deduce a contradiction (reduction to the absurd).

In propositional logic, an alternative method is to transform an argument into its corresponding formula, and build a truth table. If the formula turns out to be a logical truth, then the argument is valid. This is because the deduction theorem and its converse are valid, but also because the propositional logic is decidable, and therefore always admits of an algorithmic procedure to determine whether any formula is a logical truth or not.

{\ displaystyle {\ begin {array} {c | c || c | c | c | c} p & q & (p \ lor q) & \ neg p & (p \ lor q) \ land \ neg p & [(p \ lor q) \ land \ neg p] \ to q \\\ hline V & V & V & F & F & V \\ V & F & V & F & F & V \\ F & V & V & V & V & V \\ F & F & F & V & F & V \\\ end {array}}}

Syntactic Method
In the syntactic method, an argument scheme is said to be valid when there is a deduction of the conclusion from the premises of the argument and the axioms of the system, using only the allowed inference rules.

In a natural deduction system, it is like the set of axioms is empty, an argument scheme will be valid when there is a deduction of the conclusion from the premises, using only the allowed rules of length.

A statement can be called valid, i.e. logical truth, if it is true in all interpretations.

Validity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:

All animals live on Mars.
All humans are animals.
Therefore, all humans live on Mars.

The problem with the argument is that it is not sound. In order for a deductive argument to be sound, the deduction must be valid and all the premises true.

Model theory analyzes formulae with respect to particular classes of interpretation in suitable mathematical structures. On this reading, formula is valid if all such interpretations make it true. An inference is valid if all interpretations that validate the premises validate the conclusion. This is known as semantic validity.

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

In a false-preserving validity, the interpretation under which all variables are assigned a truth value of ‘false’ produces a truth value of ‘false’.

Preservation properties Logical connective sentences
True and false preserving: Proposition  • Logical conjunction (AND, \land  )  • Logical disjunction (OR, \lor  )
True preserving only: Tautology ( \top  )  • Biconditional (XNOR, \leftrightarrow  )  • Implication ( \rightarrow  )  • Converse implication ( \leftarrow  )
False preserving only: Contradiction ( \bot  ) • Exclusive disjunction (XOR, \oplus  )  • Nonimplication ( \nrightarrow  )  • Converse nonimplication ( \nleftarrow  )
Non-preserving: Negation ( \neg  )  • Alternative denial (NAND, \uparrow  ) • Joint denial (NOR, \downarrow  )