Formal Logic/Sentential Logic/Validity

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Satisfaction and validity of formulae

Satisfaction

In sentential logic, an interpretation under which a formula is true is said to satisfy that formula. In predicate logic, the notion of satisfaction is a bit more complex. A formula is satisfiable if and only if it is true under at least one interpretation (that is, if and only if at least one interpretation satisfies the formula). The example truth table of Truth Tables showed that the following sentence is satisfiable.

$\mathrm {P} \land \mathrm {Q} \rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )$

For a simpler example, the formula $\mathrm {P} \,\!$ is satisfiable because it is true under any interpretation that assigns $\mathrm {P} \,\!$ the value True.

We use the notation ${\mathfrak {v}}\vDash \varphi$ to say that the interpretation ${\mathfrak {v}}\,\!$ satisfies $\varphi \,\!$. If ${\mathfrak {v}}$ does not satisfy $\varphi ,$ then we write ${\mathfrak {v}}\not \vDash \varphi .$

The concept of satisfaction is also extended to sets of formulae. A set of formulae is satisfiable if and only if there is an interpretation under which every formula of the set is true (that is, the interpretation satisfies every formula of the set).

A formula is unsatisfiable if and only if there is no interpretation under which it is true. A trivial example is

$\mathrm {P} \land \lnot \mathrm {P} \,\!$

You can easily confirm by doing a truth table that the formula is false no matter what truth value an interpretation assigns to $\mathrm {P} \,\!$. We say that an unsatisfiable formula is logically false. One can say that an unsatisfiable formula of sentential logic (but not one of predicate logic) is tautologically false.

Validity

A formula is valid if and only if it is satisfied under every interpretation. For example,

$\mathrm {P} \lor \lnot \mathrm {P} \,\!$

is valid. You can easily confirm by a truth table that it is true no matter what the interpretation assigns to $\mathrm {P} \,\!$. We say that a valid sentence is logically true. We call a valid formula of sentential logic—but not one of predicate logic—a tautology.

We use the notation $\vDash \varphi$ to say that $\varphi$ is valid and $\not \vDash \varphi$ to indicate $\varphi$ is not valid.

Equivalence

Two formulae are equivalent if and only if they are true under exactly the same interpretations. You can easily confirm by truth table that any interpretation that satisfies $\mathrm {P} \land \mathrm {Q} \,\!$ also satisfies $\mathrm {Q} \land \mathrm {P} \,\!$. In addition, any interpretation that satisfies $\mathrm {Q} \land \mathrm {P} \,\!$ also satisfies $\mathrm {P} \land \mathrm {Q} \,\!$. Thus they are equivalent.

We can use the following convenient notation to say that $\varphi \,\!$ and $\psi \,\!$ are equivalent.

$\varphi \,\!$ $\psi \,\!$

which is true if and only if

$\vDash \ (\varphi \leftrightarrow \psi )\,\!$

Validity of arguments

An argument is a set of formulae designated as premises together with a single sentence designated as the conclusion. Intuitively, we want the premises jointly to constitute a reason to believe the conclusion. For our purposes an argument is any set of premises together with any conclusion. That can be a bit artificial for some particularly silly arguments, but the logical properties of an argument do not depend on whether it is silly or whether anyone actually does or might consider the premises to be a reason to believe the conclusion. We consider arguments as if one does or might consider the premises to be a reason for the conclusion independently of whether anyone actually does or might do so. Even an empty set of premises together with a conclusion counts as an argument.

The following example show the same argument using several notations.

Notation 1

$\mathrm {P} \,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$

Therefore $\mathrm {Q} \,\!$

Notation 2

$\mathrm {P} \,\!$

$\mathrm {P} \rightarrow \mathrm {Q} \,\!$

∴ $\mathrm {Q} \,\!$

Notation 3

$\mathrm {P} \,\!$

${\underline {\mathrm {P} \rightarrow \mathrm {Q} }}\,\!$

$\mathrm {Q} \,\!$

Notation 4

$\mathrm {P} ,\ \mathrm {P} \rightarrow \mathrm {Q} \,\!$   ∴   $\mathrm {Q} \,\!$

An argument is valid if and only if every interpretation that satisfies all the premises also satisfies the conclusion. A conclusion of a valid argument is a logical consequence of its premises. We can express the validity (or invalidity) of the argument with $\Gamma \,\!$ as its set of premises and $\psi \,\!$ as its conclusion using the following notation.

(1)    $\Gamma \vDash \psi \,\!$

(2)    $\Gamma \ \not \vDash \ \psi \,\!$

For example, we have

$\{\mathrm {P} ,\ \mathrm {P} \rightarrow \mathrm {Q} \}\ \vDash \ \mathrm {Q} \,\!$

Validity for arguments, or logical consequence, is the central notion driving the intuitions on which we build a logic. We want to know whether our arguments are good arguments, that is, whether they represent good reasoning. We want to know whether the premises of an argument constitute good reason to believe the conclusion. Validity is one essential feature of a good argument. It is not the only essential feature. A valid argument with at least one false premise is useless. Validity is the truth-preserving feature. It does not tell us that the conclusion is true, only that the logical features of the argument are such that, if the premises are true, then the conclusion is. A valid argument with true premises is sound.

There are other less formal features that a good argument needs. Just because the premises are true does not mean that they are believed, that we have any reason to believe them, or that we could collect evidence for them. It should also be noted that validity only applies to certain types of arguments, particularly deductive arguments. Deductive arguments are intended to be valid. The archetypical example for a deductive argument is a mathematical proof. Inductive arguments, of which scientific arguments provide the archetypical example, are not intended to be valid. The truth of the premises are not intended to guarantee that the conclusion is true. Rather, the truth of the premises are intended to make the truth of the conclusion highly probably or likely. In science, we do not intend to offer mathematical proofs. Rather, we gather evidence.

Formulae and arguments

For every valid formula, there is a corresponding valid argument having the valid formula as its conclusion and the empty set as its set of premises. Thus

$\vDash \psi \,\!$

if and only if

$\varnothing \vDash \psi \,\!$

For every valid argument with finitely many premises, there is a corresponding valid formula. Consider a valid argument with $\psi \,\!$ as the conclusion and having as its premises $\varphi _{1},\varphi _{2},...,\varphi _{n}\,\!$. Then

$\varphi _{1},\ \varphi _{2},\ ...,\ \varphi _{n}\vDash \psi \,\!$

There is then the corresponding valid formula

$\varphi _{1}\land \varphi _{2}\land ...\land \varphi _{n}\rightarrow \psi \,\!$

There corresponds to the valid argument

$\mathrm {P} ,\ \mathrm {P} \rightarrow \mathrm {Q} \,\!$   ∴   $\mathrm {Q} \,\!$

the following valid formula:

$\mathrm {P} \land (\mathrm {P} \rightarrow \mathrm {Q} )\rightarrow \mathrm {Q} \,\!$

Implication

You may see some text reading our arrow $\rightarrow \,\!$ as 'implies' and using 'implications' as an alternative for 'conditional'. This is generally decried as a use-mention error. In ordinary English, the following are considered grammatically correct:

(3)    'That there is smoke implies that there is fire'.

(4)    'There is smoke' implies 'there is fire'.

In (3), we have one fact or proposition or whatever (the current favorite among philosophers appears to be proposition) implying another of the same species. In (4), we have one sentence implying another.

But the following is considered incorrect:

There is smoke implies there is fire.

Here, in contrast to (3), there are no quotation marks. Nothing is the subject doing the implying and nothing is the object implied. Rather, we are composing a larger sentence out of smaller ones as if 'implies' were a grammatical conjunction such as 'only if'.

Thus logicians tend to avoid using 'implication' to mean conditional. Rather, they use 'implies' to mean has as a logical consequence and 'implication' to mean valid argument. In doing this, they are following the model of (4) rather than (3). In particular, they read (1) and (2) as '$\Gamma \,\!$ implies (or does not imply) $\psi \,\!$.

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