5.1.8 Valid Arguments
Now, validities are clearly logical in a certain sense; they are descriptions featuring a cast-iron guarantee of satisfiability. But logic has traditionally appealed to the more dynamic notion of valid arguments, a movement, or inference , from premises to conclusions.
Valid arguments
Suppose 
, and 
are a finite collection of first-order formulae. We then call the argument with premises 
and conclusion a valid argument if and only if the following is true for this argument: Whenever all the premises are satisfied in some model using some variable assignment, then the conclusion is also satisfied in the same model using the same variable assignment. The notation
means that the argument with premises 
and conclusion is valid.
Terminology
There is an extensive terminology when it comes to talking about valid arguments, allowing us for example to refer to as a valid inference from the premises , or to as a logical consequence of .
Note that if the premises and the conclusion are all sentences the definition of valid arguments can be rephrased as follows: an argument is valid if whenever the premises are true in some model, the conclusion is true as well. The truth of the premises guarantees the truth of the conclusion.
An Example
Let's have a look at an example. The argument with premises 
and 
and the conclusion 
is valid. That is:
The truth of the premises and guarantees that of the conclusion .
As the reader may suspect, there is a connection between the validity of this argument and the fact that
Deduction Theorem
The example suggests that with the help of the Boolean connectives 
and 
we can convert valid arguments into validities. This is exactly what is stated by the deduction theorem .