Valid inference forms
The letters—X, Y, and Z—are called variables, like the in algebra, and by which an inference form is produced by uniformly replacing all the variables in it with expressions that make sense in the context.
Every instance of the inference form will be logically valid if any variable that makes the premises true also ensures the truth of the conclusion. In other words, a valid inference form is one for which no instance of it can have true premises and a false conclusion.
In contrast, the following inference form is not valid:
- Every X is a Y.
- Some Z’s are Y’s.
- Therefore, some Z’s are X’s.
because there are instances in which the premises are true but the conclusion is false. For example,
- Every dog is a mammal.
- Some winged creatures are mammals.
- Therefore, some winged creatures are dogs.
Formal logic identifies and validates inference forms and derives the relations that hold among valid ones.
Valid proposition forms
A proposition form is a combination of propositions and it is logically valid if it is true for all instances of the propositions, for example,
- Everything is X or not X
Formal logic involves proposition forms and inference forms.
System of logic
A system of logic is made up of:
- a symbolic apparatus comprising a set of symbols, the rules for combination of symbols into formulae and the rules for manipulation of formulae
- definitions of these symbols and formulas
A system of logic without symbolic and formulaic definitions is called uninterpreted or purely formal.
A system of logic with symbolic and formulaic definitions is called interpreted.
An axiomatic system of logic is one which is based on unproved formulas called axioms from which further formulas called theorems are proved. The proof of a theorem depends solely on which formulas are taken as axioms and the rules for deriving theorems from axioms, and not on the meaning of the axioms or theorem.