The standard signature σ for fields consists of two binary function symbols + and ×, a unary function symbol -, and the two constants 0 and 1. Thus a structure (algebra) for this signature consists of a set of elements A together with two binary functions, a unary function, and two distinguished elements; but there is no requirement that it satisfy any of the field axioms. The rational numbers, the real numbers and the complex numbers, like any other field, can be regarded as σ-structures in an obvious way. But the ring of integers, which is not a field, is also a σ-structure in an obvious way. In fact, there is no requirement that any of the field axioms hold in a σ-structure.
A signature for ordered fields needs an additional binary relation such as algebraic structures in the usual, loose sense of the word.
The ordinary signature for set theory includes a single binary relation ∈. A structure for this signature consists of a set of elements and an interpretation of the ∈ relation as a binary relation on these elements.
Examples of Implications
“If this lecture ends, then the sun will rise tomorrow.” True or False?
“If Tuesday is a day of the week, then I am a penguin.” True or False?
“If 1+1=6, then George passed the exam.” True or False?
“If the moon is made of green cheese, then I am richer than Bill Gates.” True or False?
Examples of Implication Wording
If John is in L.A., then he is in California
To be in California, it’s sufficient for John to be in L.A.
To be in LA, it’s necessary for John to be in
You will get an A if you study hard.
You will get an A only if you study hard.