Bridge to Abstract Mathematics: Mathematical Proof and Structures

72 LOGIC, PART I: THE PROPOSITIONAL CALCULUS Chapter 2

that, up to now, has been used in every known derivation of the conclusion

of the theorem.

ENGLISH-LANGUAGE TRANSLATIONS OF THE CONDITIONAL

AND BICONDITIONAL CONNECTIVES

As seen already, certain uses of the propositional calculus require that we

translate sentences expressed in English into precise symbolic form. Since

there are many ways of expressing an idea in English, the following list of

translations may prove helpful.

REMARK 1 The following three lists provide translations between English

sentences and symbolic representation of those sentences.

1. p + q may be interpreted in any of the following ways:
   If p, then q (q if p)
   p implies q (q is implied by p)
   Whenever p, then q (q whenever p)
   p is stronger than q (q is weaker than p)
   q unless -p ( - p unless q)
   If not q, then not p (p only if q)
   Not q implies not p (not p is implied by not q)
   p is sufficient for q (-q is sufficient for -p)
   q is necessary for p (-p is necessary for - q)
   Either not p or q.


2. p t, q may be interpreted as:

(a) p is equivalent to q

(b) p if and only if q

(c) p is necessary and sufficient for q

(d) p implies q and q implies p

(e) If p, then q and if q, then p

-.... ,, - 3. The following miscellaneous correspondence's are also valid:

Sentence Symbolic translation

(a) p or q or both PVq

(b) p or q, but not both -(p +-+ q)

(c) p, but not q PA -q

(d) p unless q -4+P

You should convince yourself of the reasonableness of these various
i translations and representations, in particular, noting explicitly the role of

various equivalences from Theorem 1 as justification for the translation

(Exercise 8).