Chapter Four, Tutorial One
Basic Symbolizations and Expressive Completeness

In this chapter we need to use what we have learned about the semantics of SL (chapter three) in order to better understand symbolization (something we have already seen a good deal of in chapter two).

Obviously, SL comes with only five connectives. Still, with these five connectives one can symbolize many different English connectives. For example, we have used SL to symbolize English sentences of the form:

(*) P only if Q

You will recall that an English statement of form (*) can be symbolized as either

P>Q
  or
~Q>~P

(Either way to symbolize is fine: the two are logically equivalent in SL, as a truth table will quickly show. And both make sense of "only if". Think about "I'm at school only if it's Thursday"; this might be understood as "If I'm at school, then it's Thursday" or as "If it's not Thursday, I'm not at school".)

As a reminder of the many different English connectives and their symbolization in SL, let's repeat the table summary from chapter two.

Equivalent English Forms (Each table element – i.e., box – below gives English forms. Instances of each can be symbolized by a sentence of any SL form on its right. Please note that there are many more English forms than can be covered below.) Equivalent SL Forms (Each table element below gives SL sentence-forms to guide in translating English sentences of forms found on the left. Please note that this is an incomplete list of possible symbolizations.) Example Applications (Each of the table elements below shows a way to apply the table elements on their left.)

If P, then Q.
If P, Q.
Provided P, Q.
Were P to hold, Q would be true.
Should P be true, Q.
P only if Q.
P is a sufficient condition for Q.
P implies Q.

P>Q
~Q>~P 		If there is fire, then there is Oxygen" or "There is fire only if there is oxygen" may both be symbolized as 'F>O' or equivalently as '~O>~F'.
P if Q.
P provided Q.
P is a necessary condition for Q.
 Q>P
~P>~Q 		"Water is a necessary condition for life" or "there's water if there's life" may both be symbolized as 'L>W' or equivalently as '~W>~L'.
P if and only if Q.
P just in case Q.
P is necessary and sufficient for Q.
 P=Q
(P>Q)&(Q>P) 		"An argument is sound if and only if it is both valid and has true premises" may be symbolized as either 'S=(V&T)' or '[S>(V&T)] &[(V&T)>S]'

Both P and Q.
P and Q.
P but Q.
Q and P.
Q but P.
P however Q.
P although Q.
P moreover Q.

 P&Q
Q&P 		"Sandra is both brave and careful", "Sandra is brave, moreover she is careful' or "Sandra is brave but careful" can all be symbolized as 'B&C' or 'C&B'.
Either P or Q.
Either Q or P.
P or Q.
Q or P.
At least one of P, Q.
 PvQ
QvP 		"Either the other team will score and tie up the game, or we win!" can be symbolized as '(S&T)vW'.
P unless Q.
Q unless P.
Unless P, Q.
Unless Q, P.
 PvQ
~Q>P
~P>Q 		"We win unless the other team scores" can be symbolized as 'WvS', '~W>S', or '~S>W'.
Neither P nor Q.
Not-P and not-Q.
~(PvQ)
~P&~Q 		"They neither scored not tied the game" may be symbolized as either '~(WvT)' or '~W&~T'.
It's not the case that both P and Q.
Not both P and Q.
Either not-P or not-Q.
 ~(P&Q)
~Pv~Q 		"Sandra is not both brave and careful" may be symbolized as either '~(B&C)' or '~Bv~C'.

Now, using this table as a reminder if needed, say which of the following are proper symbolizations of

Jason will bike with us tomorrow unless Rhonda talks him into helping her move.

Click on all correct answers below. (Use 'B' to symbolize "Jason will bike with us tomorrow" and 'M' to symbolize "Rhonda talks him into helping her move".)

    1. B&M
    2. ~B&M
    3. BvM
    4. B>~M
    5. ~B>M
    6. ~M>B