And/or not?
One aspect that can be especially difficult in translating statements into symbolic form is trying to figure out the role negations play in conjunctions and disjunctions. However, this can be easily resolved. You will find the following charts useful:

The key to translating the above statements is to understand how negations work, and the use of the word "both." A negation only affects whatever it is in front of. In the first two examples in the chart above, the only variable that is being negated is "A." Neither statement indicates that the negation should be carried over to B.

In the second two examples, the word "both" indicates that "B" is affected by the negation. Why are the two statements translated differently? The third example has "not" in front of ‘both." This indicates it is not possible for both statements to be true at the same time: "they are not both true." Therefore, we need to first claim that they are both true: A • B. We then need to deny that claim. Since we are denying the entire claim, we put it in parentheses and then place the negation outside: ~(A • B).

In the fourth example, the word "not" occurs after the word "both." "Both are not." This indicates that both variables are not true. In other words, they are both false. How do we indicate a simple statement is false? By placing a negation directly in front of the statement. This gives us ~A • ~B.

We can sum this up with two rules:

Much the same is true for negations in disjunctions, as the following chart illustrates:

The main differences here are the use of the word "either" instead of "both," and the use of disjunctions rather than conjunctions. Note that "neither/nor" is the same as "not either." "Neither" is a contraction of "not either" and has the same meaning.

Again, we can sum this up with two rules:

Flipping over conditional statements
Another puzzle when first trying to translate into symbolic form are conditional statements. Students are confused over the differences between "if," "only if," and "if and only if." Your textbook has a chart on p. 293 covering this, and I advise you to memorize this. This is something I particularly look for on tests. I want to try to clarify how to deal with conditional statements.

The basic idea is that when we translate something as a conditional statement, we want the antecedent to come first and the consequent to come second. This is the order that is found in "if/then" statements, such as "If today is Monday, then tomorrow is Tuesday." Such a statement is translated in the order in which it is found:

M > T

However, not all conditional statements have an "if/then" form. Sometimes, "if" will occur in the middle of a conditional statement. For example:

You have bad taste if you like Michael Bolton

In translating this statement into symbolic form, we must make sure that the antecedent is listed first. The antecedent is the sufficient condition, the guarantee. Consider: does having bad taste guarantee you like Michael Bolton? No. You might hate Michael Bolton, yet love the Britney Spears. On the other hand, does liking Michael Bolton guarantee you have bad taste? Oh, yes! So "liking Michael Bolton" is the sufficient condition and must be the antecedent, although the English statement lists this part second. The correct translation is:

M > B

Where M stands for "you like Michael Bolton" and "B" stands for "you have bad taste." (Notice that I did not use "Y" to symbolize either part of the statement. "You" is such a common pronoun that it is avoided when symbolizing. Much the same is true for words such as "the" or "a.") So, when "if" is in the middle of a conditional statement, you flip the order of the variables.

On the other hand, when a conditional statement has "only if’ in the middle, you do not flip the order of the variables. Why? Because "only if" designates a necessary condition, and necessary conditions always are found in the consequent. For example:

You graduated only if you enrolled.

Enrolling is a necessary condition for graduating. Necessary conditions are found in the consequent. As noted above, the consequent should come second. In this example, it is listed second. Therefore, when we translate this, we have no need to flip the order of the variables:

G > E

Given all of this, it follows that sometimes with conditional statements you will need to flip the order of the variables, and sometimes you will not. The charts below will help serve as a guide to this.

This deals with conditional statements. But students often confuse "if" or "only if" with "if and only if." Where "if" and "only" are implications, the phrase "if and only if" requires the use of a biconditional. A biconditional is really two conditional statements combined. Think of the phrase, "If and only if." That combines "if" with "only if." So for a statement of this type, the triple bar symbol is used:

Batman finds the Riddler if and only if the Riddler leaves clues.

B = R