TIPS ON SYMBOLIZING IMPLICATIONS
Because the logical relation expressed by material implication forms a keystone of much reasoning, it is common to find premisses and conclusions in arguments expressed as implications.
The most common, basic way this relationship is expressed in English is through the use of the "if...,then...." construction. The "if" clause expresses the antecedent and the "then" clause expresses the consequent.Thus, "If P, then Q." is directly symbolized: P => QIn the logical symbolism the order of terms in the implication is of crucial importance; however, it is possible to reverse the order of the clauses in English. The main point to remember is the "if" always introduces the antecedent (i.e. the term on the left side of the implication symbol), no matter where the "if" clause appears in the sentence. Thus in English, the same proposition can be written as either:
"If P, then Q." or as "Q, if P." Both are symbolized: P => QFailure to get the order of the terms correct, i.e. getting the implication "backwards," is the most common error in symbolizing implications!
Moreover, English can express this same relationship in a variety of other ways, such as:P implies Q.We also frequently find the language of "necessary" or "sufficient" conditions, which are the reverse of each other.
P entails Q. both are also symbolized: P => Q
or it may be expressed in the passive voice as
Q is implied by P.
Q is entailed by P. both are also symbolized: P => Q
NOTE: The introduction of the word "only" in front of the "if" has the effect of reversing the order of the implication!!!
It is useful to think of the expression "only if" as a single "logical operator" which has the function of introducing a consequent.
Whenever a clause is introduced by the phrase "only if" it MUST be in the consequent of the implication (to the right of the horseshoe symbol). Where the phrse appears in the sentence is not important, what matters is which clause "only if" introduces; that clause must be the consequent. Failure to heed this point is a very common error committed by introductory students.
Thus "P only if Q." and
"Only if Q, P" are both symbolized P => Q
"P is a sufficient condition for Q."A sufficient condition is always an antecednt.
A necessary condition is always a cconsequent.
"Q is a necessary condition for P." both are also symbolized: P => Q
The logical word "if" can be thought of as a way of stating that something is a sufficient condition.
The word "only if" is a way to state that something is a necessary condition.
Disjunctions and conjunctions can be either antecedents or consequents or both. Thus:
If either P or Q, then R. is symbolized ( P v Q ) => RNegations can also enter implications, either on the specific terms of the implication or on the whole implication.
If P, then either Q or R. " " P => ( Q v R )
If both P and Q, then R " " ( P . Q ) => R
If P, then both Q and R. " " P => ( Q . R )
If either P or Q, then either R or S. ( P v Q ) => ( R v S )
If either P or Q, then both R and S. ( P v Q ) => ( R . S )
If both P and Q, then both R and S. ( P . Q ) => ( R . S )
If both P and Q, then either R or S. ( P . Q ) => ( R v S )
Thus for the cases where the terms are negated we get:If not-P, then Q. symbolized as ~P => QWhere the whole implication is negated we get:
If P, then not-Q. " " P => ~Q
If not P, then not-Q. " " ~P => ~QWhen the whole implication is negated, the symbolic expression for that implication must be put into parentheses (or brackets or braces) and the tilde be put on the outside of the parentheses.It is false that if P, then Q.
P does not imply Q.
P does not entail Q. all of which are symbolized as ~ ( P => Q )
P is not a sufficient condition for Q.
Q is not a necessary condition for P.
The most complicated cases arise when an implication has further implications as its antecedents or its consequents.The case where the consequent of an implication is itself an implication (i.e. where one implication implies another) is a very common case, and should always be symbolized by grouping terms to the right:
If we have a good defense, then if we have a good offense, then we have a good team.
If P, then if Q then R. is symbolized P => ( Q => R )
[Note that the second "if" in this format follows the first "then".]
The opposite case, where the antecedent is an implication, is relatively rare. Generally this relationship cannot be easily expressed using two "If...,then...." constructions, so generally the first of the implications, the one forming the antecedent, will be expressed using some alternative words:
All the following forms:It is alwayts a good idea to remember that symbolizing is not a mechanical process and that you must first understand the proposition in question and then symbolize it.If P implies Q, then R.are all symbolized as ( P => Q ) => R
If P entails Q, then R.
If P is a sufficient condition for Q, then R.
If Q is a necessary condition P, then R.
P implies Q, implies R.