TIPS ON SYMBOLIZING IMPLICATIONSBecause the logical relation expressed by material

implicationforms a keystone of much reasoning, it is common to find premisses and conclusions in arguments expressed asimplications.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

and the "then" clause expresses theantecedentconsequent.Thus, "If P, then Q." is directly symbolized:In the logical symbolism theP => Qorder of termsin 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 => Qor 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 ofreversing 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 aconsequent.

Whenever a clause is introduced by the phrase "only if" it MUST be in theconsequent 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 avery commonerror committed by introductory students.Thus "P only if Q." and

"Only if Q, P" are both symbolizedP => Q

"P is a sufficient condition for Q."Asufficientcondition is always anantecednt.Anecessarycondition is always a cconsequent.

"Q is a necessary condition for P." both are also symbolized:P => QThe 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 anecessary condition.

Disjunctionsandconjunctionscan be either antecedents or consequents or both. Thus:

If either P or Q, then R. is symbolized ( P v Q ) => R

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 )Negationscan also enter implications, either on the specific terms of the implication or on the whole implication.Thus for the cases where the

termsare negated we get:Where theIf not-P, then Q. symbolized as ~P => Q

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 theoutsideof the parentheses.wholeimplication is negated we get: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 animplicationhas furtherimplicationsas its antecedents or its consequents.The case where theconsequentof an implication is itself an implication (i.e. where one implication implies another) is avery commoncase, and should always be symbolized by grouping termsto 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

followsthe first "then".]The opposite case, where the

antecedentis an implication, isrelatively 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:

Allthe following forms:It is alwayts a good idea to remember that symbolizing isIf P implies Q, then 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.are all symbolized as ( P => Q ) => R

not a mechanical processand that you must firstunderstandthe proposition in question and then symbolize it.