When we make a logical inference or deduction, we reason from a antecedent (hypothesis or assumption) to a consequent (conclusion). One of the most familiar form of compound mathematical proposition is "If p, then q." When we combine two propositions by the words "if ..., then ...", we obtain a compound proposition which is denoted as an implication or a conditional proposition. The subordinate clause to which the word "if" is prefixed is called antecedent, and the principal clause introduced by the word "then" is called consequent.
Let p and q be propositions. A sentence of the form
|If p then q [or p implies q]|
|is denoted by|
|p → q|
where p is called the antecedent (hypothesis or assumption) and q is called the consequent (conclusion.) Note that the conditional operator, →, is a connective, like ∧ or ∨, that can be used to join propositions to create new propositions.
The following have the same meanings [memorize these]:
|p → q|
|If p, then q,|
|p implies q,|
|q if p, p only if q,|
|q provided p,|
|q whenever p, q when p,|
|p is a sufficient condition for q,|
|q is necessary condition for p.|
To define "conditional" is not an easy job and we will see the problems associated with this concept under the heading of implication. At this point, it is enough to say the definition of the conditional operator causes distress to many logicians and mathematicians. To give you a taste of this, consider the following. According to the general rule that we will adopt (at least at this point) what is called material implication (as opposed to formal implication), a conditional will be said to be false if, and only if, it has a true antecedent and a false consequent. That is,
|p → q if, and only if, p → q has a true antecedent and a false consequent.|
This simply means that
|When q is true, then p → q is true no matter whether p is true or false.|
|p||q||p → q|
Even more paradoxically,
|If p is false, then p → q is true no matter whether q is true or false.|
|p||q||p → q|
A problem with this concept is that it is common to permit the intrusion of a psychological element, and to consider our acquisition of new knowledge by its means. To understand this consider an example. Suppose, I say:
If he's a logician, then I'm a two-headed calf.
Here, I am making an assertion that I wish to be accepted as a true proposition. I hope that you will notice the falsehood of the consequent, I'm a two-headed calf, that from this "false consequent" you will infer the falsehood of the antecedent, he's a logician, and so come to understand that the person under discussion is no logician.
Now the problem gets really sticky in the following situation. Suppose, I say to you:
You're hanged if you do, and you're hanged if you don't.
It is easy to see that this proposition has the form:
(p → q) ∧ (~p → q)
For the above proposition to be true, each of the conditionals must be true. But, p and ~p cannot both true, so one of the presumably true conditionals has a false antecedent.
I hope that the foregoing discussion has made the following definition of the conditional more acceptable and pleasant (In any event, we will talk about the philosophy of implication and differentiate material and formal implication after the study of argument.)
|if p (antecedent) and q (consequent) be proposition variables (or sentential variables), the conditional of q by p is "If p then q" or "p implies q." We symbolize the conditional by p → q, and frequently read "p implies q" or "p only if q". It is false when p is true and q is false; otherwise it is true.|
By asserting an implication one asserts that it does not occur that the antecedent is true and the consequent is false. An implication is thus true in any one of the following three cases:
|Case 1.||Both antecedent and consequent are true.|
|Case 2.||The antecedent is false and the consequent is true.|
|Case 3.||Both antecedent and consequent is false.|
Truth table for p → q is: (Try to combine above tables into this one.)
|p||q||p → q|
As we can see from the above table, the conditional p → q is false only when the antecedent p is true and the consequent q is false. Accordingly, when p is false, the conditional p → q is true regardless of the truth value of q.
In other words, what we are saying here is that whoever accepts an implication as true, and at the same time accepts its antecedent as true, cannot but accept its consequent; and whoever accepts an implication as true and rejects its consequent as false, must also reject its antecedent.
Let us take another example, this time from a different perspective. Suppose that your friend made the following statement:
"If I behold a rainbow in the sky, then my heart leaps up."
To determine when the proposition "p implies q" is false, ask yourself in which of the four cases you would be willing to call your friend a liar.
Case 1. This case occurs when he does behold a rainbow in the sky and his heart does leap up. Clearly, your friend has told the truth and you can't call your friend a liar.
Case 2. This case occurs when he behold a rain in the sky, and yet his heart did not "leaps up", as your friend said it would. Liar Liar Liar ! Here your friend has not told the truth.
In Case 3 and Case 4, he does not behold a rainbow in the sky. Now in these two cases, you would not really want to call your friend a liar. The reason is that your friend clearly said that something would happen only if he did behold a rainbow in the sky.
Therefore, the proposition:
"If I behold a rainbow in the sky then my heart leaps up."
is true in cases 1, 3, and 4; and false in case 2. Hence, the truth table for implication.
Note that cases 3 and 4 are true by default. A conditional that is true by virtue of the fact that its hypothesis is false is called vacuously true or true by default.
Logical Equivalence Involving Conditional
If we know that a sentential variable p is true or that a sentential variable q is true, we can deduce the truth of a sentential variable r by showing following two things:
1. the truth of r follows from the truth of p, and
2. the truth of r follows from the truth of q.
Then no matter whether p or q is the case, the truth of r must follow. The division into cases method of analysis is based on the following logical equivalence:
p ∨ q → r ≡ (p → r) ∧ (q → r)
The following truth table shows that p ∨ q → r and (p → r) ∧ (q → r) have the same truth values. Hence, the two propositions forms are logically equivalent.
|p||q||r||p ∨ q||p → r||q → r||p ∨ q → r||(p → r) ∧ (q → r)|
Since, column 7 and column 8 have the same truth values and so proposition p ∨ q → r ≡ p ∨ q → r.
Representation of Conditional as Disjunction
In the Principia Mathematica, Whitehead and Russell defined implication in terms of the basic symbols as follows:
p → q = (~p ∨ q)
In the Principia Mathematica, the "=" denotes "is defined to mean." Using this denotation, the above expression can be read:
"p implies q is defined to mean that either p is false or q is true."
The following truth table shows the logical equivalence of "If p then q" and "not p or q":
|p||q||~ p||p → q||~p ∨ q|
Same truth values in column 4 and in column 5 and so p → q ≡ ~p ∨ q.
Negation of a Conditional
By definition, p → q is false if, and only if, its hypothesis, p, is true and its conclusion, q, is false. It follows that the negation of "If p then q" is logically equivalent to "p and not q."
This can be restated symbolically as follows:
~(p → q) ≡ p ∧ ~q
We can show this as follows:
Since p → q ≡ ~p ∧ q
Taking the negation of both sides to obtain
~(p → q) ≡ ~(~p ∧ q)
≡ ~(~p) ∧ (~q)
≡ p ∧ ~q