Conditional and Biconditional Connectives Discrete Mathematics
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Implication
We shall talk implication in a separate section, in depth. Here, we simply define and talk about the meaning in a general sense. We interpret the meaning of Implication, i.e. "p implies q," or "if p, then q" as "Unless p is false, q is true," or "either p is false or q is true," where p and q are proposition variables. That is to say, "p implies q" is to mean "~p or q"; its truth-value is to be truth if p is false, likewise if q is true, and is to be falsehood if p is true and q is false. The fact that "implies" is capable of other meanings does not concern us: For the time being this is our "meaning" and we are sticking to it.
Accordingly, the truth table for the "implies" or "p → q" is shown below:
p q p → q
T T T
T F F
F T T
F F T
Needless to say we will introduce other operations of propositions too, for example, biconditional, joined falsehood, "~p and ~q," etc. But for now, the above five operations (or sentential operators) will suffice. It is easy to see that negation differ from other four in being a monary (or unary) operator, whereas each of the others is a binary operator, involving two propositions (not necessarily different) for its use. But all five operations agree on this, that their truth-value depends only upon that of the propositions which are their arguments. Given the truth or falsehood of p, or of p and q (as the case may be), we are given the truth or falsehood of the negation, disjunction, incompatibility, or implication. A proposition which has this property is called truth-function. We shall talk about the truth-function shortly.
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
T T T
F T T
Even more paradoxically,
If p is false, then p → q is true no matter whether q is true or false.
p q p → q
F T T
F F T
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.
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