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IMPLICATION |
EQUIVALENCE |
Implication and
Equivalence
When we studied IF conditions, we used the truth
value of one statement to conclude the truth value of another logically
related statement. It's quite common in reasoning to draw some
meaning from one logical relationship to indicate or suggest another.
The logical standards for establishing such linkages are implication and
equivalence.
In a casual context, implication signifies an
intended interpretation of an idea not explicitly stated. The
person making the implication hopes that the recipient will infer the
intended meaning. The problem is that the speaker and the listener
(or the writer and the reader) are sometimes not quite in tune with each
other's thoughts, and any vagueness in the message can lead to
misunderstanding. Sometimes plain language seems to hint at things
that really aren't intended, and a miscued inference can make for a
strange interpretation. Other times, there's an intended meaning,
but the recipient either infers it differently or fails to catch it at
all. For example, if Albert comments on Alice's outfit, noting
with a wink, "Hey, that dress is tight," he's implying a
compliment that Alice is attractively dressed; however, if Alice fails
to catch Albert's gesture or mistakes it for a wince, she might infer
from his remark that he thinks she's getting fat. In other words,
implication and inference in the ordinary sense often have to do with
subtle hints, suggestions, and guesswork. It's different in logic,
where an implication is a conclusion that's virtually certain, and an
inference reflects an idea that appears highly likely for lack of
credible alternatives.
Note:
\
signifies implication, "therefore;" :: signifies equivalence, "is
equivalent to."
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IMPLICATION |
EQUIVALENCE
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IMPLICATION
in logic, implication occurs when the meaning of
an explicitly stated relationship contains all the necessary conditions
to conclude another, implicit, relationship. It's not just
a suggestion; it's a deductive certainty. For example, if we know
that the statement "both Passepied and Quodlibet are musicians" is true,
then we can also be sure that "Passepied is a musician" is true and that
"Quodlibet is a musician" is true. This is an example of
simplification, the implication of a true conjunction that each of
its conjuncts taken individually must also be true: if p * q
is true, then p must be true and q must be true.
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We've already
studied a number of logical implications. (Note that the
accepted abbreviation for each operation accompanies its name.
The abbreviations are useful in constructing proofs.) |
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Hypothetical syllogism
(HS) |
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If p is true then q
is true, and if q is true then r is true;
so, if p is true, then r is true.
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(p → q) * (q
→ r)
\
p → r |
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Disjunctive syllogism
(DS) |
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Either p or q is
true, and p is false;
so, q is true.
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(p ˅
q) * ~p
\ q |
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Modus ponens (MP) |
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If p is true then q
is true, and p is true;
so, q is true.
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(p → q)
* p
\
q |
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Modus tollens (MT) |
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If p is true then q
is true, but q is false;
so, p is false. |
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(p → q)
* ~q
\
~p |
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Here are a few more.
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Simplification (Simp) |
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Both p and q are
true;
so, p is true
(or so, q is true).
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p *
q
\
p
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When a conjunction as
a whole is true, each of its conjuncts is also individually true. |
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Addition (Add) |
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p is true;
so, either p or q is true.
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p
\
p ˅ q |
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So long as p
is true, it doesn't matter whether q is true or false; when
p is in fact true, any disjunction having p as one of
its disjuncts must also be true. Of course, the same holds if
q is known to be true and the truth value of p is
unknown. |
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Constructive Dilemma
(CD) |
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Either p or q is
true, and if p is true then r is true, and if q
is true then s is true;
so, either r or s is true. |
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(p ˅ q) * (p → r) * (q → s)
\ r
˅ s |
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A disjunctive
relationship can be conveyed through a pair of conditions, with each
condition using one of the disjuncts as its antecedent of one
condition, and the other disjunct as the antecedent of the other
condition. Incidentally, the same would be true of a
conjunctive relationship, using its conjuncts as the antecedents of
the conditions. |
As a rule, implication does not usually work in reverse.
While the conjunction p * q implies p individually
and q individually, neither p nor q by itself
implies p * q. Knowing only that p is true,
without knowing whether q is true or false, does not permit us to
deduce p * q, because if it should turn out that q
is false, then p * q would false as well. The same
applies, of course, if we know only that q is true without knowing whether
p is true. Only if we know that p is true individually
and that q is also true individually can we deduce that p * q
is also true.
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IMPLICATION |
EQUIVALENCE
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EQUIVALENCE
Despite that implication is usually a one-way
deal, there are special instances in which each of two statements
implies the other. In such cases, the statements are logically
equivalent. Logical equivalence occurs when two logical
relationships are exactly interchangeable. For example, "Pauline
is married to Quncy" has exactly the same logical meaning as "Quincy is
married to Pauline" (under the conventional concept of monogamous
marriage). Each statement exactly implies the other, with
absolutely nothing added or lost either way. Furthermore, both are
equivalent to "Pauline and Quincy are married to each other."
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We've
already encountered several examples of logical equivalence.
Here's a quick refresher. |
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Association (As) |
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"p and q, and
also r"
is equivalent to
"p, and also q and r,"
and also to
"all of p and q and r." |
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(p * q) * r :: p * (q * r)
:: p * q * r |
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Consecutive
conjunctions can be grouped in any way, or treated as a unit.
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"Either p or q,
or r"
is equivalent to
"p, or either q or r,"
and also to
"any of p or q or r."
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(p ˅ q) ˅ r
:: p ˅ (q ˅ r)
:: p ˅ q ˅ r |
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Consecutive
disjunctions can be grouped in any way or treated as a unit. |
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Commutation (Com) |
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"Both p and q"
is equivalent to "both q and p." |
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p * q :: q * p |
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A conjunction's
conjuncts are interchangeable. |
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"Either p or q"
is equivalent to "either q or p." |
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p ˅ q
:: q ˅ p |
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A disjunction's
disjuncts are interchangeable. |
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"If and only if p then
q"
is equivalent to
"if and only if q then p."
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p ↔ q :: q ↔ p |
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A bi-condition's
conditions are interchangeable. |
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Contraposition (Cont) |
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"If p is true then q
is true" is equivalent to "If q is false then p is
false."
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p → q ::
~q → ~p |
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A condition's
antecedent and consequent can be interchanged if and only if both
are negated. Contraposition provides the rational basis for
modus tollens. |
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Double negation (DN) |
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"It's false that p is
false" is equivalent to "p is true." |
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~~p :: p
~(~p) :: p |
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An odd number of
negations of a claim is equivalent to a single negation; an even number of
negations of a claim is equivalent to no negation. |
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Here are some more. However, the plain-language versions are
omitted, because the relationships strain the capacity of plain
language to express clearly. Still, the meanings aren't
difficult to grasp in symbolic form, once one has the knack of it.
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Distribution (Dist) |
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p * (q ˅ r)
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:: |
(p * q) ˅ (p * r)
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A conjunction is
distributed through a disjunction by applying the odd conjunct to
each of the disjuncts. |
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p ˅ (q * r) |
:: |
(p ˅ q) * (p ˅ r)
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A disjunction is
distributed through a conjunction by applying the odd disjunct to
each of the conjuncts. |
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DeMorgan's laws (DeM) |
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~(p * q) |
:: |
~p ˅ ~q
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The negation of a
conjunction can also be expressed as the disjunction of the
negations of its terms. |
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~(p ˅ q) |
:: |
~p * ~q |
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The negation of a
disjunction can also be expressed as the conjunction of the
negations of its terms. |
Equivalences provide optional ways of expressing
relationships. Sometimes one way is clearer and more efficient,
and sometimes the other way has the edge, depending on how the rest of
the argument is structured.
Implications and equivalences have been known
for centuries, but in symbolic form have found new applications in
computer science.
It isn't necessary (and might well be fruitless)
to commit all of these equivalences to memory, since the less frequently
used ones would probably be forgotten long before an opportunity to
apply them presented itself. However, the reader might wish to
copy them for future reference, because they do come in handy from time
to time.
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IMPLICATION |
EQUIVALENCE |
TERMS
In this lesson we've developed a more
comprehensive appreciation for the general power of logic. In the
following lesson, we'll acquire the key to link that power to real-world
issues, by understanding how the immense variety of plain-language
expressions translates to our five standard logical operations.
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Next: Simplifying Plain Language
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IMPLICATION |
EQUIVALENCE
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