part of the conditional.)
To take a trite textbook example, "If it is raining (in the normal way), then the streets are wet
(from the rain)".
(The caveats in quotes are there to bypass some smarty who says, "Maybe there's a big umbrella
over the streets to keep them from getting wet.)
That conditional example is sometimes used in the most powerful device of Logic known by
the Latin name of "modus ponens" (MP) of "tautology of asserting the precedent" (explanation
of the latter phrase in another file).
If it is raining, then the streets are wet.
It is raining.
Therefore, the streets are wet.
Let "R" denote the statement, "It is raining.", and let "W" denote the statement, "The streets
are wet." The conditional is usually denoted by a right-arrow, while conjunction
("and") is denoted by ampersand. So the above argument can be denoted:
((R Þ W) & R) Þ W.
The biconditional (a.k.a. equivalence) is of the form "If, and only if, ____, then
____". This means that the first statement implies the second statement, and the second
statement implies the first statement, that is, the two statements are equivalent: say the
"same thing". This is the best form for a definition, thus, "two sets are equipotent
if, and only if, they have the same members".
The other useful function of the biconditional is to "lead us not into temptation (of saying
something foolish)".) Example, with the biconditional denoted by a double arrow:
x - 1 = 0 Û x ¹ 0.
We can assert that linear algebraic equation if, and only if (iff), "x" is not zero. Why? Substitute
zero in the equation to obtain:0 - 1 = 0 or -1 = 0.
If you allow that, you "destroy" arithmetic, since very number equals every other number.
Logicians and mathematicians say, regarding "If A, then B", that precedent "A" denotes the
sufficient part of the conditional, while consequent "B" denotes the necessary
part. (In the Venn diagram, you see that the precedent circle or rectangle is inside the
consequent one. It is sufficient to know that a point is within the inner structure to know it is
inside the outer one; it is necessary but not sufficient to know a point is inside the outer
structure, otherwise it cannot be in the inner structure.) In the biconditional, both
statements are necessary and sufficient (the most desirable case, when we can attain it).
You glean some understanding of the difference between conditional and biconditional by
Venn diagrams of logical arguments using them. A Venn diagram uses a rectangle for "The Universe
of Discourse", and circles or smaller rectangles (perhaps overlapping or concentric) for statements. Here
is the Venn diagram for that (conditional) MP about "raining":
------------------------------------------
| Universe |
| _________________________________ |
| | | |
| | W | |
| | ___________________________ | |
| | | | | |
| | | R | | |
| | | | | |
| | |__________________________| | |
| | | |
| |________________________________| |
| |
|________________________________________|
The set of cases of raining is included in the set of cases of streets being wet, since
the streets can be wet from a fireplug releasing water or for some other reason. (This Venn diagram
represents MP in general: ((P Þ C) & P)
Þ C, where P denotes precedent
statement and C denotes consequent statement. The "direction"
of the logic formula is reverse that in the Venn diagram. In the formula,
the direction goes from presedent to consequent; in the Venn diagram, the
precedent set (R) is inside the consequent set (W), that is, R Ì W. So, any point in the inner circle/rectangle (of R) is also in
the outside circle/rectangle (W). Dig?)
Here is the Venn diagram for the "equipotent" definition, with "E" denoting "equipotence" and
"S" denoting "same membership":
_________________________________________
| |
| |
| _________________________________ |
| | | |
| | E S | |
| | | |
| |________________________________| |
| |
|_______________________________________|
The set of "equipotent" cases is the set of "same membership". Dig?
Incidentally, one of the most common confusions is daily speech or in the literature is that
involving "implying" and "infering". Example: "Are you infering that I'm stupid?" Should be,
"Are you implying that I'm stupid?". The form "If it's raining, then the streets are wet" is
implying. But MP (Modus Ponens) is an inference. Dig? You can imply with a
single statement, but you need more than one statement to infer.
TAUTOLOGIC
I noted above that Logic cannot produce Truth, only preserve Truth inserted into a logical form.
The criterion of logic is validity. I said above that modus ponens (MP) is
the most useful argument in logic since it's enough to prove all of statement logic. But
validities have a "flip side", as we now see.
The word "logic" derives from the ancient Greek word "logos" for "language". (The term "logorrhea"
describes someone afflicted by "diarrhea of the mouth" -- say William Buckley or Joan Rivers.) And
that indicates both the strength and the limitation of Logic. The canonical form of logic
is "the tautology", meaning "sameness of language", since the prefix "tauto-" means "the same". As
such, the term is often used pejoratively to accuse the speaker with redundancy.
This means that every validity in Logic is equivalent to a tautology such as "If it is
raining, then it is raining". (Big hairy deal! as Garfield says.) So we need something more than
Logic to aid our decision-making -- aid which we find in subsequent files.
Another limitation of standard logic is bivalency: any statement has one of two
values, "True" or "False". We owe this to Chrisippus the Stoic. (It is incorrectly attributed
to Aristotle (384-322BC), so that this is often mislabeled "Aristotlean logic".) And bivalency allows a
nonconstructive method of proof known by the Latin label, "reductio ad absurdum" (reduction
to an absurdity) -- a. k.a. prood by contradiction. If you cannot prove an argument directly,
you assume its contradiction -- that it's not so -- then follow the consequences of this;
if this leads to another contradiction, you argue that you have proven your original assertion.
It is comparable to "double negative means positive"' or a two-pole light switch wherein two flips
return the switch to its original state, whether OFF or ON.
One person who argued against "reductio" proofs was the Dutch mathematician, L. E. J. Brouwer
(1881-1966). He questioned: If a criminal, in committing a crime succeeds in destroying the only
evidence that could convict him, must we declare him innocent? Brouwer said, "No!", asking for a
third judgment, "Not proven". Scottish jurisprudence has just such a verdict, and it was discussed
in the trial in Lockerbee, Scotland, about the people killed by a bomb on a plane. During the
discusssions about impeaching President Clinton, Pennsylvania Senator Spector suggested this form
of verdict. A 1950 film, "Madeleine", sometimes seen on TV, directed by the great director, David
Lean, ends with a "Not Proven" verdict. (Based on a true 19th century story, Madeleine is accused
of poisoning her faithless lover by arsenic in a cup of chocolate. The Crown can't prove she actually
put the arsenic in the drink, but the jury does not consider her innocent, so brought in a verdict
of "Not Proven", sending Madeleine away "under a cloud of suspicion".) Regarding nonconstructiveness
in mathematics, I published a fable.
Lawyers for gangsters have regularly taken advantage of our bivalent justice by forcing trials
of their clients before the presecution is ready. Then, when insufficient evidence is cited, the
jury finds the "accused" innocent, since it cannot find him guilty; and then he cannot be tried again
because of "double jeopardy". This (along with other quandaries of decision-making) has motivated
some logicians to introduce several forms of multi-valued logic into the literature.
Another limitation of Logic is that it obtains its bivalency by considering only the
declarative form of speech -- excluding the interrogative, subjunctive, petitive forms.
At another website, I describe a vectorlogic bypassing this limitation.
Still another crucial limitation of Logic (but little discussed) is that it is monotonic,
meaning that it can increase in value or remain the same but cannot decrease in value.
However, many have noted that this goes against history and experience. Information turns up that
"take away some of the certainty we had for an argument". So, it is desirable to have a quasi-logical
procedure which is nonmonotonic. And we shall discover this in another file.
In summary, Logic has its useful functions, so we incorrigibly do not change it. As with
mathematics, we turn to another model -- in subsequent files of this present website.