Illustrating Formal Logic with
Take two sentences: "A" - "It's cloudy at 10 Downing Street" and "B" - "The Dali Lama
is on a mat." Now, both or either may be true, or both false. The logical combinations
of such sentences make up propositional logic, as illustrated by the diagram further
along on this web page.
It takes an act of imagination to understand just what the diagram below means.
The concept is that all possible situations or states of affairs (or "possible
worlds" - really meaning possible universes - if you like) have been crowded into
the rectangle, each point being one such way things could be imagined to be. The rectangle
contains different sections teeming with such possible situations. The area
inside the circle labeled "A" below contains all the possible worlds
(or if you prefer, situations or "states of affairs") in which the proposition
"A" is in fact true. Similarly the circle "B" encompasses all those states of
affairs/situations/possible worlds in which the statement "B" is accurate.
All in all, all possible states of affairs show up in one section or another
(and just one section) of the Exclusion Diagram.
Now, if you let your pointer hover for a moment over an area on the diagram,
without clicking it yet, a caption will pop up saying just what kinds of possible
worlds that particular section of the diagram contains.
Now place your pointer over an area and
click, and that area will toggle between white and black - between being possible
(white) or excluded from consideration/said not to be so (black). Immediately, the Exclusion
Diagram, and therefore its meaning given by the caption, will change. By clicking an area
to black, you are eliminating all the possible worlds it contains from
consideration - you are saying that the real world or the actual state of
affairs isn't one of those possibilities, isn't contained in that section of the diagram.
Keep clicking, to form different complex statements concerning our original statements
"A" and "B". Clicking repeatedly on an area toggles it back and forth from black to white.
Try it - click on the areas within the diagram.
(Then click again to turn black to white.)
Meaning is exclusion. As more situations (or if you like, possible worlds)
are excluded - i.e., blackened out or "removed from consideration" by clicking on
that area - you are saying progressively more.
You start with an all white table says nothing at all - it's a tautology, anything's
possible and you've been told nothing yet. Either it's cloudy or it isn't at
10 Downey Street, to use our example. When you click areas to black, by excluding
at least some possibilities from consideration, your
language becomes more specific and your assertions more meaningful. Some situations
or states of affairs or possibilities have been ruled out - you've said that the
real world isn't to be found in this or that section of the diagram. Of course,
one can go too far. Turn the whole diagram black, and you've contradicted yourself,
that is, you've said no possible ways things could be is the way they are. Just as
the tautology (pure white diagram) is always true because it says nothing, the all
black diagram is clearly false, always, because there is a way things are. It's cloudy
(even if just partly cloudy) at 10 Downey Street, or it isn't, according to logicians.
There's a rule about this in formal logic called the "Excluded Middle" - either
a proposition (sentence) is true, or false. No in-between. Whether ordinary English
works this same way is a moot point, as logicians want to construct a somewhat
artificially language, however simple, that they can fully understand. Notice too that
"or" here is the "inclusive or" meaning that one or the other or both
sentences "A" and "B" could be so.
In this view, language can never point out anything specifically, only
eliminate sets of possibilities ("possible worlds" for the modern philosopher or
logician) from our consideration. That is, language - and therefore logic - can
only say what isn't the case. And that no matter how many possibilities were
excluded by language, i.e., how specific our language, an infinite number would
still remain (a now well-known property of infinite sets.) If, for example,
we say that a friend of ours has red hair, someone listening to us knows that
our friend doesn't have black or light blonde colored hair, but not what precise
shade, of all the infinite shades of red that are possible, their hair is. Nor do
they know from what we've said how tall, or heavy, or witty our friend is.
The possibilities are still infinite.
"Isn't this an Euler Diagram?" or "Isn't this a Venn Diagram?"
Euler diagrams and Venn diagrams look identical to one another, and to
Rxclusion diagrams, since all illustrate inclusion and exclusion of one kind or
another, but each performs a different task, dividing up different kinds of things.
It's possible to think of Euler Diagrams as a special case of Venn Diagrams, and
perhaps the same can be said of Exclusion Diagrams, but what's being considered is
different in each case. Each fulfills it's own pedagogical purpose. Exclusion
Diagrams bear a much closer (logical) relationship to
Indeed, it can be said that Exclusion Diagrams are simply a more intuitive way of
presenting Truth Tables. The diagrams do this in a way that makes an ancient Buddhist philosopher's
point clearly (see below): that meaning, in human language and logic, is always
a matter of excluding
or eliminating possibilities, never a matter of directly presenting or pointing
singularly to what is the case, despite the Swiftian illusion we often have, or manage to
convince ourselves of, that language is so specific.
Buddhist anticipations of "meaning as exclusion".
It might be noted that long before truth tables, formal propositional logic, or the diagrams shown above
existed, Hindu and Buddhist philosophers were debating the nature of language.
Their debates over the nature of universals ("cow" and "red" would be
examples of universals - as opposed to particulars) predate Christian Europe's
Nominalist controversies by 800 years. The Buddhist philosopher and logician
Dignaaga (A.D. 480-540) proposed the
principle of exclusion ("Apoha" in Sanskrit) as an explanation of how universals work.
Quickly said, the theory is that a "cow" is recognized by excluding all the things that
are not cows, based on the rules set by a "cow"'s intrinsic characteristics.
(Arguably Nagarjuna's earlier assertion of the non-existence of intrinsic
characteristics makes even this view of Dignaaga's too concrete, too ontologically complex.)
Such a denial of the existence of abstract entities (to put it in terms of the
much later debate between Berkeley and Locke) avoids speculation about the
"essence" of cows, or humans; which is consistent with the central Buddhist concept of
anatta - the denial of an essence or soul to individual human beings that extends
beyond a mere collection of circumstance and causation. Put more traditionally:
"everything originates in dependence and thus lacks self-nature"
(a position referred to as "svabhaavas'uunya").
believe that this theory of universals has been blown up out of misunderstandings of comments by Dignaaga and
does not represent his own views. In any case, the principle of exclusion was not extended to
individual statements, or the nature of language in general.
Logic images and diagramming concepts © copyright Russell Johnston and LogicTutorial.com 1987
text ©1987, 2001 and 2006