Logic Tutorial .com - an interactive visual lesson in formal logic

Illustrating Formal Logic with
Exclusion Diagrams

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.)

Exclusion Diagram or Johnston Diagram (similar to Venn diagram) for Propositional Logic
Captions for logic diagrams


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 "truth tables". 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"). Some scholars 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.

  • NEXT: The sixteen things you can now say...

  • An interactive view of deduction with “AND”

  • An interactive view of deduction with “OR”

  • A discussion of meaning as exclusion

  • So our minds work by exclusion? Not so fast!

  • Relevant Buddhist philosophy of logic

  • Essays in Applied Logic

  • The Logic and Illogic of “Occam's Razor”
    or “How Occam Helped Wreck Our Environment”

  • A Modest Contribution to the Philosophy of Torture

  • “The Meaning of Life”, Logically Considered

  • Email logictutorial.com at (add seventy to the 103):
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  • Logic images and diagramming concepts copyright Russell Johnston and LogicTutorial.com 1987
    text 1987, 2001 and 2006

    Worlds where neither A or B is so Worlds where neither A or B is so Worlds where neither A or B is so Worlds where neither A or B is so Worlds where A is so but B is not so Worlds where A and B are both the case Worlds where A isn't but B is the case Worlds where neither A or B is the case