Seeing Deduction Happen by Exclusion

with Interactive Diagrams

from LogicTutorial .com


logic diagram one
Captions for logic diagrams

AND

logic diagram two
Captions for logic diagrams

means


logic diagram three
Captions for logic diagrams

(the deduction)



Click within the first two diagrams to create the third.



The sum of what is excluded by either statement A or B is excluded by the "AND" statement joining them. That is, the AND statement excludes whatever either A or B excludes; this summation of what has now been excluded from possibility is what may be deduced, the conclusion - and is the logical equivalent of the statement "A AND B".

If meaning is the exclusion of possibilities, we should be able to see this in the way in which statements imply other statements, and how deductions are made, and we can literally see this by playing with the diagrams above.

Here the first two statements diagrammed, taken together, imply the third - actually the first two imply many statements, any that exclude no areas not excluded by the first two statements. But the third statement/diagram shown is the most that is or can be implied by the sum of the first two statements. That is, the most you can conclude from just the first two statements. In other words, if the first two statements are true then the third must be, and it is also the logic equivalent of the first two together - that is, it says at least what the first two do, and at most what they say.

Toggle the small areas within either of the first two diagrams and you will change those statements and diagrams, but you wil also likely change the conclusion, which the computer automatically calculates simply by summing up all of the areas excluded by either of the first two statements/diagrams into a single diagram of exclusion - the third diagram. In other words, if an area is excluded by either of the first two statements, it is excluded from the third, the conclusion. Deduction is that simple. From this mechanical process, you can understand why Boole, who pioneered formal logic more than a century ago, thought of logical ands (which we are illustrating on this page) and ors as forms of addition and subtraction. The same principle can be seen at work, only a little less easily, in truth tables - not surprising, since our Johnston diagrams are logically equivalent to truth tables, and can be said to be just a handy or intuitive way of visualizing truth tables.

So, if the meaning of each of the first two statements is just what they exclude, what they each say isn't the case, then the full meaning of the two taken together is the whole sum of the areas now excluded from being the case, namely the statement which excludes every area excluded by either of the first two statements, and no other areas. (Since to exclude any extra areas, would be to say more than the first two statements do, since you'd be saying something - excluding some area - that neither of the first two statements do.)

If all possibilities, or all possible worlds, are excluded by one or other of the first two statements, then the result is that a contradiction is created - since there really is a world, this one, it must be wrong to exclude all possible worlds! (Or situations.) Likewise, notice that if one of the first two statements is the contradiction, the conclusion is also the contradiction, since nothing more can be said than to exclude every possibility, if meaning is exclusion. The contradiction says too much, but it also says as much as could possibly be said. It is the strongest statement (but always false.)

If you imagine the first two diagrams as being printed on transparent plastic, then if you picked the first one up and put it down on top of the second, you'd get the third diagram. (Perhaps one day I'll do some fancy Java or AJAX programming and shown this in action.) Deduction isn't any more mysterious than that, nor is meaning, remarkably. Meaning is just what we exclude. That is, meaning what we exclude from consideration - what we say isn't the case, the sum of what we have excluded as not possible.

And of course, no matter what statements we combine, there are still only 16 things we can possibly say about A and B, using logic this way.




  • NEXT: An interactive view of deduction, with "OR"
      

  • A discussion of meaning as exclusion.
      

  • So our minds work by exclusion? Nope!
      

  • Relevant Buddhist philosophy of logic.
      

  • Back to: the sixteen things we can say...
      

  • Back to the first page...
      



  • 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 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