Numerical Arguments

The "Use Numbers" feature must be activated to evaluate numerical arguments using ReasonLines. Then when a premise schematic is entered a place for its numerical value appears bearing the traditional value of 0 or 1 on it by default. This value can then be changed to match the value of the premise by touch/holding it until a number pad is brought up.

If AutoSolve is on, the app will provide the correct conclusion if the premises entail one; else it will indicate that they do not entail one. If AutoSolve is not on, the user will first need to determine whether the premise forms entail a conclusion—i.e., to determine whether the extreme terms are connected by a green arrow path in a head-to-tail configuration on the schematics. If they are properly connected then the correct conclusion schematic should be put in place and its default numerical value changed to the one calculated by one of the two formulas below. If a wrong conclusion card is entered, "Incorrect conclusion" will appear below it, but "Incorrect number" will appear if the correct card is entered bearing the wrong numerical value.

Formula 1. If the premise schematics are all predominantly green (=if they are all universals) then the numerical value of the conclusion is the sum of the numerical values of the premises:

Premises

Conclusion

For example, both numerical arguments below, viz.,

are instances of this one general, valid form. The argument to the left is the traditional syllogism where both x and y are instantiated with 0. In the one to the right, x is instantiated with 123 and y with 135; but they could be instantiated with any other nonnegative numbers whatever.

Moreover, this rule of determining the numerical value of the conclusion holds for all sorites composed entirely of universal propositions, such as the following one:

Premises

Conclusion

Here if we let w=10, x=11, y=12, and z =13, the conclusion is "At least ∀-46 E are A."

Formula 2. If there is a predominantly red schematic for a premises (=if there is a particular premise), the numerical value for the conclusion is the value of the predominantly red schematic minus the sum of the values of the other premises. (Again, premises with more than one predominantly red schematic do not yield any conclusion.)

Premises

Conclusion

The argument to the left below is the traditional, valid syllogism where x is instantiated with 1 and y with 0; in the one to the right, x is instantiated with 1000 and y with 7; but infinitely many more instantiations are possible.

The numerical value of the conclusion for a sorites with a particular premise follows the same rule, viz., the exception for the conclusion equals the exception of the predominantly red schematic minus the sum of the exceptions of all the others.

Premises

Conclusion

Here if we let x=2, y=8, and z=1 the conclusion is "At most ∀-5 D are A".

Earlier it was pointed out that any category, A, could be displayed as having membership by entering "Some A are A" as a premise. Now that we are dealing with numerically flexible quantities we can enter the existence of any number of members for a category by the same method. For example, we may know that all but 5 of a class of Logic students are also members of a Math class and that all but 4 of these Logic students are also students of Art. But from this information we cannot deduce whether any math students are also art students, as the schematic below shows no head-to-tail green arrow path between the extreme terms.

Premises

Conclusion

However, if we learn there are at least 20 students in the Logic class then we can enter it as a premise on the schematic and draw the conclusion that at least eleven math students are also art students, as follows:

Premises

Conclusion

Also, often the membership of a category is given at the beginning of an argument, such as:

There are 20 Teenagers at a particular summer camp. At most 4 of them are Honors students and at least all but 3 of the nonHonor students are Athletes. Therefore, at least 13 of the camp's Teenagers are Athletes.

Premises

Conclusion