More on restrictions in deductive reasoning

By way of reminding ourselves of the restrictions on deductive reasoning discussed informally the other week, we are told that once we’ve made an assumption (sort of like turning away from our opponent for a bit of shadowboxing just to find the punch that works best before we resume the fight), that assumption must not be generalized until it is discharged. To see why, we observe what happens in the following argument:

Let Ex - x is even, Dx - x is evenly divisible by 4, Universe of Discourse: all integers

1. (x)Ex ⊃(x)Dx
2. * Ex ......... Assumption
3. *(x)Ex ......... ERROR!
4. *(x)Dx
5. *Dx
6. Ex ⊃ Dx ......... Assumption discharged
7. (x)(Ex ⊃ Dx)

Our premise on line (1) says that ‘If every number is even, then every number is evenly divisible by 4.’ The sentence is true because the antecedent is false – not every number is even, of course. We need not worry about the consequent ‘every number is evenly divisible by 4,’ because whatever the consequent, if the antecedent is false, the sentence is always true. The conclusion, however, is false, because it says ‘Every even number is evenly divisible by 4.’ This is plainly not the case with 2; the number 2 is not evenly divisible by 4.

The error was made on line (3) where we generalized our assumption from line (2) ‘a number picked at random is even.’ There is no problem with making that kind of assumption, it is just that we are prohibited from generalizing it to ‘every number is even.’

Our second restriction says that if we introduce into our reasoning an individual specified by name after we have picked another individual at random, then the relation between the latter and the former cannot be reversed.

Let Lxy - x is larger than y, a - the integer 2; UD: all integers

1. (x)(∃y)Lxy
2. (∃y)Lxy
3. Lxa
4. (x)Lxa ......... ERROR!
5. (∃y)(x)Lxy

Our premise on line (1) says ‘Every number is larger than some number.’ This sentence is obviously true. We drop the universal quantifier to get something like ‘A number picked at random is larger than some number,’ – still true. Then, we drop the existential quantifier to get ‘A number picked at random is larger than 2.’ This sentence can very well be true. However, when we now restore the quantifiers (we restore them in reverse order or else we wouldn’t be getting very far with our reasoning), on line (4) we get ‘Every number is larger than 2.’ This is patently false.

Finally, when we generalize a variable of a particular kind, we must generalize all of them at once.

Let Dxy - x is evenly divisible by y; UD: all integers

1. (x)Dxx
2. Dxx
3. (x)Dxy ......... ERROR!
4. (y)(x)Dxy

Line (1) says ‘Every number is evenly divisible by itself.’ This is true. Line (2) says ‘A number picked at random is evenly divisible by itself.’ This is also true. Then, on line (3) we generalize only one variable to get ‘Every number is evenly divisible by a number picked at random,’ and we run into trouble.

The idea, as usual, is to avoid being led from true premises to a false conclusion.