More Derived Rules

More! As we have seen, derived rules are very handy. So it’s worth getting some more of them. Let’s start of with the Argument by Cases rule. It looks like this:

Argument by cases Or, in words, whenever you have X ∨ Y, X ⊃ Z, and Y ⊃ Z on a particular scope line, the Argument by Cases rule licenses us to write Z underneath them on that scope line. There’s an intuitive rationale for this rule (if either X or Y is true, and for both of them if they are true then Z is true, then Z must be true too). But we need more than an intuitive rationale – we need a proof.

Argument by cases

Argument by cases, second form We can now derive a second form of the Argument by Cases rule: X ∨ Y : X : Z : Y : Z Z AC Or in words, if you have X ∨ Y on a particular scope line, and you assume X in a subderivation of the outer scope line and derive Z, and then assume Y in a subderivation of that same outer scope line and derive Z as well, then you may conclude Z on the outer scope line.

Argument by cases, second form Again, there is a nice intuitive rationale for this rule, and you can see how it is connected to the first rule. If X ∨ Y is true, and Z follows from the assumption of both X and Y, then Z must be the case. But again, we need more than an intuitive rationale – we need a proof from the primitive rules. And we can do that quite easily, by using our first form of the Argument by Cases rule.

Argument by cases, second form

Argument by cases, second form That last line of the proof utilizes the first form of the Argument by Cases rule. But isn’t that a problem? Weren’t derived rules supposed to be derived from the primitive rules of inference?

Deriving rules from derived rules The answer is no – it’s not a problem! Because the proof of the first derived rule shows us how to turn any proof using that rule into a proof that uses only the primitive rules. So we know that we could, if we wanted to, write out the full proof of the second form of the rule using only primitive rules. So long as all our derived rules have proofs, deriving rules from derived rules is not a problem.

Argument by cases Ok so that’s argument by cases. It’s a particularly important rule, because it closely models a pattern of reasoning we very often make in real life. The pattern is roughly this: ‘Well, either A or B is true. If A is true, then C would follow. But C would also follow from B! And since at least one of A or B must be the case – it follows that C must be the case.’

More! But there are many more useful derived rules. Remember for all of these – they are not primitive rules, but rather handy shortcuts that abbreviate set patterns of employing those primitive rules.

Biconditional rules There are a couple of new introduction and elimination rules for the biconditional, that allow us to cut out the middle man of the conditional:

Disjunction elimination And a nice disjunction elimination rule that allows us to skip fiddling around with negations:

Denying the consequent (or ‘Modus Tolens’) This rule is particularly important in that it again represents a very common pattern of reasoning in everyday thought:

Reductio Ad Absurdum And likewise for the reductio rule – it models a common form of argument that runs like this: ‘If X were false, then Y would be. But Y is not be true! So X must be true.’

Rules from laws of logical equivalence Teller also introduces a series of rules that are intended as rough analogues of some of the laws of logical equivalence. These include De Morgan’s rules (DM): ~(X ∨ Y) ~X ∧ ~Y ~(X ∧ Y) ~X ∨ ~Y ~X ∧ ~Y ~(X ∨ Y) ~X ∨ ~Y ~(X ∧ Y) (We can express these rules in abbreviated form by just saying that ~(X ∨ Y) and ~X ∧ ~Y are mutually derivable, and likewise that ~X ∨ ~Y and ~(X ∧ Y) are mutually derivable.)

Contraposition rules X ⊃ Y and ~Y ⊃ ~X are mutually derivable. (CP) These are all variations on contraposition, with some adjustments for negations to save time.

Contraposition Rules I.e.: X ⊃ Y ~Y ⊃ ~X ~Y ⊃ ~X X ⊃ Y ~X ⊃ Y ~Y ⊃ X ~Y ⊃ X ~X ⊃ Y X ⊃ ~Y Y ⊃ ~X Y ⊃ ~X X ⊃ ~Y

Conditional Rules And finally some rules pertaining to the laws of the conditional. X ⊃ Y and ~X ∨ Y are mutually derivable. (C) ~(X ⊃ Y) and X ∧ ~Y are mutually derivable. (C)

Conditional Rules I.e.: X ⊃ Y ~X ∨ Y ~X ∨ Y X ⊃ Y

Derived rules We could go on, introducing more and more derived rules. But that’s enough to be getting on with – these are the most useful ones, and these are the only ones we’ll be using in the course…