#rules of inference

[documentassion]

Mel Bochner (b 1940), ‘Rules of Inference’, 1974, aquatint, 56.5 x 78.9 cm.

[theautologicalmetaphor]

Mel Bochner, Rules of Inference. Charcoal and gouache on paper. 1973.

[jordan-jetson-blog]

Free Class: Classical Logic 101 (Part IVa: Rules of Negation)

Now we're ready for the rules of inference. There are several of them, so I will break them up into sub-categories. Rules of inververse apply to the following relational operators: negation (¬), disjunction (∨), conjunction (∧), implication (⊃), and equivalence (≡). We'll go in this order.

Rules for Inference - Negation - Reductio Ad Absurdium

For example, 'if it can be proved that the limit approaches five at f(0), and it can be proved that the limit approaches negative five at f(0), then we have proven that there isn't a limit at f(0)'.

Or to take an example that doesn't involve mathematics we'll reference the game of Clue, if it can be shown that a if the lead pipe is the murder weapon then Ms. Scarlet is the murderer, and it can also be show that if the lead pipe is the murder weapon, then Ms. Scarlet is the murderer not, then we can conclude that the lead pipe is not the murder weapon since Ms. Scarlet cannot both be the murderer and not the murderer.

This is known as a reductio ad absurdium. Essentially, the argument is deconstructed until its realized that if it were proven to be true, then it would also prove that contradictions exist, violating the law of non-contradiction, therefore the argument is proven false.

We're going to introduce a new symbol for proveability (⊢). So, 'p proves q' is written like this:

⊢ p q

And, the reductio ad absurdium is written like this

∵ ⊢ p q ∵ ⊢ p ¬q ∴ ¬p

Because, p proves q, and because also p proves not q, therefore not p. Essentially you show that the argument argues from the absurd therefore proving the argument to be false.

Rules for Inference - Negation - Noncontradiction

Essentially, what this rule states is that if a set of contradictory premises, if taken to be true, can imply anything. 'if up is down, and up is up, then there are 12 months in a year.' is true, and 'if up is down, and up is up, then there are 13 months in a year.' is also true, because in classical logic, if we allow contradictions, then anything could follow from that, which is why contradictions are not allowed. The statement 'contradictions do not exist' is a law of classical logic. If we contradictions did exist, we couldn't really say anything is false. This is known as an 'explosion'.

∵ ⊢ p ∵ ⊢ ¬p ∴ q ∵ ⊢ p ∵ ⊢ ¬p ∴ ¬q

'because it can be proved p, and it can also be proved not p, q' 'because it can be proved p, and it can also be proved not p, not q'

Rules for Inference - Negation - Double Negation Elimination

Lets introduce a coin that has two states, heads or tails. If it is not heads, it's tails, and visa versa. If it is not not heads (something gramatically frowned upon in English, but not in logic), then it can be proved that it is not tails, therefore it is heads. So, we can equivalently eliminate the double negation signs as they are trivial.

∵ ⊢ ¬¬p ∴ p

We can draw an equivalence here.

𠪪p p

Rules for Inference - Negation - Double Negation Introduction

Conversely, we can trivially introduce double negation signs. If the same coin is proven to be heads, then we can conclude that it is not tails, and also that since tails is not heads, it is not not heads.

∵ ⊢ p ∴ ¬¬p

These are also said to be equivalent,

≡ p ¬¬p

That's all for the rules of inference regarding negation. Next we'll discuss the rules of inference regarding disjunction.