Philosophy 112
New Rules
We have three new connectives, and six new rules.
First, we have two new rules for conjunction (‘and’).
The first rule is Simplification (S), which allows us to infer either conjunct from a conjunction. Because you can infer either conjunct, the rule has two forms:
\({\mathord{\wedge}}\)
S
\({\mathord{\wedge}}\)
S
Simplification is obviously valid. If this is not immediately obvious to you (and there is no reason it should be) play around by plugging some sentences into the box and circle, until you are convinced that any values that make the premise true will also make the conclusion true.
The second rule is Adjunction (Adj), which allows us to infer the conjunction when we have both conjuncts separately. So,
\({\mathord{\wedge}}\) ADJ
Again, this is obviously valid, but again, you might wish to play around by plugging in some sentences, to get a better sense of why it is valid.
Next, we have two rules for disjunction (‘or’).
The first rule is Addition, which allows us to infer a disjunction from either disjunct. Again, this rule has two forms.
\({\mathord{\vee}}\) ADD
\({\mathord{\vee}}\) ADD
The second rule is kind of like MP and MT. We will call it Modus Tollendo Ponens (Latin for ‘the way of putting by taking’). It says that from a disjunction and the negation of one of the disjuncts, you can infer the other disjunct. Again, the rule has two forms:
\({\mathord{\vee}}\)
\({\mathord{\sim}}\)
MTP
\({\mathord{\vee}}\)
\({\mathord{\sim}}\)
MTP
Play around with these forms, to get a sense of why they are valid.
Finally, we have two rules for the biconditional (‘if and only if’). The two rules track the fact that the biconditional, is equivalent to \({\mathord{\leftrightarrow}}\) , is equivalent to the conjunction of both conditionals, ( \({\mathbin{\rightarrow}}\) ) \({\mathord{\wedge}}\) ( \({\mathbin{\rightarrow}}\) ).
The first rule is called Conditionals to Biconditional (CB), and says that, if you have both conditionals, you can infer the biconditional.
\({\mathbin{\rightarrow}}\)
\({\mathbin{\rightarrow}}\)
\({\mathord{\leftrightarrow}}\) CB
The second rule has two forms. It says that, from the biconditional, you can infer either conditional. We call it Biconditional to Conditional (BC):
\({\mathord{\leftrightarrow}}\)
\({\mathbin{\rightarrow}}\) BC
\({\mathord{\leftrightarrow}}\)
\({\mathbin{\rightarrow}}\) BC