Talk:Formal Logic/Sentential Logic/Informal Conventions

Association of connectives

 * We will let a series of the same binary connective associate on the right. For example, we can transform the official


 * $$(\mathrm{P^0_0} \land (\mathrm{Q^0_0} \land \mathrm{R^0_0}))\,\!$$


 * into the informal


 * $$\mathrm{P} \land \mathrm{Q} \land \mathrm{R}\ .\,\!$$


 * However, the best we can do with


 * $$((\mathrm{P^0_0} \land \mathrm{Q^0_0}) \land \mathrm{R^0})\,\!$$


 * is


 * $$(\mathrm{P} \land \mathrm{Q}) \land \mathrm{R}\ .\,\!$$

Is this correct? Why can't it be written as


 * $$\mathrm{P} \land \mathrm{Q} \land \mathrm{R}\,\!$$

? --193.196.13.2 17:20, 19 September 2006 (UTC)

Sorry to be slow to respond. The two formulae


 * (1)   $$(\mathrm{P} \land \mathrm{Q}) \land \mathrm{R}\,\!$$
 * (2)   $$\mathrm{P} \land (\mathrm{Q} \land \mathrm{R})\,\!$$

are equivalent (both are true if and only if all three conjuncts are true. However, our rules state that we associate otherwise ambiguous formulae such as


 * (3)   $$\mathrm{P} \land \mathrm{Q} \land \mathrm{R}\,\!$$

(where all the connectives are the same) on the right. Thus only (2) disambiguates (3) according to our rules. Suppose we write (1) as (3) and then disambiguate. The result is (2), thus making our abbreviation rules inconsistent.

We could have written the rules differently. We could have written the rule so as to associate on the left. Nothing much rides on our decision since the truth condidtions for (1) and (2) are the same. But if we are going to have some abbreviations, we need to pick a set of rules and stick to it. --JMRyan T E C 21:13, 2 October 2006 (UTC)