# Is the conjunction of all necessary statements sufficient? What about the converse?

A necessary condition for consequent $q$ is a proposition $p$ such that:

$$\neg p \implies \neg q$$

let $P:= \{p_i: \neg p_i\implies \neg q\}$

What I want to know is if

$$\bigwedge_{p_i\in P} p_i \implies q$$

Here's my attempt at this problem:

Let $P$ be defined as above, which means that $P$ is the set of all necessary propositions for $q$.

Now, if we assume that $$\bigwedge_{p_i\in P} p_i \implies q$$

is false, is means that

$$\exists r\notin P: \neg r \implies \neg q$$

However, this is means that $r\in P$ since it is also a necessary condition. We have deduced a contradiction by denying the truth of the statement "the truth of all necessary conditions implies the truth of the consequent".

Conversely, a sufficient statement is defined as:

$$p \implies q$$

For this, I am interested in the statement:

$$\neg \bigvee_{p_i \in Q} p_i\implies \neg q$$

Where $Q:= \{p_i: p_i \implies q\}$

Along similar lines, it appears that denying the statement "if no sufficient conditions hold, then not q$requires that there be an additional statement whose truth implies q...but this is a contradiction, since we area assuming that we have already identified all sufficient conditions. Question If we assume that the universe is deterministic and that$q$is not trivially true or false (a priori), then are my arguments correct? ## 1 Answer It seems that you are considering "causality" and some sort of "natural law"; if so, propositional logic and the conditional ($\to$) are ot the "right tools" for your job. Simplifying the example, from the fact that$(p_1 \land p_2) \to q$is false, we can derive that:$p_1 \land p_2 \land \lnot q$is true. From this you conclude that there must be some "missing" condition$r$responsible for the fact that$q$has not "happened". In terms of logic, this is not necessary; if$q$is an "impossible" condition (a contradiction), you can "add" a false premise$t$whatever and what you get:$(p_1 \land p_2 \land t) \to q\$

will be true.

The truth-functional conditional is used in the analysis of the relation of logical consequence, but gives "unpleasant" results if we try to use it to analyse the causation relationship between events.

• +1: Thanks for pointing out the error. As someone with a scientific background, I think causal reasoning pervades my logic. I've read that the material conditional is problematic, and now I can see why! – user237392 May 22 '16 at 12:29
• @Bey - you can see Counterfactual Theories of Causation and Conditionals. – Mauro ALLEGRANZA May 22 '16 at 12:33