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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?

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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.

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  • $\begingroup$ +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! $\endgroup$ – user237392 May 22 '16 at 12:29
  • $\begingroup$ @Bey - you can see Counterfactual Theories of Causation and Conditionals. $\endgroup$ – Mauro ALLEGRANZA May 22 '16 at 12:33

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