There are supposed to be paradoxes of material and strict implication. If there are, why is there no paradox of conjunction? And if there is no paradox of conjunction, why are there paradoxes of material and strict implication? With apologies to the friends and family of Dennis Wilson, the ill-starred original drummer of the Beach Boys, let's take this as our example:
1. Wilson got drunk, fell overboard, and drowned.
Translating (1) into the Propositonal Calculus (PC), we get
2. Wilson got drunk & Wilson fell overboard & Wilson drowned.
Now the meaning of the ampersand (or the dot or the inverted wedge in alternative notations) is exhausted by its truth table. This meaning can be summed up in two rules. A conjunction is true if and only if all of its conjuncts are true. A conjunction is false if and only if one or more of its conjuncts is false. That is all there is to it. The ampersand, after all, is a truth-functional connective which means that the truth-value of any compound proposition formed with its aid is a function (in the mathematical sense) of the TVs of its components and of nothing besides. You will recall from your college calculus classes that if f is a function and y = f(x), then for each x value there is a unique y value.
Now are the conjuncts of (2) related? Well, they are related in that they all have the same truth-value, namely True. But beyond this they are not related qua components of a truth-functional compound proposition. The 'conjuncts' -- note the inverted commas! -- of (1), however, are related beyond their having the same truth-value. For it is because Wilson got drunk that he fell overboard, and it is because he fell overboard that he drowned. So causal and temporal relations come into play in (1), relations that are not captured by (2).
Note also that the ampersand has the commutative property. But this is not so for the comma and the 'and' in (1). Tampering with the order of the clauses in (1) turns sense into nonsense:
3. Wilson drowned, fell overboard, and got drunk.
We should conclude that the ampersand abstracts from some of the properties of occurrences of the natural language 'and' and cognates. Despite this abstraction, (1) entails (2), which means that (2) does capture part of the meaning of (1), that part of the meaning relevant to the purposes of logic. But surely there is no 'paradox' here. Any two propositions can be conjoined, and the truth-value of the compound can be computed from the TVs of the components. It is the same with material implication: any two propositions can be connected with a horseshoe or an arrow and the TV of the result is uniquely determined by the TVs of the component propositions. Thus we get a curiosity such as
4. Snow is red --> Grass is green
which has the value True. This is paradoxical only if you insist on reading the arrow as if it captured all the meaning of the natural language 'if' or 'if...then___.' But there is no call for this insistence any more than there is call for reading the ampersand as if it captures the full meaning of 'and' and cognates in ordinary English.
What I am suggesting is that, just as there is no paradox of conjunction, there is no paradox of material implication either.
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