# Logial Entailment vs. Material Conditional: binding free variables?

I think that now I DO understand basic logic manipulations. I DO understand why (camels have feathers) -> (Michigan has a lot of great lakes) is true.

Nevertheless, we can still write $\mathscr{A}$ for (camels have feathers) and $\mathscr{B}$ for (Michigan has a lot of great lakes), then why is it NOT true that $$\mathscr{A} \implies \mathscr{B}$$ Or maybe it is true. That's why I'm confused.

QUESTION: What does this have to do with "binding free variables" and "substitutions"? And how do we know when a logical sentence needs "bindings" or "substitutions"?

• That's one of the so-called paradoxes of material implication. – Hunan Rostomyan Nov 1 '13 at 7:11
• – The Chaz 2.0 Nov 1 '13 at 7:12
• Von Neumann famously said that in mathematics you don't understand things, you just get used to them. Although I don't think that's always good advice, this might be one of the times when it is. – MJD Nov 1 '13 at 7:14
• Your question is common and hence not stupid. Your title however... – fhyve Nov 1 '13 at 7:14
• possible duplicate of Assumed True until proven False. The Curious Case of the Vacuous Truth – MJD Nov 1 '13 at 7:16

## 4 Answers

Before turning to the symbols, think first about the difference between something being true, plain and simple, and something being logically true (true as a matter of logic).

It is plain true that I'm less than six foot tall. But that's not a matter of logic: it's just a contingent fact about how things have turned out. (A few growth supplements when I was small and things might have turned out differently!) Compare the proposition that I'm not both less than six foot tall and taller than six foot. That uninformative tautology is true as a matter of logic. However things go with the contingent facts it will remain the case that I can't be both less than and taller than six foot tall.

Similarly, it is true that if I press this switch the light will go on. But that's contingent, it just happens that the wiring goes like that, there isn't a power cut and so forth. It's not a matter of logic that light-switches work. Compare: it is logically true that if I press the light switch and turn on the coffee machine then I press the light switch. There is no logically possible way that the antecedent of that conditional can be true and the consequent false.

Similarly again here. Define the so-called material conditional $A \to C$ to be equivalent to $\neg A \lor C$. [And let's not tangle now with the question of quite how the material conditional relates to the "if ... then ..." of ordinary language, as this isn't the key thing that is being asked for here.] Then with $A$ for camels have feathers and $C$ for Michigan has a lot of great lakes, it is plain true that $\neg A \lor C$ since both conjuncts are true and $\lor$ is inclusive, and hence -- trivially, by definition -- $A \to C$ is true too. But $A \to C$ is not logically true. We could imagine a world I guess where camels evolve to have a feathery coat, and the topography of Michigan is different so it has a lot of little lakes instead.

In logic we are concerned with what follows from what; so we are going to be interested in cases when, if a given $A$ is true, then $C$ has to be true as well as a matter of logic. Let's use $A \Rightarrow C$ to express this strong relation between $A$ and $C$. Then, at least as a first approximation, we have $A \Rightarrow C$ just when it is logically true that either we don't have $A$ or we $C$ together with $A$, which is equivalent to

$A \Rightarrow C$ just in case it is logically true that $A \to C$.

Now, with the camels/lakes readings for $A$ and $C$, as we saw, although it is plain true $A \to C$ is not logically true. So $A \Rightarrow C$ comes out false.

Of course, we'll want to fancy up our definition of $A \Rightarrow C$ so as not to rely on the intuitive notion of logical truth, so we'll perhaps want to define it in terms of their being no interpretation/model on which $A$ comes out true and $C$ false, where we give a fancy account of interpretations. Or we might give a definition along the lines of, for every substitution for the non-logical vocabulary in $A$ and $C$, $A \to C$ remains true. But that's fine tuning. We've already said enough to see how $A \Rightarrow C$ and $A \to C$ [in one use of the notations] can peel apart.

An implication $P \Rightarrow Q$ is false only when the antecedent $P$ is true while the consequent $Q$ is false. In every other case the implication is true.

• This doesn't really answer the question I asked - but thanks anyway. – user86971 Nov 1 '13 at 7:30

Von Neumann made a good point, but consider what it is that logicians are giving us: one truth or two? If "I switched on the light" is true because of the way the world is, why is "If I switched on the light then the moon is round" logically true? It is logically true, and yet somehow hanging on the world, but then so is the simple claim "I switched on the light". Propositionally speaking, you don't get to have it both ways. If "I switched on the light" is conversational or narrative, then it is true in virtue of simply turning on the light. However, if that is taken as a proposition, then it is a tacit measure of affairs by an implicitly analytical list of conditions. When those conditions are met, then the calculation comes out true. That is to say that, epistemologically, ALL TRUTH IS TRUTH-FUNCTIONAL.

Finally, let me address the first question that started this thread. If $P$ then $Q$ means what it means truth-functionally granting that $P$ and $Q$ are variables for any proposition whatsoever. Your question mixes up a specific value with the general one. You have in effect given two propositions "pet names" with single symbols. I am Robert, but some call me Bob. You want to replace camels have feathers with another symbol. Go ahead. The calculation works the propositional value of the "real name". The stand-in symbol has no weight on its own. If you fail to provide a legend for such cases, then it would look like just a different set of variables than the tradition $P, Q, R$, etc.