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Defining the conditional statement "$\phi \implies \psi$" so as to capture what is meant by implication requires that we ignore the issue of causation.

From Keith Devlin's Introduction to Mathematical Thinking:

$\quad$The truth or falsity of a conditional will be defined entirely in terms of the truth or falsity of the antecedent and the consequent. That is to say, whether or not the conditional expression $\phi \implies \psi$ is true will depend entirely upon the truth or falsity of $\phi$ and $\psi$, taking no account of whether or not there is any meaningful connection between $\phi$ and $\psi$.
$\quad$The reason this approach turns out to be a useful one, is that in all cases where there is a meaningful and genuine implication $\phi$ implies $\psi$, the conditional $\phi \implies \psi$ does accord with that implication.
...
$$ \begin{array}{ccc} \phi & \psi & \phi \implies \psi \\ \hline T & T & T \\ T & F & F \\ F & T & ? \\ F & F & ? \\ \end{array} $$ ...
$\quad$Once we get used to ignoring all questions of causality, the truth-values of the conditional seem straightforward enough when the antecedent is true...[b]ut what about when the antecedent is false?

Intuitively, we can see that $\phi$ implies $\psi$ means that it does not happen that $\phi$ is true and $\psi$ is false, as this captures the hypothetical aspect of implication, since we would like to consider an implication as true when the consequent does follow from the antecedent, even if the antecedent turns out to be false. Consider the following conditionals in each of which the antecedent is false:

$\quad$If it is true that polygamy is legal in New York, then marriage laws in New York are not identical$\quad$with those in Nevada. (Here the consequent is true).

$\quad$If it is true that polygamy is legal in New York, then a man living in New York can legally have $\quad$three wives. (Here the consequent is false).

Few would deny the truth of the preceding conditionals. From Introduction to Logic by P. Suppes:

Although we have already admitted that the notion of connection or dependence being appealed to here is too vague to be a formal concept of logic, in choosing examples which will force upon us, within our truth-functional framework, a truth value for implications with false antecedents it is reasonable to pick an example...for which our intuitive feeling of dependence is strong rather than an example...for which it is weak.

And the reason why this is a reasonable thing to do is because there is a hypothetical aspect of implication that we would like to capture in our defining the conditional. Thus, we can agree that the truth table for $\phi \implies \psi$ and $\lnot (\phi \land \lnot \psi )$ should have the same truth-value for each particular combination of truth-values for $\phi$ and $\psi$. $$ \begin{array}{cccc} \phi & \psi & \lnot (\phi \land \lnot \psi) & \phi \implies \psi \\ \hline T & T & T & T \\ T & F & F & F \\ F & T & T & T \\ F & F & T & T \\ \end{array} $$

First question: Is this convincing and appropriate to use as an explanation for the definition of the conditional? Also please provide your own approach.

Again from P. Suppes:

The truth functional demand...has no undesirable effects, since conditional sentences whose antecedents and consequents are unrelated and whose antecedents are false play no serious role in systematic arguments.

Even though this is the case, I feel that it is important for 1) the conditional to be defined in all cases and 2) to provide an explanation for this definition. However, it is often said that the assignment of truth values for the case when $\phi$ is false is arbitrary, but that this produces a useful logic. This seems to imply that the conditional could have a different truth table and that we need not provide an explanation that appeals to our intuition of how implies should work. Second question: Is this true?

Response to Derek Elkins:
At this point in my self education (I am looking at Modern Introductory Analysis by Dolciani, a precalculus text) I am looking for a solid explanation about implication and how we come to define the conditional. Most of your answer contains material that is currently out of my depth, and I am seeking an answer that will put to rest my doubts about my understanding of implication in propositional logic. In fact, I'm only even aware of first-order logic and propositional logic not as theories, but more as a way to make communicating about mathematics precise. Given your extensive knowledge and my lack thereof, and bearing in mind that I am self-studying mathematics at the precalculus level, can you provide me with an explanation that you found satisfying when you were at this level? Also, can you please share with me online resources, or books, that helped you in your education? Thank you.

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This seems to imply that the conditional could have a different truth table and that we need not provide an explanation that appeals to our intuition of how implies should work. Second question: Is this true?

The first issue is that treating $\Rightarrow$ as a truth function already removes much of our intuition about implications, because now the truth of the implication can only depend on the truth or falsity of the sentences involved, not on their actual meaning.

In any case, if we accept that $T \Rightarrow T$ is true and $T \Rightarrow F$ is false, there are four possible truth tables for $\Rightarrow$, which I will list with ugly formatting.

$$ 1.\quad \begin{matrix} & T & F \\ T & T & F \\ F & T & T \end{matrix}\\ 2. \quad \begin{matrix} & T & F \\ T & T & F \\ F & T & F \end{matrix} \\ 3. \quad \begin{matrix} & T & F \\ T & T & F \\ F & F & T \end{matrix} \\ 4. \quad \begin{matrix} & T & F \\ T & T & F \\ F & F & F \end{matrix} $$

  • Option 1 is the usual definition of $P \Rightarrow Q$
  • Option 2 would make $P \Rightarrow Q$ be the same as just $Q$
  • Option 3 would make $P \Rightarrow Q$ be the same as $P \Leftrightarrow Q$
  • Option 4 would make $P \Rightarrow Q$ be the same as $(P \text{ and } Q)$.

As you can see, there are already names for the other three options, and none of them acts like our intuition says that implication should act. The truth table we use may not seem ideal, but it turns out to be better than the alternatives.

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  • $\begingroup$ Although not explicitly stated in my post, I too, agree that the truth table is better than the alternatives. In fact, I think it captures a whole lot about genuine implication. However, I am interested in being confident in my explanation of why we should consider a conditional statement as true whenever the antecedent is false. What was the thinking behind this? Was it because it captures the hypothetical aspect of implication, or because it is not the case that the antecedent is true and the antecedent false? $\endgroup$ – user185744 Feb 19 '16 at 4:05
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A definition doesn't have to be convincing or appropriate; you can define things however you want. Of course, it is better to have a convincing argument on why a definition is appropriate and corresponds to what is being modeled. You can definitely have good and bad definitions. If we did change the result for $\phi$ false, you'd get logical equivalence.

If you define your logical connectives in terms of truth tables, then you don't have the choice of not defining implication for all inputs. However, truth tables are a semantics for propositional logic and just because a proposition is satisfiable doesn't mean that it is (syntactically) entailed. Of course, for the standard rules of propositional logic we have soundness and completeness theorems that state exactly that these two notions coincide. If we change the rules of logic, though, we can formulate a logic where $\phi \implies \psi$ is not provable (i.e. not syntactically entailed) when $\phi$ is false, for example relevance logic. This is roughly what "not defining implication for false antecedents" would mean at the syntactic level.

To directly answer your first question: within the context of standard propositional logic the answer is unambiguously "yes". Soundness requires that two logically equivalent formulas have the same truth table. However, this just pushes the problem back to "should those two formulas be logically equivalent?" To which many people, including me, say "no, not in general". This is (one of) the difference(s) between classical and non-classical (and in particular constructive) logic. Ironically, that logical equivalence that you present as a possible justification for $\phi \implies \psi$ is true when $\phi$ is false and $\psi$ is true, doesn't hold in (non-classical) constructive logics, but the statement itself does hold in most constructive logics (particularly in ones that are not substructural [note that relevance logic is a non-classical, substructural logic.]) Since classical logic is just a special case of a constructive logic, this means that there is a better, more fundamental reason why implication should behave the way it does. The BHK interpretation provides one approach to this. If we think of propositions as true when we have a proof object for them, then logical connectives like implication are operations on these proof objects. In particular, in the BHK interpretation implication is interpreted as a function that takes proof objects of $\phi$ and produces proof objects of $\psi$. With this interpretation, the fact that implication can be true when its antecedent is false is merely the fact that functions can ignore some of their arguments. Indeed the analog of the BHK interpretation for relevance logic is exactly a lambda calculus where functions are not allowed to ignore their arguments. The Curry-Howard correspondence, by grounding logic in computation, provides, for me, the real explanation and justification for logical notions. Admittedly, this is a fairly radical view.

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  • $\begingroup$ Thank you for your answer. Please see my response above. $\endgroup$ – user185744 Feb 18 '16 at 19:21
  • $\begingroup$ You haven't stated your overall goal so what answer is most appropriate is not clear. For example, if your goal is merely to verify your understanding of standard propositional logic, then if you understand truth tables, you understand everything. If learning about formal logic is your goal, then much can be said but it requires a broader perspective than propositional logic. If you are interested in mathematical reasoning, then you might be taking an approach akin to learning a natural language by reading dictionaries and grammar texts when you could be having conversations. $\endgroup$ – Derek Elkins Feb 19 '16 at 0:43
  • $\begingroup$ Well, specifically, how do you think of the conditional when the antecedent is false? Do you consider it true because it's reasonable to consider conditionals in which there is some connection and the consequent really does follow from the antecedent. Or, do you consider it true because it cannot be falsified? (i.e., it is not the case that $\phi$ is true but $\psi$ is false.) These two options are very different. Or is this something that I shouldn't even be concerned with? $\endgroup$ – user185744 Feb 19 '16 at 2:13
  • $\begingroup$ As Carl says, by choosing a "truth-functional framework" as the references you cite do, they've already eliminated much room for nuance. The question is then just a matter of choosing names for Boolean functions. The way I, personally, think of it is essentially the BHK interpretation I gave. Another perspective is the semantics of first-order logic, where a proposition is represented by a set of things for which it holds. $P\Rightarrow Q$ means $P\subseteq Q$ and False implies True is just the fact that the empty set is the subset of any set. $\endgroup$ – Derek Elkins Feb 20 '16 at 5:53
  • $\begingroup$ The principle of explosion is a purely logical reason for it, albeit it is not much more than a statement of it. It can be shown, for a very broad class of "logics", that if implication behaves like "normal", then the principle of explosion holds. (For the technically minded, it corresponds to $A^0\cong 1$ which holds in any monoidally closed category with an initial object.) Paraconsistent logics, including relevance logic, don't have the principle of explosion. In many cases, the principle of explosion is the definition of falsity. $\endgroup$ – Derek Elkins Feb 20 '16 at 6:17

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