The truth-function conditional, also called the material conditional, is a binary boolean truth function whose values are as you described in the truth table; it is true for all inputs except when the antecedent (left hand side) is true and the consequent (right hand side) is false. The conditional is written with a variety of symbols, e.g., $P \to Q$, $P \supset Q$, and $P \Rightarrow Q$.
There is an inference rule called modus ponens which says that from a conditional $P \to Q$ and $P$, infer $Q$. This can be written in a number of ways, e.g.,
- $P$
- $P \to Q$
- $Q$ by modus ponens from 1 and 2.
or
$$ \begin{array}{c} P \quad P \to Q \\ \hline Q \end{array} $$
Because modus ponens is so important, and because in some axiomatic systems it is the only inference rule, it is sometimes called “the inference rule.” Most of the time, that usage should probably be avoided, because in practice (i.e., outside of those specific axiomatic systems), there are plenty of other inference rules, and modus ponens is just one among many.
Asserting that the conditional $P \to Q$ is true is simply to assert that either:
- a. $P$ is true and $Q$ is true; or
- b. $P$ is false and $Q$ is true; or
- c. $P$ is false and $Q$ is false.
Using the inference rule modus ponens lets us affirm that $Q$ is true, based on the prior assertion that $P \to Q$ is true and that $P$ is true.
Modus ponens is a sound inference rule because whenever all the premises are actually true, then the conclusion is also actually true. To see why this is the case, consider the premises to modus ponens. These are a conditional $P \to Q$ and $P$. If $P \to Q$ is true, then one of the three cases (a, b, c) described above must also be true. $P \to Q$ being true does not, by itself, ensure that $Q$ is true, because there is one case (c) in which $P \to Q$ is true, but $Q$ is false. However, with the additional requirement that $P$ is true, we are restricted to the case (a) in which $Q$ is also true. Thus, if the premises to modus ponens are true, then so is its conclusion.
Now, it's worth considering how this applies to the example that you gave. The example of “all men are mortal, Socrates is man, therefore Socrates is mortal”. It uses first-order reasoning, and is not actually a case of modus ponens, neither premise is a conditional. However, the proof does use _modus ponens_, in that it requires us to make the following inference.
- (Premise) If Socrates is a man then Socrates is mortal.
- (Premise) Socrates is a man.
- (Conclusion) Therefore, Socrates is mortal.
There are two important concepts to consider: soundness and validity. These mean slightly different things for inference rules and for arguments.
An inference rule is sound if whenever its premises are true, then its conclusion is true. As we saw earlier, modus ponens is a sound rule of inference. The term valid is not used concerning inference rules. We do not say that an inference rule is valid or invalid.
An argument is valid if each reasoning step is an application of a sound inference rule. This means that each sentence in the argument must be true if the earlier sentences that it is based on are true. It does not make the claim that those earlier sentence are true, but just that if they are true, then the current sentence is true. An argument is sound if its premises are, in fact, true. If an argument is both sound and valid, then its conclusion must be true.
So, both
- (Premise) If Socrates is a man then Socrates is mortal.
- (Premise) Socrates is a man.
- (Conclusion) Therefore, Socrates is mortal.
and
- (Premise) If Italy is a man then Italy is mortal.
- (Premise) Italy is a man.
- (Conclusion) Therefore, Italy is mortal.
are valid arguments, because they use only valid inference rules (namely, modus ponens). The first argument is sound because both of its premises are true. The second argument is unsound because one of its premises its second premise, “Italy is man,” is not true.
If the conclusion of an argument is not true, it means that the argument is either invalid or unsound. (Of course, it could also be both.)
Your distinction between "true" and "logical value 1" is not one that formal logic generally observes. Here "1" and "true" are synonyms for the same concept.
The meaning of the $\Rightarrow$ connective is what its truth table says it is, neither more nor less -- the truth table defines the connective (in classical logic). Fancy words such as "implication" or "if ... then" are just mnemonics to help you remember what the truth table is, and what the connective is good for -- but when there's a conflict between your intuitive understanding of those words and the truth table, the truth table wins over the words.
The important thing to realize is that $\Rightarrow$ is designed to be used together with a $\forall$. If you try to understand its naked truth table it doesn't seem very motivated -- certainly it can't express any notions of cause and effect, because the truth values of $p$ and $q$ just are what they are in any given world. As long as we're only looking at one possible state of the world, there's not much intuitive meaning in asking "what if $p$ held?" because that implies a wish to consider a world where the truth value of $p$ were different.
The device of standard formal logic that allows us to speak about different worlds is quantifiers. What we want to say is something like
In every possible world where I put in a coin, the machine will spit out a soda.
(though that is a little simplified -- we want to consider a "possible world" to be one where I made a different decision about my coins, not to be one where the machine had inexplicably stopped working even though it does work now. But let's sweep that problem aside for now).
This is the same as saying
In every possible world period, it is true that either I don't put in a coin, or I get a soda.
which logically becomes, using the truth table
For all worlds $x$, the proposition (In world $x$ I put in a coin) $\Rightarrow$ (In world $x$ I get a soda) is true.
Since there's a quantification going on, the truth value of the whole thing is not spoiled by the fact that there are some possible worlds with a broken machine where the $\Rightarrow$ evaluates to true. What interests us is just whether the $\Rightarrow$ evaluates to true every time or not every time. As long as we're in the "not every time" context, the machine is broken, and that conclusion is not affected by the "spurious" local instances of $\Rightarrow$ evaluating to true in particular worlds.
The construction that models (more or less) our intuition about cause and effect (or "if ... then") is not really $\Rightarrow$, but the combination of $\forall\cdots\Rightarrow$.
Unfortunately in the usual style of mathematical prose it is often considered acceptable to leave the quantification implicit, but logically it is there nevertheless. (And to add insult to injury, many systems of formal logic will implicitly treat formulas with free variables as universally quantified too, so even there you get to be sloppy and not call attention to the fact that there's quantification going on.)
Note also that this is the case even in propositional logic where there are no explicit quantifiers at all. To claim that $P\to Q$ is logically valid is to say that in all valuations where $P$ is true, $Q$ will also be true -- there's a quantification built into the meta-logical concept of "logically valid".
Best Answer
If the question was instead phrased as “If you were Sally, is your favorite color blue?” then it's natural to interpret it as “Is Sally's favorite color blue?”, whose answer is No.
Technically, it is unclear whether “your” is pointing back at Bob or still at Sally (I discussed such ambiguity of variable recycling in this answer).
Let's convert the given question into a statement:
“If Bob is Sally, then their favorite color is blue.”
With the implicit premise that Bob is Sally, regardless of whether “their” refers to Bob or Sally, the above (implication) statement is vacuously true, and the argument valid, since its associated conditional $$P\to\big(\lnot P\to Q\big)$$ is a tautology.