The wikipedia article on logical tautologies, begins with the typical example of “A or not A” is given. Then in the section on “verifying tautologies” the following example is given:
((A & B) -> C) <=> (A -> (B -> C))
I get the “math” of this sufficient that I could have created the truth table that follows. But can anyone here construct a sample English sentence that exhibits this formulation?
What about the one in the section on “substitution”
((C or D) & (c->E)) or (!(C or D) or (!(C->E))
If we had ham, we could have ham and eggs, if we had some eggs.
“The siren will go off whenever the alarm is set and the door is opened, so if you set the alarm, then if the door is opened the siren will go off.” Something like that?
I love that expression … do you know where it came from? Also, I thought it was used to state an impossibility not a tautology (though it’s been many years since I sat for formal logic).
Chronos’s example is nice; if we are to make it more slavishly correspond to this, then we might say something like “Ham and eggs are sufficient for a full lunch precisely when having having ham is sufficient for having eggs as well to be sufficient for a full lunch.” (Where A is the proposition that one has ham, B is the proposition that they have eggs, and C is the proposition that they have a full lunch)
Thudlow Boink’s example is also nice, but he basically treats the “<=>” as just “->”. Easily remedied, of course.
But what is your goal here? Why is it that you want to see English formulations of these? Is it that you think they will help you better develop an intuition for why these should be formal tautologies, or that it will help you understand the use of the propositional calculus, or is it something else?
See, this is why formal logic makes my brain feel fuzzy.
From dim memory and the Wiki page:
From Chronos
Why isn’t this statement false if we have neither ham nor eggs (or pie, for that matter)?
That’s what I meant by having assumed it was spoken in exasperation in times of impossibility (not that it is impossible per se, just that it pleasantly fits a lot of situations where you want something but are absolutely missing critical components).
Because if we had neither ham nor eggs, then we couldn’t have ham and eggs. So it’s true.
It comes from my grandmother. Before that, though, I don’t know.
And the implication is that one doesn’t have ham nor eggs, but it remains a tautology, because, as the statement says, if we did have those foods, we’d have them. A conditional statement is not rendered false if the condition is false: In fact, it’s rendered true.
I guess it’s mainly to develop an intuitive sense. It’s really because I’m writing an essay (its details unimportant) and in one section I thought a discussion of tautologies may be illustrative. Probably ill-advised because, like Rhythmdvl, my head started going all swimmy and I quickly got out of my depth. My curiosity remains, though, about how one could “say” that expression.
Starting with (A & B) -> C, I tried
“If all animals that produce milk are called mammals and if a cow produces milk, then a cow is a mammal”
But I stopped there because to my ill-developed intuitive sense of logic, that appears to be a tautology all by itself. Whether or not milk producers are called mammals, and whether or not cows produce milk, that statement is true - because of the ifs.
Clearly I must have made an error in translation.
Show off! 
Can you remedy it for me?
Sure. But that’s because of the specific details of what you picked to place in for A, B, and C. You’ve put more structure into it than was just in (A & B) -> C; your statement isn’t true in virtue of that form alone.
The reason the whole statement ((A & B) -> C) <=> (A -> (B -> C)) is a tautology is because, no matter what specific propositions you insert for A, B, and C, the result will be true, by force of its logical form alone. (A & B) -> C alone doesn’t have that property; e.g., it’s not true that “(cows are mammals & England is in Europe) -> unicorns exist”.
Basically, a tautology is any statement such that you can tell it’s true, even you don’t know what any of the words in it mean except for the logical connectives. In that sense, “If all animals that produce milk are called mammals and if a cow produces milk, then a cow is a mammal” is a tautology, but not because it has the form (A & B) -> C. Rather, it’s because it also has the more specific form (X -> Y & Y -> Z) -> Z that it is a tautology.
Whoops, I fucked up that last line. I should have written (X -> Y & Y -> Z) -> (X -> Z).
But I don’t think all this symbol-shuffling is necessarily particularly helpful for your purposes. What exactly is it that you intend to discuss about tautologies? You might well not need to invoke this level or style of formalization at all; like I said, probably the best guide to what a tautology is is something which you would know to be true, even if you didn’t know what any of the words in it meant except the logical connectives. E.g., “Either John blorgs or he doesn’t blorg” is a tautology; I don’t need to know what blorging is to see this to be true.
Something with the form ((C or D) & (c->E)) or (!(C or D) or (!(C->E)) is never going to come out of someone’s mouth in ordinary language just like that, so you probably need pay it little attention (not to mention all the difficulties about when typical English usage does and does not correspond to naive formalization in this system; e.g., many of the instances of “if… then” in English aren’t actually instances of the material conditional “->” in this particular mathematical system).
But, anyway, as for Thudlow Boink’s example, you can remedy it by making it into something like “The siren will go off whenever the alarm is set and the door is opened if and only if whenever the alarm is set, if the door is opened the siren will go off”.
English “or” is also far from logical disjunction. Usually it is an exclusive or (“either or”).
Eh, I think “usually” takes it rather too far; there’s a pretty good rebuttal to that commonly voiced idea here.
Pullum holds that even those few cases where plain “or” carries a flavor of exclusive disjunction only acquire this through conversational implicature; I don’t know if I’d necessarily hold to that, per se, but I would say that the examples I can think of are usually complicated by the inclusion of modalities beyond those captured in the pure propositional calculus (e.g., the waiter who says “You can have coffee or tea” is not saying “Of coffee and tea, there is precisely one which you can have and one which you can’t have, though I’m not going to tell you which is which”. You could validly infer from the waiter’s statement “I can have coffee”, which you couldn’t do if this was merely one disjunct of the proposition he was expressing. This situation is better analyzed with the additive conjunction of linear logic rather than with any traditional disjunction). But, the fit between natural language and formalization in some particular logical calculus is never that tight, so we could argue about alternative analyses as well.
That’s why I said I must have lost something in translation - or as you more accurately pointed out, added something in translation. Thank you for (X -> Y & Y -> Z) -> (X -> Z)
It was a tangent to the main thrust of the rest the discussion. I’ve already decided not to pursue it but as I mentioned my curiosity remains.
The tangent began with the notion of being able to prove anything from a false premise. I recall reading (I think it was in the Bathroom Reader) a joke where a philosopher made that claim and was challenged. “Okay, smart guy, 2 + 2 = 5. Now prove I’m the King of England.” The philosopher, subtracted 3 from each side, showing that 1 = 2 then concluded, “Therefore you and the King are one.”
The back and forth between the logical/algebraic and English intrigued me. There is a minor plot point in Asimov’s Foundation where some government official made a grand speech that sounded as if it was full of promises and assurances. But the speech was run through a symbolic analyzer and its finding was the entirety of the speech said nothing.
Trying to imagine how such a speech could be constructed triggered thoughts of tautologies. I came across an interesting English language example illustrating tautologies: A journalist writes “All mainstream U.S. Senators agree that the House bill is unacceptable” then later in the article writes “Senator K disagrees, and has therefore shown himself to be outside of the mainstream,” revelaing the journalist’s definition of “mainstream” meant “agreement with the bill” and thus the first sentence is a tautology.
Really, though, this tangent has consumed time that should have been spent finishing the damned piece.
This is usually attributed to Bertrand Russell, when he was at Trinity, Cambridge. He made the claim that anything follows from a contradiction, to which McTaggart replied if “2+2=5, then prove I am the Pope”. The rest follows as you say.
Are you dissing Schroedinger’s cat?! :eek:
Maybe, maybe not.
As for the Russell witticism, of course, in general, proving that anything follows from a contradiction is less humorous; if C is a contradiction and P is a consequence you want to prove follows from it, you just note something like “C cannot possibly be true. Thus, it cannot possibly be the case that C is true while P is false. Thus, P holds whenever C does; i.e., given C, I can conclude P”.
The proof does not bother to establish any real connection between C and P; it just plays on the fact that the material conditional is defined to be true whenever its antecedent is false. In so-called “relevance logic”, this principle is abandoned, and not everything follows from a contradiction, but that it something of a “non-traditional” logic.