# Fun With Conditionals

In logic, a conditional is an if-then statement. “If it rains, then I will go to the movies,” that’s a conditional. The question is, how should we assign a truth value to such a statement? This is a question of some importance to mathematicians, since every theorem is ultimately an if-then statement.

In some situations it seems easy enough. If it actually rains and I do go to the movies, then we would say the statement is true. If it rains and I don’t go to the movies then the statement was false. Simple! But what if it doesn’t rain? You might be inclined to say that we just should not assign a truth value in this case, since the statement was never properly tested. Sadly, classical logic does not permit this option. Every proposition must be either true or false. No exceptions!

Actually, it gets worse. Classical logic, by which I mean the sort of logic we teach to beginning math majors in discrete mathematics classes, treats conditionals as “truth-functional.” That’s a fancy way of saying that the truth value of the full if-then statement should depend solely on the truth values of the individual parts. This means that some convention has to be adopted for assigning a truth value to the if-then statement in each of the four possible cases. This includes those cases where the first part is false.

The standard convention is that a conditional statement in which the first part is false is automatically true, regardless of whether the second part is true or false. Put differently, the only way for a conditional to be false is if the first part is true and the second part is false. We figure that if the first part of an if-then is false then I didn’t actually lie to you.

Now, this has some weird consequences. It implies that the statement, “If Santa Claus exists then the Moon is made of green cheese,” is true. That’s weird, but whatever. This is the kind of thing you just accept when you’re doing logic.

But there are other problems. Quite a few of them, actually. Here in the real world, as opposed to on planet abstraction, both parts of an if-then could be true while the whole statement is false. Keeping in mind that I am writing this in Virginia, how about the statement, “If I am not in France then I am not in Spain.” Both parts of the conditional are true, but it would be absurd to say the whole statement is true.

Here’s another amusing consequence of the standard convention: All historical counterfactuals have to be considered true. For example, “If Neil Armstrong had not walked on the moon, then someone else would have.” Making such a statement presupposes that Neil Armstrong really did walk on the moon. If we adopt the standard convention, then we would have to declare the if-then to be true since its antecedent is false. Of course, that would apply with equal force to the statement, “If Neil Armstrong had not walked on the moon, then no one else would have.” This is all very strange.

Here’s another problem. In classical logic we have the “principle of strengthening the antecedent.” In other words, if you grant the statement “If P then R,” then you must also grant the statement “If (P and Q) then R.” Since (P and Q) can only be true if P is true all by itself, and since P implies R, we must grant that this enhanced conditional is true. But that’s not how it works in the real world, is it? “If it rains, then I will go to the movies,” therefore, “If it rains and I break my leg falling down the stairs, then I will go to the movies.” Or how about, “If I put sugar in my tea it will taste good, so if I put sugar and diesel oil in my tea it will taste good.” That doesn’t seem quite right.

Treating conditionals as truth-functional also pays no attention to whether the antecedent has any relevance to the conclusion. “If 2+2=4 then Paris is in France.” Are you really comfortable saying that’s true?

This is all good fun. The question is what to do about it. Mathematicians take the easy way out. We don’t worry about it. The classical account works perfectly well when applied to the sorts of conditionals we care about. Good enough! Philosophers, though, are hunting bigger game, and I’m sure you will be shocked to learn that they have produced a vast and dense literature trying to figure out what’s going on. Maybe we need to make a distinction between indicative conditionals and subjunctive conditionals. Maybe there’s a distinction between truth and assertibility, so that a statement could be true in some technical logical sense but still be unassertible given various conversational conventions. Maybe conditional probability can come to our aid, so that the truth of “If P then Q,” has to do with the conditional probability of Q given P. Maybe modal logic is the way to go, so that we need to refer to possible worlds to get a grasp on what’s going on.

Good stuff! But that’s all definitely to be saved for a different post…

1. #1 Frank Wappler
http://spherical.friction-field-and-moment-of-inertia-less.bull
July 11, 2013

jrosenhouse wrote (July 11, 2013):
> “If it rains, then I will go to the movies,” therefore, “If it rains and I break my leg falling down the stairs, then I will go to the movies.”

Good stuff!
Reminds me on having to make the distinction
“If I put sugar in my tea then it will taste good, except if I also put diesel in my tea.” vs.
“If I put sugar in my tea then it will taste different, even if I also put diesel in my tea.”

p.s.
> The question is what to do about it.

Honey-mustard! Quick!! …

2. #2 Neil Rickert
July 11, 2013

“If I am not in France then I am not in Spain.” Both parts of the conditional are true, but it would be absurd to say the whole statement is true.

It is this kind of speech that persuaded me that natural language expression is not logic and should not be confused with logic.

3. #3 TheDawgLives
http://www.suckitdown.org
July 11, 2013

When you assert “If I put sugar in my tea it will taste good” in natural language, does P not include an implied “and nothing else”?
How do you handle conflicting conditionals of:
“If I put sugar in my tea it will taste good”
and
“If I put diesel oil in my tea it will not taste good”
Both of those are true, so what happens when you combine them?

4. #4 Eric Lund
July 11, 2013

@Neil: But saying that natural language is not logic doesn’t help you. An equivalent way of stating that proposition is, “If I am not an element of set A then I am not an element of set B.” (Where in this case sets A and B consist of people located in France and Spain, respectively, at the time the statement was made.) According to the rules of sets, this should only be true when B is a subset of A (e.g., it would be OK if we replaced “France” with “Europe” in the original proposition). But under the rules of formal mathematical logic, the only requirement for this proposition to be true is that I cannot be in set B unless I am also in set A. I’m sure that people who do this kind of thing for a living have come up with resolutions to issues like this, but I don’t know what they are.

5. #5 Truthspew
Providence, RI
July 11, 2013

Then of course there are the case constructs. Multiple possibilities to act on a piece of data.

6. #6 Jonathan Lubin
http://www.math.brown.edu/~lubinj/
July 11, 2013

“If Santa Claus exists, then I’m monkey’s uncle”, and “If Marc Rich was honest, I’ll eat my hat.”

7. #7 Richard Wein
July 11, 2013

It seems to me that when you adopt the “standard convention” you stop speaking English and start speaking Logicese. Unfortunately English and Logicese use the same words, but with subtly different meanings, so they’re easily confused. When our English-trained brains read a sentence of Logicese, we can’t help interpreting it as an English sentence, i.e. misinterpreting it. And we shouldn’t expect a sentence of Logicese to have the same truth value as the identical-looking sentence of English, since they have different meanings.

8. #8 Verbose Stoic
July 11, 2013

Mathematicians take the easy way out. We don’t worry about it. The classical account works perfectly well when applied to the sorts of conditionals we care about. Good enough! Philosophers, though, are hunting bigger game, and I’m sure you will be shocked to learn that they have produced a vast and dense literature trying to figure out what’s going on.

Jason, can you PLEASE start limiting these statements to “Philosophers of logic”? Again, philosophers in other areas don’t really care. For example, the sort of condition that most other philosophers are looking at would be something like this:

If Santa Claus exists, then the Moon is made of green cheese.
Santa Claus exists.

Therefore, the Moon is made of green cheese.

This logical argument is valid, meaning that if all of the premises are true then the conclusion is also true (limited only to the premises here). But it is clearly not sound, and so no one needs to care about the fact that the first premise is true, as the second one is clearly false.

Note that for the first premise, what most philosophers would do on reading that is the equivalent of trying to build an argument like I just outlined showing that it is indeed necessarily the case that if the first proposition is true, that therefore the second must also be true, and so not just by assigning the truth values to it. If THAT turns out to be false, then it seems to that that conditional is false regardless of the truth values of each component, because the relation doesn’t hold.

So, yes, this is very interesting for philosophers working in logic, and particularly those working in symbolic logic. All philosophers — and computer scientists, for that matter — take and care about symbolic logic to some degree, not in general not enough to care about these sorts of examples other than as interestingly odd outcomes, like you do here.

The reason, BTW, that I’m bringing this up again is that phrasing it as you do fosters the strong attitude about philosophy being impractical and wrapped up in such abstract and esoteric problems, and your direct comparison to mathematicians just dropping it and getting on with their lives seems to be almost a direct comment of that sort. But this is a false impression of philosophy and I think contributes to the idea that philosophy is useless and produces nothing worth noting, which philosophers have to increasingly struggle with.

9. #9 Verbose Stoic
July 11, 2013

Let me highlight what I mean above with the premises.

I take a conditional statement, which is of this form:

“If P then Q”.

Now, I need to assign it a truth value. One way to do this is to do exactly what you’re doing here, and assign truth values to each component. And doing this purely symbolically, that’s all you can do: there is no content to evaluate for a specific truth value, and so it’s about as interesting as doing algebra with equations of the form: X squared + 2X +1 = 0 (solve for X). And in that, we come up with some rules of implications for conditionals, which are all fine and dandy, and potentially lead to the sorts of problems you cite here.

Except that there’s another way to assign a truth value to that statement: assign it to the conditional as a whole, regardless of the truth values of the components. Which means that I can assign a truth value, in a sense, to the “implies” operator as a whole: if the statement does not meet the conditions of the implies operator, then the whole statement, I argue, is false.

So, let’s take the oddest one:

“If 2+2=4 then Paris is in France”.

Now, before even EVALUATING the components, I have to ask “Does the implication apply to those statements?”. And it doesn’t; there is no reason why 2+2 = 4 means that Paris is in France. Therefore, the implication does not apply, and so the conditional is false … regardless of the truth values of the components.

As I pointed out above, this is what philosophers actually do when they try to evaluate arguments. The problem you cite here is only a problem when you’re doing symbolic logic, and can’t ask if the conditional/implication actually applies. But that means that since symbolic logic can’t help itself to the solution I’ve raised here, it either has to find a way to do that or else admit that it can’t cover all of the things that we need to do in evaluating an argument. Fortunately, most people who care about symbolic logic already admit that; symbolic logic doesn’t care about the details of the propositions and so can easily derive some very ridiculous statements, but the job of others is to go about and show that it is wrong.

10. #10 couchloc
July 11, 2013

I’ll make a comment about this post which I wasn’t sure I was going to make. But since VS has raised the issue maybe it’s appropriate at this point. I’m sort of waiting to see where the next post in this series goes since I’m not sure what moral Jason intends to draw from this helpful discussion, and it almost appears that he likes the work by philosophers coming out of studies on conditional (at the end, he says, “Good Stuff!”). So I’m not sure he’s intending to impugn philosophers who work on these sorts of issues in logic, which is what VS is potentially worried about.

One thing I will note in case this comes up later, is that there’s one part which is misleading in what is said. When you say in the post that “The classical account works perfectly well when applied to the sorts of conditionals we care about. Good enough! Philosophers, though, are hunting bigger game…” this is not quite right. What it should say is that the classical account works well when applied to “some of the sorts of conditionals we care about.” This is because there are conditionals in ordinary language that we often make and are important, but that don’t fit into the classical system. Take the sentence:

If the player swung the bat well, the ball went far.

This claim is naturally read as expressing a causal relation between the antecedent and consequent: the swinging of the bat caused the ball to go far. Causal claims occur in ordinary discourse and science and come up all over the place, and certainly are something that we care about. But they are not anything treated well in the classical system and explain some of the interest in other systems. Again, I’m not sure what moral you intend to draw from work on conditionals but it is worth keeping this point in mind.

11. #11 Verbose Stoic
July 11, 2013

couchloc,

I want to clarify that I don’t think that Jason is disparaging philosophy here personally, but he does seem to be portioning it off into these sorts of abstract considerations as if this is a major problem for philosophy in general and, therefore, that this is what philosophy is mainly concerned with … while most philosophy doesn’t care about it any more than most mathematicians do.

12. #12 Peter
July 12, 2013

“Implication” in symbolic logic means something different and more specific than if-then does in normal English. For one thing, symbolic logic doesn’t encode causality, and isn’t meant to, but normal English if-then often does. It doesn’t strike me that there’s any deep puzzle here, but it is important that students get that when they’re learning discrete math, etc.

If you want to do a better job of expressing a normal English if…then, you might want to try predicate logic. To take Eric Lund’s example:

A is a subset of B can be written:
A(x) = x is in A
B(x) = x is in B
(For all)x (A(x) -> B(x))

However, even if A is not a subset of B, it may still make sense that

~B(c) -> ~A(c)

for some reason specific to this case. But that’s not weird when you express it in predicate logic:

~[(For all)x (A(x) -> B(x))] AND (~B(c) -> ~A(c))

I.e., it isn’t really weird that just because the inference doesn’t always work, it does work in some cases.

Also, note that the standard convention for assigning truth values to implication isn’t arbitrary. The rule for transforming between IFs and ORs is *useful*.

13. #13 Jason Rosenhouse
July 12, 2013

Richard —

It seems to me that when you adopt the “standard convention” you stop speaking English and start speaking Logicese.

But Logicese is meant to be derived from English. For example, when someone speaking Logicese says that an and statement is only true if all of its conjuncts are true, he says that because that’s how the word and is used in everyday language. The problem is that conditionals resist any such easy treatment, at least in general.

Verbose Stoic and couchloc —

I wasn’t trying to disparage philosophers, but I was rolling my eyes a bit. My opening post was based heavily on Chapter Three of Stephen Read’s book Thinking About Logic As I alluded to in the post it’s a very dense and technical chapter. To me it seems like an awful lot of hard work for very little reward. I doubt if anyone really understands conditionals any better for invoking modal logic or conditional probability. Read talks frequently about “giving an account” of conditionals, but why should there be just a single account that captures everything there is to capture?

This is how I often feel about philosophy. I find it very interesting, which is why I keep reading work written by philosophers, but often their whole project seems quixotic to me. Of what value are endless discussions trying to give precise definitions of knowledge, or of truth, for example? No one has trouble applying these notions in everyday life, and I don’t think the difficult arguments summoned forth in the professional literature really clarifies much about them. Why is it even desirable to have sharp definitions for these ideas?

14. #14 MNb
July 12, 2013

“To me it seems like an awful lot of hard work for very little reward.a”
That’s why I have disliked formal logic since the first time I was confronted with it. About the only thing I remember is the distinction between if-then conditions and the if-and-only-if distinction.

15. #15 Verbose Stoic
July 12, 2013

Jason,

Let me start by saying that I’ve decided to order Read’s book and read it myself. When I do, I’ll write out some comments on it and let you know.

Moving on …

But Logicese is meant to be derived from English. For example, when someone speaking Logicese says that an and statement is only true if all of its conjuncts are true, he says that because that’s how the word and is used in everyday language. The problem is that conditionals resist any such easy treatment, at least in general.

I don’t think this is true. The less formal phrasings of logic — which might be called “Logicese” — are supposed to facilitate translation between everyday speech and symbolic logic. We want to do this so that we can evaluate arguments that contain specific content for validity without having to worry about whether the propositions are true — or, in some sense, even make sense. Most philosophers of logic that I’ve encountered are perfectly comfortable with the mappings not being exact. For example, take or. In normal language, “or” always refers to “exclusive or”; when we say “I am going to the store or I am going to the park”, we almost always don’t mean that the two things are both going to happen. But “or” in both “Logicese” and symbolic logic is ALWAYS inclusive or. As far as I know, no one is in any way bothered by this disconnect, mostly because as long as you make the right translations, and translate to exclusive or when it needs to be exclusive, the translations work fine.

The issue with conditionals is that it seems that the translations are off, which risks the symbolic analysis of validity being off. But as long as it is only in these cases, again it is a concern for philosophers of logic and not for anyone else.

This is how I often feel about philosophy. I find it very interesting, which is why I keep reading work written by philosophers, but often their whole project seems quixotic to me. Of what value are endless discussions trying to give precise definitions of knowledge, or of truth, for example? No one has trouble applying these notions in everyday life, and I don’t think the difficult arguments summoned forth in the professional literature really clarifies much about them. Why is it even desirable to have sharp definitions for these ideas?

The problem is that this applies to pretty much any formal field. I would estimate that at least 90% of what mathematicians do isn’t really useful or required for everyday reasoning, but it is important to mathematics or for other formal cases. In fact, I’d posit that for everyday purposes the mathematics we learn through primary school is probably more than sufficient. I know that I’ve never used the calculus or abstract algebra that I learned in high school and in university.

The same thing applies to science. We don’t need to learn the intricate physical details and the calculus to catch a ball, so why should physics care so much about it? But it is important to their fields, and some of the things they do end up being of interest to more every day concerns.

Take knowledge as an example. By default, most people tend to treat knowledge as being certain; if they know something, they are not wrong. But we don’t have certainty and can’t get it, so knowledge can’t require certainty. You can see how this is relevant to the atheism/theism debate, as neither side wants to have to have certainty in order to know something. It’s also relevant for science; since science cannot have certainty, it has better not need that to produce knowledge.

So it doesn’t strike me that this is a particular problem for philosophy as opposed to other fields. And I’ve done at least some of science, mathematics, social sciences and philosophy. I find philosophy interesting because the issues they go after are indeed fundamental and important in that sense, but admit that it isn’t all that relevant to my every day life. But neither is most of the science or mathematics I’ve learned either.

16. #16 couchloc
July 12, 2013

Jason,

“it’s a very dense and technical chapter. To me it seems like an awful lot of hard work for very little reward. I doubt if anyone really understands conditionals any better for invoking modal logic or conditional probability.”

First, it seems to me someone probably once said that Frege’s work on the foundations of arithmetic and modern logic was “dense and technical” and that nobody “really understands Aristotelian logic any better for invoking the concept script or existential quantifiers Frege introduced.” We should have just stuck with Aristotle’s syllogism since all this obscure work by professionals in their academic papers doesn’t fit well with our common knowledge or predilection for what’s familiar?!? I think this would be the wrong attitude towards work which is trying to improve our understanding of some difficult issues. I don’t work on this particular area myself since it’s a matter for logicians, nor do many philosophers, as VS has already suggested, but this doesn’t lead me to think there’s no reason for people to work in this area. It is important to improve our understanding of conditionals and what we can and can’t do with certain logical systems.

Second, you raise the question whether work by philosophers on conditionals/modal logic/probability has been useful. I would suggest you read some more. The influential theory of causality and explanation in the sciences that has been getting attention recently is Judea Pearle’s book, _Causality: Models, Reasoning, and Inference_ This work is widely discussed in the social sciences, computer science, and elsewhere. This is certainly an important area of recent work. But anyone who knows Pearle’s book knows that it develops a view of counterfactual conditionals in giving his analysis of causality, and that in doing this it appeals to work by the philosopher David Lewis (Princeton) who wrote a whole book on the subject (called _Counterfactuals_) based on modal logic and possible worlds. Pearle doesn’t accept everything Lewis says, but develops his account in response to Lewis’s system. So there’s already evidence that the study of conditionals and modal logic by philosophers has contributed directly to this important work. Maybe it’s hard for someone outside the field to follow these discussions, but that doesn’t mean there’s not interesting work being done in the area.

For reference you can look at this article by Pearle and Halpern (both computer scientists). Click on the title, “The logic of counterfactuals in causal inference.”

http://philpapers.org/rec/PEATLO-2

Notice that this is an article offering us an analysis by two computer scientists of the notions of “causality and explanation”—two concepts which are absolutely fundamental to our understanding of science and what it aims to do. Look especially at section 3 and 3.1 on causality and explanation. Cited in the article is work by philosophers Carl Hempel, David Lewis, Wesley Salmon, Michael Scriven, and Jim Woodward who worked on conditionals. So, clearly, this work is already seen by those like Pearle as useful and not simply idle trifling or something.

17. #17 couchloc
July 12, 2013

Let me just make clear that I think VS is basically right in saying that it seems Jason is focusing on some specific concerns of philosophers of logic and blowing them up as if all philosophers are worried about them. I have suggested that there is useful work being done in the area, but, in general, focusing on these issues isn’t a helpful way to view what most philosophers are concerned with in my view.

18. #18 Jason Rosenhouse
July 12, 2013

VS —

We want to do this so that we can evaluate arguments that contain specific content for validity without having to worry about whether the propositions are true — or, in some sense, even make sense.

But we don’t come to logic with a blank slate. We start with certain ideas about how language is used and about which inferences are valid and which are not, and then we formulate our rules of logic accordingly.

Most philosophers of logic that I’ve encountered are perfectly comfortable with the mappings not being exact. For example, take or. In normal language, “or” always refers to “exclusive or”; when we say “I am going to the store or I am going to the park”, we almost always don’t mean that the two things are both going to happen. But “or” in both “Logicese” and symbolic logic is ALWAYS inclusive or.

That’s not correct. There are many, many instances in everyday speech where or is used in the inclusive sense. That’s precisely why the convention in symbolic logic is what it is. But, as you note, logic also has an exclusive or. The convention is simply that you assume the inclusive or is intended unless it is specifically stated otherwise. So I’m not sure what your point is with this example. “Or” has two distinct meanings in English, and our principles of logic capture both of those meanings.

The reason conditionals provoke such controversy is, again, that we start with some notions about when conditionals are true and when they are false, and then we try to devise rules of logic that capture those notions. But the classical account, which works well in some cases, does not work well in others. We start off knowing that, “If I am not in France then I am not in Spain,” is false, and then decide the classical account is incomplete, if not simply wrong, because it can’t handle this.

The problem is that this applies to pretty much any formal field. I would estimate that at least 90% of what mathematicians do isn’t really useful or required for everyday reasoning, but it is important to mathematics or for other formal cases. In fact, I’d posit that for everyday purposes the mathematics we learn through primary school is probably more than sufficient. I know that I’ve never used the calculus or abstract algebra that I learned in high school and in university.

I agree completely with the first part of this! It’s one of the reasons I’ve gotten pretty sour on large swaths of academic research, including in my own discipline. But I also think you’re misunderstanding my objection. I’m not complaining simply that the work on conditionals is dense and abstruse. Nor am I complaining that this is material that most people can safely ignore. My complaint is that it’s unsuccessful on its own terms. I don’t think it really clarifies much of anything.

Conditionals don’t become confusing until you try to find a single account that covers everything. So stop trying to find such an account! Why should we even expect to be able to give a single account of conditionals? The examples I gave in the opening post are humorous precisely because we know exactly how to assess their truth or falsity without any help from the philosophy of logic. Why isn’t that enough?

19. #19 Jason Rosenhouse
July 12, 2013

couchloc —

Second, you raise the question whether work by philosophers on conditionals/modal logic/probability has been useful. I would suggest you read some more. The influential theory of causality and explanation in the sciences that has been getting attention recently is Judea Pearle’s book, Causality: Models, Reasoning, and Inference This work is widely discussed in the social sciences, computer science, and elsewhere. This is certainly an important area of recent work.

I’m not familiar with Pearle’s book, but I’m pretty sure that not many scientists or social scientists could tell you anything at all about modal logic. I don’t think they are confronting practical problems about causality in their work, find themselves getting confused, and then whip out the philosophical machinery to bring clarity. So I would question how influential or widely discussed this work could possibly be.

I have no problem with dense and technical material, and I don’t think everything has to be defended based on its practical value. I’m all in favor of knowledge for its own sake, and, since you never know where the next great idea is coming from, I’m all in favor of people investigating whatever happens to catch their interest. My problem with this particular material, as I suggested to VS, is that I don’t think it really is clarifying anything. It’s all very ingenious, but at some point things get so abstract and abstruse that it just isn’t helpful anymore.

Let me just make clear that I think VS is basically right in saying that it seems Jason is focusing on some specific concerns of philosophers of logic and blowing them up as if all philosophers are worried about them.

Well, that’s fair enough. I’ll be more careful next time.

20. #20 couchloc
July 12, 2013

As I said, I don’t think that focusing on logicians’ worries about conditionals will give one a good sense of the typical concerns of philosophers. But since this issue has been raised, and I’ve suggested there is some worthwhile work in this area, I will address this. You write:

“I don’t think they are confronting practical problems about causality in their work, find themselves getting confused, and then whip out the philosophical machinery to bring clarity. So I would question how influential or widely discussed this work could possibly be.”

http://en.wikipedia.org/wiki/Judea_Pearl

“[Pearl] is credited for developing a theory of causal and counterfactual inference based on structural models….He is the 2011 winner of the ACM Turing Award, the highest distinction in computer science, for fundamental contributions to artificial intelligence through the development of a calculus for probabilistic and causal reasoning….”

“He is interested in the philosophy of science, knowledge representation, nonstandard logics, and learning….His work on causality has “revolutionized the understanding of causality in statistics, psychology, medicine and the social sciences” according to the Association for Computing Machinery.”

Pearl discusses conditionals in his papers and book and is certainly familiar with possible world semantics and philosophical theories about counterfactual conditionals (note his interest in “nonstandard logics”). He considers them in his work, although he thinks there are problems with such accounts. I’m not suggesting that lots of scientists are getting confused and then, as you say, whipping out their modal logic textbooks or something. Nobody is suggesting that’s how the influence works in this case. But Pearl himself takes work on conditionals seriously. I think this is sufficient to justify the work that’s been done in this area because it is being picked up and applied in useful ways. In some ways I’m not in disagreement with you, because much of this work is technical and of interest to specialists and not your average person. But this is just a point about this particular area of research being done that you are focusing on.

I agree that sometimes the abstruse aspects of philosophical discussions can make seeing the broader picture difficult after a while. Maybe it would help to read something that keeps the relevant issues in focus. I’ve heard the book by Simon Blackburn called “Truth: A Guide” is good, though I haven’t read it. This is on a different topic but one you mentioned before. If he can’t make philosophy seem relevant, then nobody can.

21. #21 Verbose Stoic
July 14, 2013

Well, that’s fair enough. I’ll be more careful next time.

Well, that’s what I wanted for the most part, so thanks for that.

I could write more about the other parts, but let me just address this one last thing:

Conditionals don’t become confusing until you try to find a single account that covers everything. So stop trying to find such an account! Why should we even expect to be able to give a single account of conditionals? The examples I gave in the opening post are humorous precisely because we know exactly how to assess their truth or falsity without any help from the philosophy of logic. Why isn’t that enough?

Well, following on from the first part, it is indeed enough for most purposes. It ISN’T enough for … philosophy of logic, which is trying to build a logic that will let us, in fact, determine their truth or falsity inside a symbolic logic or Logicese structure, and finds at least some cases where that’s problematic. So they want to resolve it. So here you diagnose the problem as trying to find a single account of conditionals — I assume — that covers everything, and wonder why they want to do that. But they don’t. They want to find a single LOGIC that covers everything we might want to argue about, or failing that be able to discover what other logic is necessary and how to know precisely when to use one logic or the other. There are already tons of alternate logics to use that were invented in response to problems like these, but it isn’t clear how to do that for conditionals.

I’ll admit that I find that a lot of the time in philosophy things get bogged down in trying to fix problems and end up with far more complicated theories than we should need. But for most of them, I’ve come upon them as part of an overarching course of study, and so I can see what problems they were invented to solve, and why they did that, and so I find them understandable, even as I disagree with them. I’ll have to wait for Read’s book to see if he manages to do that or not; I’m not, myself, well-versed in philosophy of logic so it should be an eye-opener for me.

22. #22 Another Matt
United States
July 15, 2013

Anyone who wants to learn about conditionals like these needs to read Nelson Goodman’s Fact, Fiction, and Forecast. Conditionals which invoke counterfactuals, causal situations, or some kind of “dispositional property” of something have to be treated in “non-classical” ways. The 3rd chapter of Quine’s Methods of Logic has a nice introduction on this.

Of some note — in classical logic, “If A then B” is translatable to “~(A . ~B)” or “not(A and (not B))”. Clearly this doesn’t work for counterfactual conditionals (what Jason is calling subjunctive), because “A” is almost always false. Statements like this are inductive, so deductive rules don’t work.

One must also be careful about how to state causal situations as conditionals. Take for instance, “where there’s smoke, there’s fire,” which usually implies “if you see smoke, a fire has caused it.” This is not “If fire then smoke,” but rather “If smoke, then fire.” This is sometimes counterintuitive because “If A then B” often feels like A is causing B.

The same is true for couchloc’s statement here:

If the player swung the bat well, the ball went far.

This claim is naturally read as expressing a causal relation between the antecedent and consequent: the swinging of the bat caused the ball to go far.

I would rather switch antecedent and consequent here. If we’re trying to express the players swing as the cause of the ball’s travel, it should be phrased,
“If the ball went far, the player swung the bat well.”

23. #23 eric
July 16, 2013

VS:

It ISN’T enough for … philosophy of logic, which is trying to build a logic that will let us, in fact, determine their truth or falsity inside a symbolic logic or Logicese structure, and finds at least some cases where that’s problematic.

Doesn’t classical symbolic logic meet Godel’s criteria for his incompteleness theorems? IIRC, an axiom-based system in which arithmetic is possible. Well, symbolic logic is axiom-based, and (again IIRC) all its basic functions can be mapped onto algebra.

If that’s the case (and I’m truly asking, not stating that it is), then any logic they build is bound to have problems because its either going to be incomplete or inconsistent. Perhaps what we’re seeing with the conditional function is one of these issues.

24. #24 Richard Wein
July 16, 2013

Jason wrote:

But Logicese is meant to be derived from English. For example, when someone speaking Logicese says that an and statement is only true if all of its conjuncts are true, he says that because that’s how the word and is used in everyday language. The problem is that conditionals resist any such easy treatment, at least in general.

Well, that’s sort of my point. If no good treatment is available but you impose one anyway, then you’re imposing a new meaning, not giving an account of an existing meaning.

But the classical account, which works well in some cases, does not work well in others. We start off knowing that, “If I am not in France then I am not in Spain,” is false, and then decide the classical account is incomplete, if not simply wrong, because it can’t handle this.

Yes, but we don’t have to consider it an “account”. I think there’s a reason you call the standard convention a “convention” and not an account. You probably realise that it’s more of a stipulation than a description of the way things are. If mathematicians find this stipulation useful, let them use it. But we must avoid the temptation to treat this as an account of ordinary language. I think you appreciate this, but your OP playfully enticed us into conflating a mathematician’s/logician’s convention with an account of ordinary language.

Perhaps we should recognise that classical logic serves as both a stipulated logical calculus and as a tool for helping answer real-world questions. The key is to recognise that the language of the logical calculus is only an approximation to ordinary language, and some parts diverge more than others. So we should apply it to ordinary language with caution.

Of what value are endless discussions trying to give precise definitions of knowledge, or of truth, for example? No one has trouble applying these notions in everyday life, and I don’t think the difficult arguments summoned forth in the professional literature really clarifies much about them. Why is it even desirable to have sharp definitions for these ideas?

This sounds quite Wittgensteinian. (I say that with approval.) Most natural language concepts are inherently fuzzy, so there are no sharp definitions to be had.

25. #25 peter hoffman
backwoods
July 16, 2013

I’d agree with Jason’s general point if suitably interpreted, but disagree quite strongly with several of his examples.

As for his general point, I would much sooner see it expressed to say that general studies of logic by non-mathematicians since maybe 1880 (i.e. since Frege, a mathematician for our purposes, as were Russell/Whitehead in Principia) have yielded virtually nothing of real intellectual value or any other value. That failure includes almost everything related to modal logic. I would beg anyone here to come up with even a single specific example to show me wrong. Now here I do exclude two things as parts of modal logic, very valuable contributions that are sometimes called modal, but merely use similar notation and related rules, but are not part of modal logic as conceived originally and carried out ‘ad nauseam’ by people mostly working as philosophers. The first is the valuable “provability logic”, initiated by Godel I think, and carried ‘to perfection’ by Boolos, who was a mathematician no doubt, despite being in the so-called philosophy department at MIT. The second is the circle of ideas in the program verification side of computer science, dynamic logic being prominent, and again valuable, and resembling modal logic but really quite distinct. Anything else is fair game for such an example to show me intemperate and wrong on this. I would grant that the earlier formal fiddling with one or another of the 17 (or infinity!) distinct flavours of modal logic may have helped speed up the good work of George Boolos on provability and, say, David Harel on dynamic logic. But this formal fiddling in its attempts to ‘extend logic’ is of virtually no permanent value.

So I’m saying that rather than simply expressing that one should not expect there to be a single convincing treatment of conditionals in logic, Jason should have said that nothing done outside of mathematical logic in the last 130 years (amusing how the adjective “symbolic” is studiously clung to over “mathematical”—there may be some interesting psychological aspects to that in both directions!) has much permanent intellectual value.

I would also disagree with his general characterization of mathematicians as accepting the conventions re conditionals just because “it works for me”—see below.

As to Jason’s examples, there are two major points:

Firstly it seems to me that the real thing many philosophers in particular are trying to do is to somehow wedge just about every part of natural language into a logical form. That has always failed miserably, mainly because the sentences where it fails simply do not denote propositions which must be true or false in such an attempted application of formal logic. Getting down to conditionals as a major source of this misconception of logic, a big problem I can see with this

(1) in the hands of mostly philosophers,

(2) with almost all teaching of and texts for logic beginners, and

(3) with many of the responses here where, because logic is done informally by any halfwit who learned to speak, and also by those here, who are much more capable than that, so everybody has a go, usually without attempting any education in the subject,

is a complete failure to appreciate the separation between ‘formal language’ and ‘metalanguage’, that separation really begun by Frege, made somewhat confusing though not really mistaken by Russell/Whitehead, brought to a very high point by Godel and succeeding mathematical logicians.

Let me give an illustration. Here is a true statement in the metalanguage we all use here. (Call it math/english).

For any 3 formulas ‘F’, ‘G’ and ‘H’ in propositional logic (or in any other reasonable formal logic such as 1st order), the formula

‘(F —>G) OR (G—> H)’

is a propositional tautology (i.e. always has truth value Tr, never Fs).

(The formal language of propositional logic, perhaps sentential logic to some readers, is within the single quotes, and the AND would be a single symbol, maybe &, or a wedge, in a text.)

Many would say: “How can you say that you can either deduce G from F, or else deduce H from G, no matter which statements F, G and H are used? That’s ridiculous. Not only that, my prof made clear that tautologies were obvious truths, or all his examples were, and it was implied that the only non-obvious ones are that way merely because they are too lengthy to be obvious to most of us!” This combines the confusion above of languages with the failure to distinguish conditionals in the formal logic from logical arguments in the metalanguage. (The latter was mentioned by one of your respondents, but I don’t agree with much else he said!).

Those who dislike the above after verifying that it is in fact true (just consider the two possibilities for the truth value of ‘G’), make a naive mistake if they then dismiss logic as a waste of time, as done in several of the responses to Jason. I trust they will be happier once they learn that the following statement is false:

For any 3 formulas ‘F’, ‘G’ and ‘H’ in propositional logic, at least one of the formulas

‘(F —>G)’ or ‘(G—> H)’

is a tautology.

This is false despite the fact that many would mistakenly deduce this 2nd statement from the first. Note where the single quotes are, and that “or” in the metalanguage is not OR.

Even more false, if that is possible, is the statement

For any 3 formulas ‘F’, ‘G’ and ‘H’ in propositional logic, the formula

‘(F —>G)’ is true, or, for any 3 formulas ‘F’, ‘G’ and ‘H’ in propositional logic, the formula ‘(G—> H)’ is true.

And that brings up another important point: there is no attempt by anyone here it seems to distinguish quite distinct formal languages from each other (propositional versus 1st order logic here) nor their informal counterparts from each other (e.g. the respondent who generalized France and Spain as sets of people to arbitrary sets). I think this is partly caused by having little appreciation of the formal versus meta distinction which is so central to any modern (or even 1920’s) appreciation of logic.

The counterfactuals are good specific examples of this real error: that philosophers in many cases simply misunderstand what logic can be of value for. So let me use the one in the most recent PDF on the publication list of Pearle, mentioned by another respondent here. Pearle’s work I do not think he would regard as logic at all, but really CS (though nothing to do with program verification above) or also perhaps partly philosophy. The value of his work remains to be seen to me anyway, but perhaps part of that is due to my suspicion of the subject of artificial intelligence, which has had a long history of vastly over-inflated claims, such as the ‘5-year plans.’ starting as early as 1957—“in 5 years, our system will be translating Russian fluently, playing chess exactly like a grandmaster but better, etc. etc.” In any case, I don’t think any of the artificial intelligencia would regard their subject as having overlap with logic as a subject. (Nothing coherent can avoid overlap with informal logic as in “Maintaining both that God affects the physical universe, and that science and religion are disjoint, is illogical”.)

Pearle’s example of a counterfactual is:

If Oswald had not killed Kennedy, then someone else would have.

The part after the “if” and before the comma is simply not a string of English which can in any sense by understood as expressing a proposition, despite the mighty attempts to do this by some persons on the payrolls of philosophy departments.

However, there is a second source of examples here where I disagree strongly.
Jason really emphasizes an example which is not in this form of an English statement which simply is not a suitable candidate to be formalized in any reasonable form of logic. These examples are the (if true1 then true2)-form, ones which he finds hard to take to be true. I simply disagree 100% with that, so let’s go through his example. I’ll write down several sentences in a row, starting with what seems to me to be pretty decent English, then getting worse in English, but closer to formalizable versions. The first of these is obviously true. I claim the next in each case is obviously true if the previous is. (As it happens, all the middle pairs are actually logically equivalent, i.e. the truth of 2nd also implies the truth of the first.) Here we go:

(1) Jason is not in Spain.

(This we must all take to be true of course.)

(2) Whether he is or is not in France, Jason is certainly not in Spain.

(3) If he is in France or if he is not in France, Jason is certainly not in Spain.

(4) If (F or G) then H, where we have abbreviated “Jason is in France” to F, its negation to G, and “Jason is not in Spain” to H.

(5) If F then H, and if G then H, with the three symbols as above.

(Hint: (4) and (5) are logically equivalent to each other no matter what the symbols are taken to mean, as is (6) to them, by English usage.)

(6) F implies H, and G implies H.

(7) G implies H.

It is very clear that the last of these has exactly the same meaning in English as the statement “If Jason is not in France, then Jason is not is Spain”, and of course it gives me no trouble at all to assert confidently that Jason’s conditional is in fact true, not only not “obviously false” but not false at all. Exactly the same applies to the conditional “If 1+1=2, then Paris is in France”. Both may be rather useless, but they surely are true.

If anyone here cares to dispute this, I’ll be happy to entertain it. But you must tell me which in my list is the first one which you think is false. And then you must explain to me why this is so, despite the fact that you accept the fact that the one previous is true.

And don’t quibble about whether the ‘if–then’ form and the ‘–implies–‘ form should be the same in English. If you think not, just skip (6) and rewrite (7) in the former form. That makes the disagreement beside the point.

Sorry for the length of this—I’ll refrain from taking up details of where I disagree with many of the responders!

26. #26 Another Matt
July 16, 2013

Here’s why the Spain-France example seems “absurd,” as Jason put it.

“If I am not in France then I am not in Spain.”

In English we often use “I” and “you” as a universal quantifier, as in “If I were on the moon, I’d weigh less than I do on earth.” This is not a simple if-then proposition — it is not meant as a statement about me at all, but rather as a statement about the general difference in gravity on earth and on the moon. It is of the form “For all X, if X is on the moon, it weighs less than X weighs on Earth.” Which is true.

If you read the original as though it meant, “For all X, if X is not in France, then X is not in Spain,” it is clearly false.

27. #27 Another Matt
July 16, 2013

Whereas “If I am not in France then I am not in Paris,” taken universally as “For all X, if X is not in France, then X is not in Paris,” is true, because of the relation between the predicates “is in France” and “is in Paris” — “For all X, if X is in Paris, then X is in France.”

28. #28 peter hoffman
backwoods
July 16, 2013

I don’t know whether (#’s 26 and 27) is meant to specifically disagree with me, but again it is confusing propositional with 1st order, or at least confusing the fact that

“For all x, P(x”)

implies

“P(a)”

with the fact that the converse is false; i.e. the 2nd does not imply the first.

If the conditional is abbreviated to Q —-> R, there is also the ‘problem’ that the truth of each of Q and R separately here has nothing at all to do with logic, just with the empirical observation we assume has been made that Jason is sitting at that time comfortably on a chair in his office in USA, some geographical facts about where countries are, and perhaps that Jason is not so large as to be in danger of getting the back of his knees wet somewhere in the Atlantic south of Iceland. But once both the statements about being in neither France nor in Spain are accepted as true, I don’t think that anyone who speaks normally in English about factual statements, and has spent his or her time thinking carefully about it, would disagree that the implication of the 2nd by the 1st is both silly and true!

29. #29 Another Matt
July 16, 2013

I’m not confusing first-order with propositional logic. I’m saying that when it comes time to translate the English statement “If I am not in France then I am not in Spain.” into “logicese,” as others here have called it, one has to decide between first-order and propositional interpretations, and it’s not clear which to choose –though, I think in everyday speech such a statement would almost always be meant and understood as first-order, with the quantifiers implied.

The dissonance comes from this intuition — clearly the statement is false in the more usual first-order interpretation, and it’s quite hard to “make the switch” to the propositional interpretation once you’ve intuitively understood it the other way.

Another interpretation is to take it to mean, “At all times, if I am not in France, then I am not in Spain.” In this case the original is not speaking about “right now,” but instead trying to form a general rule (and one has to decide whether the rule I’m trying to form applies to me or if it applies to Spain and France, in which case it would be the same for everyone). Perhaps it turns out to be true if I never visit Spain before I die.

Again, we often use “I” and “you” to mean “someone” in ordinary English. “If someone is in France, then he/she is not in Spain” is false.

30. #30 peter hoffman
backwoods
July 16, 2013

Matt: You missed a ‘not’ in the last sentence, but I’m sure that is a typo.

I would definitely take this propositionally, and that seems to be definitely what Jason is discussing in this particular example. That is, if Q and R are definitely given as true, he maintains that sometimes Q —> R is somehow false. After all, he specifically referred to himself, not to some general variable “I”, as being in Virginia. So I think the 1st order interpretation of this is a red herring here, except possibly to explain why he (in my opinion mistakenly) thinks his conditional is not true.

31. #31 peter hoffman
backwoods still
July 17, 2013

I hope any confusion I caused, with my

“…the formula

‘(F —>G) OR (G—> H)’

is a propositional tautology (i.e. always has truth value Tr, never Fs).

(The formal language of propositional logic, perhaps sentential logic to some readers, is within the single quotes, and the AND would be a single symbol, maybe &, or a wedge, in a text.)…”

was not a problem. I inadvertently referred to AND rather than OR as the formal symbol. I guess the OR would often be the wedge, but written upside down vis a vis the AND. I cannot remember what the philosophers whose notation dates from 1910 or earlier use for the ‘inclusive or’.

Now I’m feeling the need to apologize as well for too many messages!

32. #32 Sean T
July 18, 2013

peter hoffman:

I think you hit the point exactly. If meant propositionally, the sentence “if I am in not in France then I am not in Spain” is true (assuming that the speaker is in fact not in Spain). However, many people would hesitate to call that a true sentence. The reason IS that they are implicitly giving it a first-order interpretation, much like Another Matt pointed out, such as “For any time t, if I am not in France at time t, then I am not in Spain at time t”.

Another possibility: an implicit interpretation in natural English of the sentence “if I am not in France, then I am not in Spain” might be “If I am not in France, then it is not possible that I might be in Spain,” which is clearly not logically equivalent to the original conditional.

In our natural language we tend to read more meaning into the “if … then” operator than what is there. We think we know what “implication” means, and in many contexts, we actually are dealing with a situation which does allow for a more intuitively obvious notion of “implication”. In the example above, however, our intuitive grasp of implication fails us.

33. #33 Sean T
July 18, 2013

I find that the conditional in general is very counterintuitive. Intellectuallly, I have no problem dealing with the logic of conditionals, but it just seems somehow unnatural to me. I certainly understand why the conditional is defined the way it is in propositional logic. There are only four ways it could be defined (after agreeing upon the intuitively obvious definitions of the truth value of a conditional with a true antecedent). However, three of them are not very useful, so we only really have one choice (the other choices would make the conditional logically equivlalent to a either a biconditional, a conjunction or just the conclusion).

That doesn’t help me to intuitively grasp, however, the fact that a sentence such as “If x is an even number, then x is an integer” is actually logically equivalent to “either x is not an even number or x is an integer”. The two are logically equivalent, and obviously both are true for all x, but I just can’t intuitively see the connection; they seem to mean very different things.

34. #34 Another Matt
July 19, 2013

Sean T,

My favorite discussion of the conditional is from Quine’s Elementary Logic.

IIRC, his introductory example is a character named Toby saying, “If Hawkshaw saw me, then the jig is up!”

What Toby is saying is that it can’t be that Hawkshaw saw him and yet the jig is not up. This is the usual logical translation — “if p then q” is the same as “not (p and (not q)).”

But it’s quite possible to reframe it as an “or” — “Either Hawkshaw didn’t see me, or the jig is up.” These two possibilities show the connection between the conditional and the or formulation; it’s easy to recast this as “Either Hawkshaw didn’t see me, or he did see me and the jig is up.” — Hawkshaw seeing Toby is always conjoined with Toby’s jig being up.

This wording is problematic, though, because the “either-or” can imply an “exclusive or” — there is the third possibility that “Hawkshaw didn’t see me, and yet the jig is still up (because the Feds caught me for tax evasion).” — so Toby’s jig being up is not always conjoined with Hawkshaw having seen him.

35. #35 peter hoffman
backwoods
July 19, 2013

“..it can’t be that (Hawkshaw saw him and yet the jig is not up)”

and

“…(it can’t be that Hawkshaw saw him) and yet (the jig is not up)”

have entirely different meanings. So some punctuation, to indicate those parentheses, is needed. Of course the logic conditional is logically equivalent to the first. I initially read it as the second, and I’d bet I’m not the only one!

That’s a much more salient illustration of the ambiguity of any natural language, than is the trivial fuss about the inclusive versus exciusive ‘or’.

As for the 10 billion-word fuss about conditionals in logic (as a subject where progress can actually be made), when the fuss is really entirely about English usage in places where formal logic adds little (but computer scientists and philosophers thinking deeply about it probably does add a lot), is an example unfairly used to be negative about philosophy, rather than more fairly about a number of ‘philosophers’.

36. #36 Rick
July 19, 2013

At work I had to learn and mentor others in doing Boolean searches on a full text database of documents. One of the difficulties in teaching how to use the database was the fact that the Logicese and English are, at least pragmatically, quite different languages. English “and” often means logic “inclusive-or” (“members and their spouses may use the pool” does not mean only people who are members married to other members may use the pool; “and” here means “or”.) In the document search context, if you want the documents with the word “proton” and those with the word “neutron”, forget English, use Logicese “or”. In interpreting laws and contracts, courts often note that in some context “and” means “or”.

And, as Another Matt notes, English “or” is sometimes Logicese “inclusive or” and sometimes “exclusive or” (on a menu, “soup or salad” rarely allows for both).

This is why the term “and/or” is sometimes used. The “and” prevents a simple “or” from being interpreted as “exclusive or”, while the “or” prevents a simple “and” from being interpreted as, well, “and”.

A statements such as “if I am not in France, then I am not in Spain” is in English a false statement and in Logicese a true one. Since the two are different languages, this is not a contradiction. I like the example “if Paris is not the capital of France then Paris is the capital of France”, which is also false as an English sentence but is true in Logicese.

Logicese is a formal language, that deals with the form of the statements, “if I am not in France then I am not in Spain” has the same form as “if I am not in France, then I am not in Paris”, and thus are treated the same. In English, we call upon the meanings of the words, and recognize that not being in France does not mean not being in Spain. But we have left Logicese and moved over to English.

If we want to capture the falseness of “if I am not in France, then I am not in Spain” ( or, since we are dealing only with form, ~F -> ~S ), we need to bring in some of the additional knowledge that we implicitly draw upon in the English version of the sentence. We could recognize that one cannot be in France and Spain at the same time, giving ( ~F -> ~S) & ~(F & S), which reduces to ~S (I am not in Spain). If we add the further condition ~F (I am not in France) the we can conclude that I am neither in France nor Spain.

In my experience using and helping others to learn and use even the restricted set of Boolean operators we used in the database search, it is clear that English and Logicese are different languages; in my opinion this is pragmatically and pedagogically true. And the difference is what allows philosophers to write many long and dense papers and books trying to reconcile the two.

37. #37 Explicit Atheist
July 21, 2013

Mathematics, and therefore also logic as a topic within mathematics, is focused on dealing with the addressing the question what could be true. Science, and also philosophy, is focused on addressing the question what is true. Mathematics plays a central role in science because what could be true plays a central role in constraining what is true. So the rules of logic are correct in their could be true context but they often fail in the is true context.

38. #38 peter hoffman
backwoods
July 21, 2013

Explicit:

Can you give me even a single explicit “rule of logic”, as you call them, which “…fail(s) in the is true context”? Thanks in advance.

Rick:

I agree with you if you are saying that ambiguity of English is the real problem, and is often addressed by CSers like yourself and sometimes by philosophers; but other philosophers who attempt to wedge most of natural language into logic have hardly made any progress and likely won’t. However:

(1) I’d really like to know what this “logicese” referred to by you and others actually is. If it includes anything much in modal logic, I’m afraid it does not do much. Anyway, why not be explicit about what formal logic you are referring to?

(2) The example “members and their spouses may use the pool” has all sorts of strikes against it as far as saying anything about the vices or virtues of logic. Firstly, it is not in the indicative (decarative, assertive) form, but rather the imperative. Some formal languages of theoretical CSers may deal with that, but applications of the logic Jason attempted to criticize do not. Secondly, it is plural, so surely you need to get into 1st order logic somewhat to deal with this. To say “members and their spouses do use the pool” is at least trying to state a proposition, but if you do not start it with ‘at least one married pair of…’ or ‘at least two …’ or ‘all married pairs of …’, it’s clearly nowhere near having any relation to a connection between a possible criticism of logic and the ambiguity of natural language.

(3) You say “..In English, we call upon the meanings of the words..”. I’d like to reinforce this as being almost a backhanded definition of logic when that word is used sensibly and more precisely then many here seem to wish. That is, one can sometimes learn about the truth without knowing the meaning of individual words, and that is logic, not more in general, and not less if we ever really think we’ve found it all. Even Kant seemed to think they had then found it all, and he was only more wrong when he seemed to think that Euclidean geometry was the only possible geometry!

(4) Finally you mention (somewhat glibly in my view) the ‘…the falseness of “if I am not in France, then I am not in Spain”…’. As long as that is taken as a statement of Jason with ‘I’ referring to him, and a time when he actually is not in Spain, it seems, as I said above, quite clear to me that it is true, not false (assuming as usual that any specific statement with no free variable must be one of those two). I made a list near the bottom of my lengthy rant above, (1) to (7), and I would at least hope you, now that you have said (7) is false, would tell me which in that list is the first false one, and why the truth of the immediately previous one does not contradict this. No one else has, including Jason, so I’m assuming now that they all (except you) agree that the statement is true.

39. #39 Another Matt
July 21, 2013

Can you give me even a single explicit “rule of logic”, as you call them, which “…fail(s) in the is true context”? Thanks in advance.

Have you ever read the literature surrounding the “is logic empirical?” question? The law of the excluded middle is the exemplar for a “rule of logic” (if I understand what this means) which fails in some empirical contexts (e.g. quantum states). It doesn’t mean that it fails to be true in all contexts — but it can’t be a universal law that is True with a capital T.

I’d really like to know what this “logicese” referred to by you and others actually is.

The way I think of it, a sentence in “Logicese” has exactly the same words in the same order as a sentence in English, but not necessarily the same meaning. Consider this sentence:

“I have three dogs; one of them is a poodle.”

In English, it’s totally unambiguous — in all contexts that the sentence might be uttered, it means that the other two dogs are not poodles. In Logicese, you can’t assume that at all because you don’t have the context — you can’t assume anything not in the sentence, so conservatively the meaning is the first-order “there exists a D such that D is poodle and D belongs to me.” “One of the dogs is a poodle” doesn’t say anything at all about the other two in logicese.

This is like those dumb elementary school jokes/word problems — “I have two coins in my pocket totaling \$0.20. One of them is a dime; what is the other?” In Logicese, the answer is “a dime.” In English, the answer is, “OK, wise guy, you mean ‘at least one of them is a dime, or else you’re wrong.”

Finally you mention (somewhat glibly in my view) the ‘…the falseness of “if I am not in France, then I am not in Spain”…’. As long as that is taken as a statement of Jason with ‘I’ referring to him, and a time when he actually is not in Spain, it seems, as I said above, quite clear to me that it is true, not false (assuming as usual that any specific statement with no free variable must be one of those two).

It’s usually false in English because its meaning in English would almost always be first-order, so you don’t get to make the assumptions you’re making about it. But in Logicese it reads as a straightforward proposition which would be true when uttered by anyone not currently in Spain.

Imagine peter hoffman had responded to Rick: “As long as you take it as a statement of Jason with ‘I’ referring to him, and a time when he actually is not in Spain, it seems, as I said above, quite clear to me that it is true, not false.”

In English, this can only be a statement about language and logic, not a statement about Rick. That is, it’s equivalent to the passive construction peter hoffman actually used, not to something which says, “As long as you, Rick, take it as a statement…. it is true, not false, but if Jason takes it as such a statement, it might actually be false.” It’s not clear to me that in “Logicese” one could justifiably make this inference to the “general you.”

40. #40 peter hoffman
backwoods
July 22, 2013

Re Another’s #39:

(1)
“…The law of the excluded middle is the exemplar for a “rule of logic” … which fails in some empirical contexts (e.g. quantum states)…”
Yes, I certainly at least agree with excluded middle being called a law of logic, as I would for any propositional tautology. But no, not much of an example: Many major physicists, such as Feynmann, did not think we yet have any reasonable understanding of how to interpret quantum theory. And I think major physicists today , those who think about interpretations and explanations, not just ‘shut up and calculate, reject the Copenhagen interpretation, and have a good deal of sympathy for Everett’s Many Worlds interpretation, where there is no contradiction with excluded middle (nor is there in David Bohm’s, for that matter). So thanks for that, as a good attempt at an example, but I think it ceased to be at all convincing after the roughly 1925-1950 period, if it ever was.

If you actually think that Explicit’s statement “…the rules of logic are correct in their could be true context but they often fail in the is true context…” is anything but frothy nonsense, I’d be interested in seeing how it makes any sense at all to you. It seems to me that the “could be true” would include the “is true” context if any sense could be made of this, which causes serious difficulties, unless Explicit himself is writing in a context where self-contradiction is a law of logic which can be neglected. (More on context below) Even were your example further above actually really an example, how could excluded middle then be regarded as correct in some “could be true” context, when you claim it is not correct in microphysics? Is it that somehow microphysics couldn’t be true, yet is true? This is totally confused, but you didn’t attempt to defend the frothy nonsense, and trust you won’t.

I certainly hope that my asking for an example is not misinterpreted as asserting baldly that present day standard logic is all that could ever be relevant. That would be arrogant, and is not what was intended. In future, undoubtedly we will understand the world better, and maybe something like such an example will be clearly correct. However, as explained above, getting an example makes it easier for me to make clear the nonsensical contradiction in saying something like that, where what ‘is true’ somehow is not part of what ‘could be true’.
(I hope you are no longer losing too much sleep worrying that I have entirely neglected my duty to “..read the literature surrounding the “is logic empirical?” question..” It’s a shame that more of that writing is not by scientists and mathematicians, and perhaps less of it by philosophers.)

(2)
“…a sentence in “Logicese” has exactly the same words in the same order as a sentence in English, but not necessarily the same meaning…”
That is very far from telling us, as I asked, what that, apparently precise, language actually is. Logicese seems almost to be (English spoken by someone desiring complete precision). Modern logic, if not earlier empirical evidence, shows that guy will not succeed, unfortunately.

(3)
“….
“I have three dogs; one of them is a poodle.”
In English, it’s totally unambiguous — in all contexts that the sentence might be uttered, it means that the other two dogs are not poodles…”

Of course, an accurate speaker of English would either say ‘…exactly one…’ or ‘…at least one…’ Are you certain that, say, an English speaker on a little island off the south island of New Zealand (surely all your contexts are not middle america!) does not mean the “at least” version? Or maybe a speaker in 1777 somewhere? Perhaps the modern USian speaker from Virginia cannot figure out the breed of the other two?

(4)
” In Logicese, you can’t assume that at all because you don’t have the context — you can’t assume anything not in the sentence, so conservatively the meaning is the first-order “there exists a D such that D is poodle and D belongs to me.” “One of the dogs is a poodle” doesn’t say anything at all about the other two in logicese.”

I think we agree on this pretty much. But ‘logicese’ is simply left completely unspecified by you and everyone else here. I tend to feel that there is too much vagueness here, when a very extensive subject of modern mathematical logic exists (being called “mathematical” does not refer to a restriction on its applicability, but to something else). Examples such as the above give an indication, if any was needed, of what you are trying to get at. But, for example, I could ask: Suppose you had an English text which you believe could be translated into this still mysterious more formal language. Is it then the case that it could be translated into one of the languages of 1st order logic? If so, surely doing that is far better than being so vague. If not, then the whole thing is up in the air again. Does logicese operate assuming excluded middle? etc., etc.

(5)
“….It’s usually false in English because its meaning in English would almost always be first-order, so you don’t get to make the assumptions you’re making about it. But in Logicese it reads as a straightforward proposition which would be true when uttered by anyone not currently in Spain…”

Firstly, I’m pleased you agree that Jason’s conditional about France and Spain is true, not false as he said. At risk of repetitiveness, let me quote his original blog, so that there is no doubt he is speaking propositionally: ‘…..I am writing this in Virginia, how about the statement, “If I am not in France then I am not in Spain.”..’ That “I” he uses is clearly not a variable in any sense at all.

And at risk of boring many here who know the following perfectly well, I’d also point out that all this reference to “context” is utterly irrelevant. If you wish to discuss this as a 1st order (i.e. predicate) statement in which “I” is a variable, that variable is of course free, not bound, in that statement. And basically for that reason, the statement is neither true nor false. If you really really want to precede it by “For all I’s” , the new statement now has that variable bound, and it has a definite truth value, false, in the interpretation where the variables range over the set of humans. And no babbling about context plays any role at all in any of this. But discussing it as some kind of 1st order statement is irrelevant here, since we are discussing what Jason claimed.

(6)
Your final two paragraphs are incoherent to me—-unfortunate since they refer directly to me. But they may have a subtlety beyond my ken. However, I’ve already asked for quite a bit of response, so to ask for writing them so I know what you are talking about here may be asking too much.

41. #41 Another Matt
July 22, 2013

Well, peter hoffman, all I can say is that in natural language, context is everything. I don’t think anyone thinks “Logicese” is any kind of real formal language. What it amounts to is whether you get to bring context in.

For instance, you say:

And no babbling about context plays any role at all in any of this. But discussing it as some kind of 1st order statement is irrelevant here, since we are discussing what Jason claimed.

You’re saying a first-order interpretation is irrelevant because the relevant context is Jason’s framing. What I’ve been trying to say is that this contextual information is crucial, because the sentence by itself without any other contextual surrounding would usually be taken in the first-order sense.

To translate the English sentence into a formal logic, the most probable solution is first-order, with the bound variable “I,” due to the frequent idiomatic use of “I” to mean “all I’s.” To translate the same sentence in “logicese” to a formal logic, you don’t get to use this idiomatic information — at face value it’s a simple proposition.

Misapplication of idiom is a classic mainstay of humor, especially with androids in science fiction. There’s a series of children’s books with a character who always takes idiomatic English at face value (i.e. as “logicese”) — http://en.wikipedia.org/wiki/Amelia_Bedelia

Your final two paragraphs are incoherent to me—-unfortunate since they refer directly to me. But they may have a subtlety beyond my ken.

No, I’m sure it’s my fault. Here’s a better example. How would you interpret this, if I told you:

“If you went to Venus, you’d be cooked alive.”

I would say that in English, the “you” is idiomatically understood as a general one, and so the semi-formal translation would be “For all X, if (X is a human and X is on Venus), then X is cooked alive.” In “logicese” I don’t think you get to use the idiom, so “you” has to be interpreted as the usual 2nd-person pronoun. Then the meaning would be “If peter hoffman is on Venus, peter hoffman is cooked alive,” and we are as yet agnostic about whether the same would be true of me or of Jason or Superman. The point is, to interpret it the latter way in normal English would require a ton of other contextual information that would override the normal idiom.

42. #42 peter hoffman
backwoods
July 22, 2013

Yes, thanks, I know even more resoundingly now what you mean by “context” in the one case of an English statement that might or might not be taken as 1st order because of its use of a pronoun. No one really knows exactly what that word “context” would mean, for other sentences, by you or by anyone else. So difficulties and disputes inevitably arise in understanding texts. I doubt anyone could disagree.

However an attempt to educate in, and a stronger use of, simple facts that formal logic has brought more clearly to light, especially avoiding the even worse word ‘logicese’, would certainly be good. The responses to Jason here has illustrated that quite a lot, with the spurious introduction of the 1st order interpretation of his disputed conditional being the most prominent. Words like” logicese” just promote more of the faux-knowledge about logic and about clarity of natural language, much of which has come from so-called ‘philosophy of logic’ and particularly from many lovers of modal logic.

And for the 3rd time, Jason gave it all the “contextual surrounding” it needed, when he preceded it with “…I am in Virginia..”, right?

And finally once again, when you take that statement out of context, good old plain mathematical logic gives it exactly the right truth value, sometimes true, sometimes neither and sometimes false (see my immediately previous), depending on what is the context, but done with precision, including whether you implicitly assume the quantifier or not. There is no justification at all for the original fuss about whether maybe there is a mistake in assigning ‘true’ to a statement of the form ‘if true1 then true2’, despite lots of junk in freshman textbooks by philosophers .

43. #43 Rick
July 22, 2013

Peter:

(1) I’d really like to know what this “logicese” referred to by you and others actually is.

“Logicese” is a metaphorical shorthand for the differences between logic as a language system and natural languages, such as English, as language systems. To continue the metaphor, there are as many “dialects” of logicese as there are types of logic; there is the dialect of syllogistic logic, that of propositional logic, that of first order logic, etc. There is, however, a sufficient “family resemblance” among them to make the term useful, mainly that they are all formal languages (which natural languages are not) that at least attempt to remove ambiguities by abstracting away almost all semantics. They also differ from natural languages as to their scopes of applicability. These differences allow for clearer, formal reasoning within the logic system, which natural languages do not as easily do.

As I said in my earlier comment,the context for my understanding includes using and mentoring others in the use of a particular “dialect” of logic used in a full text search of documents. It was not at all uncommon for those learning the system to write “and” when they wanted an “or” because the English version they came up with of what they wanted used the word “and”. I could not say that, in many cases, the English version they started with, with an “and”, was wrong, or even ambiguous as an English statement of what they wanted, but the translation into the logical form the search system wanted required an “or” rather than an “and”.

Now I suppose that one could argue that the pragmatics of this pedagogical context simply made it more useful to explain the manner of constructing the desired logical form as translating from one language (English) to a different language (search system logic) rather than as some deep philosophical statement as to the inherent relationship between natural language and various forms of logic.

(2) The example “members and their spouses may use the pool” has all sorts of strikes against it as far as saying anything about the vices or virtues of logic. … it is plural, so surely you need to get into 1st order logic somewhat to deal with this.

I really don’t think I am criticizing logic. Pointing out that logic as a system and English as a system are different is not criticizing one or the other.

Although the treatment of “members and their spouses may use the pool” may by some accounts be more naturally formulated in first-order logic, probably the relationship between being in France, being in Spain, being in Virginia, and being Jason probably also is more naturally formulated using quantification. But we can use the same “trick” Jason used and translate it into predicate logic by taking some individual (to pick a name, Chris) and defining the predicates M (Chris in a member), S (Chris is the spouse of a member), and P (Chris is allowed to use the pool). “Thus (M | S) (“|” = “or”) defines the condition under which Chris can use the pool. P (M | S) defines whether a particular allowance of Chris using or not using the pool is in accordance with the rule.

This is actually a technique we who had to help people learn the search system often found helpful. Some of the difficulty in translating from what they wanted into an expression that caused the system to give them what they wanted was often avoided by in effect forgetting about a big pile of many documents and focusing instead on just one document, and asking “what is it about this document that makes me want or not want it?”

(3) You say “..In English, we call upon the meanings of the words..” … one can sometimes learn about the truth without knowing the meaning of individual words, and that is logic, not more in general, …

Yes, logic is a formal language, and a lot can indeed be learned through the type of foramalization logic represents. As a language sytem logic operates by a set of transformation rules that define valid steps one may take in reasoning.

But natural language is not a formal language. Natural language operates in a different manner. A statement in natural language is understood against a background of our knowlege of the world and context of the discussion, and a set of assuptions that guide a mental construction of the meaning of a sentence. Which is why natural language “gets away with” so much ambiguity., the ambiguity is often resolved by the reasonable meaning of the words.

(4) Finally you mention (somewhat glibly in my view) the ‘…the falseness of “if I am not in France, then I am not in Spain”…’. … I made a list near the bottom of my lengthy rant above, (1) to (7), and I would at least hope you, now that you have said (7) is false, …

Now, our context here is the discussion of set of sentences, “Jason is not in France”, “Jason is not in Spain”, and “if Jason is not in France then Jason is not in Spain”. Now, as a set of logical propositions translated to ~F, ~S, (~F -> ~S), I don’t think anyone, certainly not me, would argue that, given the first two, the third statement is not true as a logical proposition. As a logical derivation your steps (1) through (7) are all correct. This is logic, all well and good.

However I find the English sentence “if Jason is not in France then Jason is not in Spain” to not be true, and obviously so. So where do I disagree with your derivation? Right after your step (7) you say “It is very clear that the last of these has exactly the same meaning in English as the statement “If Jason is not in France, then Jason is not is Spain …”. But this is the point that I am making: these two do not have “exactly the same meaning”; as a verbalizable form of the logical (~F -> ~S) and as a English sentence the two do not have the same meaning.

In English the ” if … then” construction sets forth a claim of a relationship between the “if” phase and the “then” phrase such that the “then” phrase is in some manner a result or a consequence of the “if” phrase. For instance, the first “if … then” construction in Johnathon Haidt’s “The Righteous Mind” (just because I have it close at hand), on page 12 of the Nook version (page numbers are different than the print version) is “If you think that moral reasoning is something we do to figure out the truth, [then] you’ll be constantly frustrated …”. He is setting forth that the frustration will somehow be a result of that particular belief. He is not merely alleging the logical proposition (B -> F).

Now in logic, since we are reducing everything down to form only, we abstract away the semantics of the statement to the point that the propositions are reduced down to only their truth values, and so we lose the semantic part of the meaning of the English construction that relates the meaning of the “then” phrase to being somehow related as a result or consequence of the “if” part.

Now with “if Jason not is France, then he is not in Spain” is false because his not being in Spain is not in general a reasonable consequence of his not being in France. Telling someone Jason is not in France does not reasonably inform a listener of his location vis-a-vis Spain. This is because we interpret English sentences in light of our knowledge of the world, and not, as in logic, in a manner abstracted away from such knowledge. In the real world we know France and Spain are related in such a manner that not being in France does not reasonably exclude not being in Spain. Which is why “If Jason is not in France, then he is not in Paris” is a true statement; not being in Paris is a reasonable result of not being in France.

44. #44 peter hoffman
backwoods
July 23, 2013

Rick,
Thanks for the reply, and I appreciate both the difficulties of the type of work you do, and your point of view on this. I really don’t need to add anything with much substance, but two small addenda to what you said:

(1) Yes, various forms of logic do of course involve a ‘very strict’ language; but logic is far more than merely language of course.

(2) To put the situation baldly:

“Jason is not in France” is true—both in English by observation, and in propositional logic as an assumption here.

Virtually identical is:

“Jason is not in Spain” is true—both in English by observation, and in propositional logic as an assumption here.

However:

“If Jason is not in France then Jason is not in Spain” is—false in English by what many, including you, would assert, and yet true in propositional logic as a consequence of a fundamental aspect of that logic.

It is likely unavoidable in your type of work to need to accede to this type of (what I would term a) misconception, and natural language speakers are full of those misconceptions. But as a ‘problem for philosophical logic’, which is the way several people here including Jason basically put it, I think it is nonsense. That particular sort of disagreement between careful thinking and natural language is basically harmless, but it doesn’t require going very far to get into the kind of thing which may have negative consequences in the long run.

Here is an example, still in itself virtually harmless, but indicative of a kind of mental sloppiness which is endemic. It allows people in places like Fox News and the gun lobby to easily bamboozle a large portion of people into opinions harmful to themselves and others. The example would involve 1st order, rather than propositional, logic, if analyzed by a logician. I’d be willing to bet that at least 19 0f 20 USians would say

“All baseball players are not pitchers.”

In their (bastardized) English, this is perfectly true, because they take it to mean

“Not all baseball players are pitchers.”

But of course it doesn’t really mean that, and it isn’t true. Without any pitchers, the game would have trouble getting off the ground!

Do you ever need to deal with that inability of most people (USians are just handy for that example) to mix quantifiers and negations correctly? I tend to hope that an elementary course in logic will (among other things) partly go towards rectifying this mental sloppiness. A lot of the difficulties people have with understanding the most basic theory in calculus come down to exactly that inability.

45. #45 peter hoffman
still enjoying backwoods
July 23, 2013

Forgot: in Rick’s referring to syllogistic, 1st order, propositional, … (and other people here referred to syllogistic as though it was still of importance in itself, not just as a source of employment for philosophers in small Catholic colleges), I really feel impelled to add that syllogisms of Aristotle, and all the rather frivolous embellishments by the medievals of it, are in fact a tiny portion of 1st order logic. This is seldom mentioned in even decent texts on modern logic.

46. #46 JonJ