So there is these simple modal logic sentences:

□(a → b) and a → □b

Can anyone help me with some real-life examples, because I have troubles grasping the difference?


The simpler question is this: how does □(a → b) differ from a → b?

  • As @GrahamKemp said, modal logic requires you define a set of "accessible" possible worlds, and a list of what proposition are true in each of them. If a proposition is necessarily true relative to this set, the proposition must be true in all of them; if it's not necessarily true, there are some worlds where it is false. So □(a → b) means the proposition "(a → b)" is true in every accessible world, while the proposition "a → b", if asserted in some particular world, just means that the truth values of a and b in that world make a → b true in that world, but it might be false in others.
    – Hypnosifl
    May 15 '21 at 18:27
  • BTW, do you understand the way the material conditional in logic differs from the indicative conditional of ordinary speech? With the indicative conditional, it's usually implicit that if you say something like "if A, then B", the truth of A should in some way be relevant to seeing that B is true; whereas with the material condition A → B there is no such requirement, for example A could be "Triceratops had three horns" and B could be "Jimmy Carter won the U.S. presidential election in 1976". Only if A is true and B is false is A → B false.
    – Hypnosifl
    May 15 '21 at 18:32

It helps to think of the difference between the conditionals in terms of their implications.

1. p → q     can be combined with p to detach q; 
2. □(p → q)  can be combined with □p to detach □q;
3. p → □q    can be combined with p to detach □q. 

1 carries the sense that if p holds, so does q. 2 means that if p holds, q follows by necessity. 3 has it that if p holds, q is itself something that holds necessarily.

Usually, the distinction between 2 and 3 crops up in descriptions of the modal fallacy (or modal scope fallacy). It is common in English to say, "if xxx then necessarily yyy" and this sounds like an instance of 3 when its correct form is 2.

If we interpret □ as 'necessarily' then examples of 2 are easy enough to find. Necessarily, if Jack has a brother, Jack has a sibling. Necessarily, if S knows that p, then p is true. Necessarily, if George Orwell wrote 1984, Eric Blair wrote 1984. It is harder to find good examples of 3, and under some accounts of necessity there are none (except trivially), since it implies that a necessary proposition may follow from a contingent one. We could perhaps find a kind of example of 3 in mathematics. Suppose we interpret the □ operator to mean necessarily true, specifically in the way that mathematical theorems are usually considered to be necessarily true. Now let's choose p to be an unproven conjecture, such as the Goldbach conjecture. Then we may say that if p is true, it is necessarily true, so p → □p. This would not hold in general for contingent propositions, though □(p → p) does.

We could find examples of 3 if we switch modality and interpret □ to mean 'it is obligatory that'. If p is "I make a promise" and q is "I keep that promise" then we have an instance of 3: "if I make a promise, I am obliged to keep it." This is a 3 and not a 2, because we only need the nonmodal p "I make a promise" to detach the obligatory consequent, "I am obliged to keep it". If the conditional were a 2 we would need to combine it with "I am under obligation to make a promise" in order to detach the consequent. (That said, the logic of obligation cannot properly be represented using modal logic like this.)

  • "It is harder to find good examples of 3, and under some accounts of necessity there are none (except trivially)" What do you mean by trivially here? 3 would be satisfied in any case where q is necessarily true (say, because it's a tautology), and would also be automatically satisfied in a world where p is false.
    – Hypnosifl
    May 16 '21 at 2:59
  • Yes, that is what I mean by trivially. In other words where there is no connection between p and □q.
    – Bumble
    May 16 '21 at 12:19
  • For 3 still in necessity: If we are speaking Russian, when I say /mir/ it necessarily means 'peace, world or universe'. If we are speaking English, when I say it, that necessarily means 'only or small'. Presuming languages are defined constructions, which one we are speaking is not forced by necessity, but each of the consequences is necessary once that is known. May 20 '21 at 15:31

In the "accessible worlds" interpretation: A frame of worlds are connected by an accessibility relation. The Necessity quantifier is interpreted as: "□p means p is true in all worlds accessible from the current world".

Frames may be constructed where the current world is not accessible to itself. In such a frame we have that:

□(a → b) is true if in all accessible words, either a is false or b is true. This cannot be satisfied by any frame where there is an accessible world where both a is true and b is false.

a → □b is true if either a is false in the current world, or b is true in all accessible worlds. Now, in frames where a is false in the current world, this may still be satisfied if there is an accessible world where both a is true and b is false.

So the statements are not equivalent in all frames.

  • Well, I know symbollic logic quite well, just am not really accustomed to modal logic. My question was about a real-life example May 15 '21 at 15:33

The notion of necessity cannot be reduced to another notion. In other words, there is no way to say "necessarily A" means "A is blah" where "blah" doesn't include modal language. The in thing today is to reduce the concept of necessity to the concept of possible worlds, but the notion of a possible world is a modal concept that to many of us is no more clear than the concept it supposedly explains.

The only way to really grasp the notion is to read the work of people who use the word and try to gain an intuition for how they are using it. First, it's best to start with "possible" instead of "necessary" because that is the common notion. "P is necessary" just means "it is not possible that P would be false".

You already have a primitive notion of possibility if you can understand sentences like "if it had been a snake it would have bit you" (said to someone who is looking for something and doesn't see it right by his foot). In more formal language, this means, it is possible that there might have been a snake by your foot, and since you didn't look down, it might have bit you.

You ought to have an intuition for what kinds of things might be "possible" in this sense. For example, you might possibly have been born in a different city, you might possibly have owned a cat instead of a dog, it might possibly have rained yesterday instead of been sunny, etc.

Then there are sentences where if someone says it was possible, you don't think it could have been possible. For example, someone says "I'll marry you when pigs will sprout wings and fly". You understand that it is not possible that pigs will sprout wings and fly, so you had better start looking for another mate. This is where the philosophical notion tends to differ from the common notion. A lot of philosophers would say that it is possible that pigs could sprout wings and fly. Why? Because they have a special notion of possibility developed over centuries of discussion. Of course, there are other philosophers who say that it is not possible that pigs will sprout wings and fly, and still others who say that there are different kinds of modality. In some forms of modality (related to the laws of nature), it is not possible that pigs could sprout wings and fly; in other forms (not related to the laws of nature) it is.

There are rational reasons to say that it is possible that pigs could sprout wings and fly. The laws of nature might have been different. Maybe there is a special kind of energy in the universe, and when a planet enters a beam of that energy (maybe sent out by a pulsar), it causes wing mutations and lowered force of gravity. The point isn't that there is any non-zero probability that this is true, just that it is conceivable that the universe might have been that kind of universe. Or it is possible that we are all really living in a computer simulation, and the programmers can do anything they want, including causing pigs to sprout wings and fly.

So what kinds of things are not possible? Well, the traditional answer is that violations of logic and mathematics are not possible. No matter what energies there are in the universe, no matter if the universe is a simulation, it is not possible that 2+2 could equal anything but 4. In other words, necessarily 2+2=4.

So let's get back to your question. What is the difference between, for example, "necessarily if x=2 then x+x=4" and saying "if x=2 then necessarily x+x=4"? In this example, they seem to have the same outcome, but they do mean something different. In one case it is the conditional that is necessary, in the other case it is the consequence that is necessary, but only if the antecedent is true. It might be easier to see the difference from this philosophical claim: "if God exists then he exists necessarily". This seems to mean something more than "necessarily if God exists then he exists". The first sentence seems to say something about God or God's existence, while the second sentence just says that the law P->P is necessary.


Regarding your "how does □(a → b) differ from a → b?", the second conditional is the material conditional in classic logic, and the first is called strict conditional usually needed to be expressed in modal logic such as your example in order to overcome the vague scope of the material kind of conditional. According to reference here:

In logic, a strict conditional is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions p and q, the formula p → q says that p materially implies q while □(p → q) says that p strictly implies q. Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequately express indicative conditionals in natural language.

Now a real-life example: a=Paris is in England, b=Mary knows who is Socrates. In classic logic material condition is a truth-function connective that can be used to create new propositions. So in this case a → b creates a new proposition which translates as "if Paris is in England, then Mary knows who is Socrates". We all agree a=false, b=either true or false, however, according to classic logic the truth value of the new proposition is true which actually makes sense.

But we all agree whether Mary knows who is Socrates or not should have nothing to do with whether Paris is in England or not. So how can we express only those indicative conditional to exclude the above nonsensical proposition? Lewis devised strict conditional using modal constraint as above, so you cannot form the new proposition using example above, only those indicative condition make sense. Now we can change a=Mary studies philosophy, then both a → b and □(a → b) hold. So in this modal way, □(a → b) is more restricted and much closer to being able to express natural language conditionals.

Please note strict conditional expressed in modal logic is not perfect, according to the same reference:

Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems with consequents that are necessarily true (such as 2 + 2 = 4) or antecedents that are necessarily false... Some logicians view this situation as indicating that the strict conditional is still unsatisfactory.

Finally regarding the subtle difference between □(a → b) and a → □b, this is easiest to understand in Deontic logic referenced here where □ should be interpreted as "ought to be":

When we try to formalize ethics with standard modal logic, we run into some problems. Suppose that we have a proposition K: you have stolen some money, and another, Q: you have stolen a small amount of money. Now suppose we want to express the thought that "if you have stolen some money, it ought to be a small amount of money". There are two likely candidates, (1) K → □Q (2) □(K → Q)

As explained in the reference, if choosing form (2) as your deontic logical form, then it turns out that telling someone they should not steal deontologically implies that they should steal large amounts of money if they do engage in theft while form (1) does not have such problematic entailment.

Another example can be construed in the context of de dicto and de re modal scope difference: Let a=Bachelors, b=unmarried. Then □(a → b) correctly expresses English meaning which is Bachelors are necessarily unmarried or Necessarily bachelors are unmarried as tautology interpreted de dicto (by speech), however, it's false interpreted de re (by thing) since some bachelor can marry later. But a → □b in this case can express English meaning which is Bachelors enumerated by a,b,.,z are unmarried and never will be married by the added necessity interpreted de re, since a,b,..,z are rigidly referred to the same bachelor respectively in all possible worlds. Most people are naturally against such weird de re interpretation in modal logic including Quine and Lewis, while renowned logician Kripke is a proponent for it and applied it in his work.

  • "So in this modal way, □(a → b) is more restricted and much closer to being able to express natural language conditionals"--note however that while it matches better with how we use the indicative conditional in cases where "a" and "b" are both contingent propositions that are each true in some possible worlds but not others, it differs from our intuitions if you pick an "a" that's false in all possible worlds, or a "b" that's true in all possible worlds; in that case □(a → b) is guaranteed to be true even if there seems to be no conceptual or causal connection between "a" and "b".
    – Hypnosifl
    May 15 '21 at 20:37
  • @Hypnosifl thx for your comment. I've added the cautionary criticism part of strict implication from the same reference above about your concerns. The strict conditional analysis encounters many known problems, notably monotonicity. This fact led to widespread abandonment of the strict conditional, in particular in favor of Lewis's variably strict analysis. May 15 '21 at 20:57

Aristotle's example here is about the conflict of time and free will in The Sea Battle. So sometimes the best framing is in the science-fiction of time travel.

The interpretation of box (a implies b) is that a implies b in every timeline. If a happens, then b absolutely has to happen, whatever else we change in multiple attempts to force the timeline.

If a lit match is applied to a puddle of gasoline in the presence of ordinary air, it ignites. This is true in all timelines.

Without the box, you are discussing only current temporal reality (I find it amusing that we even call this 'temporal logic'). It happens that right now, in our current timeline when a happens b happens. But we might be able to rearrange things so that b would not be the consequence of a if we rewound time and altered other conditions.

Whether a specific lit match lights a given puddle of gasoline could be changed by going back and removing all the air.

Of course you can quibble with the example. Just as the rules of science-fiction time travel vary, so do approaches to necessity in different kinds of modal logic, and these are mediated through the notion of 'accessibility'. Can you go all the way back to the Big Bang and change how chemistry works by tweaking the energy separation in the first three seconds of time? Or can you only imagine going back to the formation of the Earth and moving the continents around so that we did not have oil and there is no gasoline? From a normal standpoint, neither, and the example stands.

Theoretically, we all have the same notion of how these play out, even if we differ in our approaches to accessibility, so we can agree on the basic rules about deductions in modal logic, even if we disagree on the ultimate interpretation.

(This is just a slow-motion version of the solution via possible worlds and the accessibility of interpretations, but I somehow find it clearer in my own mind. And it accords homage to one of the greats.)

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