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This is something I am wondering about, because all the formulations I've seen of the logic seem fairly difficult to grasp, e.g. lists of abstract axioms that have a few missing versus classical logic. Yet, I've had the following idea floating around and wonder if it could work.

For me, ordinary classical logic is "more intuitive" because I can put it to a concrete evaluation algorithm, in the sense that if you know whether the elementary facts (atomic variables) of a given situation are true or false, you can easily evaluate the truth of a compound statement about them by using truth tables, which also have simple, intuitive interpretations, e.g. that "and" is only true when both its inputs are true, which makes sense from casual usage in English. Yet, as is well-known, for intuitionistic logic, not only is this not possible, but any "truth-functional", i.e. truth table-based approach must utilize a system with infinitely many truth values. Hence, what I am wondering about is - is there some non-truth functional "evaluation algorithm" for intuitionistic logic?

To illustrate what I mean. For the infinite truth values approach, one such system that works is to use the open subsets of $(0, 1)$ (I believe), where a full subset (i.e. $(0, 1)$ itself) is maximum truth, an empty subset is maximum falsehood, and the "middle ground" is every other subset, where many subsets are incomparable as truth values - we can only say one is "more truthful" than another if the former includes the latter as a subset, thus forming in effect a sort of fibrated structure to the truth values. This is, admittedly, a bit more "intuitive", but what would it mean for a realistic proposition to have, say, a truth value of $(0, 0.5)$ or $(0, 0.1) \cup (0.2, 0.9)$?

Now I suppose that, if one interprets the underlying epistemic basis of intuitionist logic as starting from that "truth means verifiability", i.e. to say that a proposition $P$ is "true" means we can prove it to be true, and to say that a proposition $P$ is "false" means we can prove it to be false, these may represent, perhaps, states of incomplete proof, i.e. they correspond to a situation in which we have some, but not all, evidence a given proposition is true, and perhaps we can understand the subsets of the interval as "coloring in" different areas where we have and don't have evidence with a suitable understanding of what the different numbers in the interval mean. After all, topologies or, better, open sets, have been described as generally specifying "semi-decidable properties", i.e. properties for which an assertion of their truth can be verified.

In addition, we perhaps could use more complicated topological spaces, too, as they will also form Heyting algebras in the same fashion, which then gives a ready example of what I'm talking about. Suppose an atomic proposition is "The car is red". Now we have a "car-shaped" topological space (i.e. a surface in $\mathbb{R}^3$ shaped like at least the hull of the car), which then induces a Heyting algebra. We've not seen the whole car, but only its back end, and seen that yes, it is red. Thus we should valuate that atomic formula to the open subset consisting only of the back end of the car. Would this make any sense? In this regard, it would become a richer version of fuzzy logic, where we can only give a percentage (0% to 100%) in which something is true.

But if it is not sensible, and also for its own sake, I will also say what I mean by a non-truth functional approach, which was the original thrust of this question. First, it would seem natural and intuitive to, given that the "excluded middle" is rejected, first expand the set of truth values to include an additional "unknown" or "unverifiable" value (U), in addition to "true" and "false". Now we have three, but not two, truth values. It is clear this is insufficient for a truth-functional interpretation. Hence, what I wonder then is if we can come up with some algorithm - which need not be efficient - that:

  1. Takes as input an intuitionistic propositional formula,
  2. Takes as input an assignment of T/F/U values to each atomic variable,
  3. Evaluates the formula to the correct T/F/U value. Most importantly, that tautological formulas always evaluate to T and contradictory ones always to F.

Is such an algorithm possible? If not, why not? If so, what does it look like?

amWhy
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    Is your question just about decidability or about semantics as well? If so, intuitionistic propositional calculus is decidable. As for semantics, in addition to the topological semantics you referred to, IPC can be inductively translated into S4, which, as a normal modal logic, has a possible world semantics (in addition to the topological one). – Greg Nisbet Jan 07 '22 at 04:31
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    Have you looked at the Brouwer–Heyting–Kolmogorov interpretation of intuitionistic logic? – Zhen Lin Jan 07 '22 at 05:04
  • You really can't go wrong with classical logic. – Dan Christensen Jan 07 '22 at 05:06
  • If you have some problems with classical logic, I suggest you work them out before shopping around for another system. – Dan Christensen Jan 07 '22 at 05:15
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    Another vote for the BHK interpretation, which I think is highly intuitive. – MJD Jan 07 '22 at 05:18
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    @MJD: Hmm I think Kripke semantics is the most intuitive for intuitionistic logic. See this question which we can answer easily using Kripke frames but (to my knowledge) not easily using the BHK interpretation. – user21820 Jan 07 '22 at 13:46
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    It is not clear to me what you imagine "the correct T/F/U value" to be. – Pilcrow Jan 07 '22 at 13:57
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    The benefits of classical logic notwithstanding, all the preceding comments about classical logic are utterly irrelevant to this thread. – user21820 Jan 07 '22 at 19:45
  • @ZhenLin Is there actually a formal interpretation of BHK that is complete for intuitionistic logic in a classical context (the same way classical Kripke frames are complete for intuitionistic logic)? – James Hanson Jan 06 '24 at 01:28

3 Answers3

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There are a variety of semantics you can give to different systems.

You seem to be arguing intuitionistic logic has a more complicated denotational semantics than classical logic. A huge motivation behind constructive logics today is the fact that they can be given useful operational semantics. It is sort of possible to give an operational semantics to law of excluded middle in terms of control flow but the matter is confusing.

Ultimately I think it is a goal of constructive logics to have more complicated denotational semantics than classical logic. Otherwise such logics could not be interpreted into more complicated structures than boolean values.

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    I am curious: what would be the operational semantics here? And what would such a more complicated denotational semantics be for intuitionistic logic? – The_Sympathizer Jan 07 '22 at 10:18
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    @The_Sympathizer so I am more familiar with the cases of substructural logics and co-intuitionistic logic because those aren't boolean but basically you can think of LEM operationally in terms of continuations, labels and exceptions. I am most familiar with stuff like denotational semantics for extra logical features like nontermination. I struggle to think of denotational semantics for classical logic vs intuitionistic. – Molly Stewart-Gallus Jan 07 '22 at 17:40
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Topological semantics is indeed one natural way to look at intuitionistic logic, but your line of thinking with the car is based on a misinterpretation of the topological semantics. You seem to be focused on the idea that intuitionistic logic has infinitely many truth values, but it really doesn't (except in a technical algebraic sense which has little relevance to your request for an intuitive understanding of intuitionistic logic). Intuitionistic propositions have two truth values, just like classical ones, namely verified and not verified (or if you will, asserted and not asserted).

To understand how intuitionistic logic is the logic of open subsets of topological spaces, we first need to understand how classical logic is the logic of arbitrary subsets of arbitrary sets. In the classical case, the ambient set is the set of all possible worlds, or state of affairs, or outcomes of some experiment or observation, or something along those lines. A proposition is determined by its being true in some possible worlds and false in others. A proposition in classical logic is therefore interpreted as a subset of the set of all possible worlds, namely the set of all those worlds where it is true. Of course, if propositions $p$ and $q$ are interpreted by some subsets $X$ and $Y$ of the set of possible worlds $W$, then their conjunction $p \wedge q$ is interpreted by $X \cap Y$, whereas the negation $\neg p$ of the proposition $p$ is interpreted by $W \setminus X$.

Sticking to your car example, the proposition "this car is red" would be interpreted by the set of all possible worlds (however you conceive of those) where the color of the car, which I suppose for the sake of simplicity to be completely uniform, lies in some specified range. We can simplify this model by saying that the only relevant feature of the world that we are interested in is the color of the car, in which case the ambient set can perhaps be represented as the interval of wavelengths measured in nanometers $(0, \infty)$ and the proposition "this car is red" would be interpreted by some subinterval. It could be a closed interval, or an open one, or neither. Let us say that it is the open interval $(650, 700)$.

Now, for classical logic, this semantics is an overkill. This is because to determine whether a complex proposition of classical logic is true at a given world $w \in W$, all you need to know is whether its atomic subpropositions are true at the given world. In other words, if you are only interested in truth in the actual world, there is no reason whatsoever to bring other possible worlds into consideration. You can always take the ambient set to be a singleton.

This is not so with intuitionistic logic. To obtain a semantics for intuitionistic logic, we assume that the set of all possible worlds is in fact a topological space, the open sets representing things that you can verify (semi-decidable properties, in your terminology). We assume that each proposition is interpreted by an open set. Now suppose that I pick a possible world $w$ and a proposition $p$, e.g. your proposition "this car is red" again. Think of $p$ as a measuring stick that I apply against $w$. What could possibly happen?

If $p$ is verified in $w$, then we are good. The car is red. In other words, the measured wavelength lies somewhere in the interval $(650, 700)$.

If $p$ never gets verified, there are two possible reasons for this. One is that we can verify that the wavelength lies outside of $(650, 700)$. How could we possibly verify that? Well, our only measuring sticks are open sets, so the only way to verify that the wavelength lies outside of $(650, 700)$ is to verify that it lies in $(0, 650)$ or $(700, \infty)$. In this case, we say that $\neg p$ is verified. More formally, $\neg p$ is true in those worlds which lie in the interior of the complement of the open set of worlds where $p$ is true.

Of course, the remaining possibility is that we can verify neither $p$ or $\neg p$. The measured wavelength lies just on the borderline. We keep measuring for more and more precision, but we get $650.0$, $650.00$, $650.000$ and so on. At no point will we verify that the wavelength lies in $(650, 700)$, but at no point will we also be certain that this is not the case. Thus, the possible worlds where neither $p$ nor $\neg p$ can be verified may be thought of as the borderline cases, if you will. I hope that clarifies things at least a little.

You could also think of this as running parallel computational processes for $p$ and $\neg p$: one outcome is that $p$ successfully terminates, another outcome is that $\neg p$ successfully terminates, but it could also happen that neither process will terminate, in which case you are left in the dark about whether the point lies inside the set or not.

tl;dr: you should not think of open sets as truth values and then boggle your mind by thinking about what it could possibly mean to talk about the truth value $(0, 0.1) \cup (0.2, 0.9)$. Rather, you should think of them as interpretations of propositions in different worlds, scenarios, states of affairs, etc. Each proposition has only two possible truth values at each world: either the world lies inside the given open set, or it doesn't.

Pilcrow
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  • Hmm. The first part, where you mention it has "only two truth values" kind of gets to what I was trying to get at in the post - is there some non-truth functional interpretation that lets us make sense of it with three truth values, which seems more natural for the case when we can verify neither $p$ nor $\neg p$? – The_Sympathizer Feb 02 '22 at 01:26
  • So then in this regard the Heyting algebra is not actually an analogue of a Boolean algebra, then, right, because the latter does have an interpretation as "truth values", while for this, that interpretation fails? – The_Sympathizer Feb 02 '22 at 01:29
  • @The_Sympathizer You can think of the Kripke semantics or the topological semantics as non-truth functional (it would be more accurate to call it non-extensional) in the sense that the truth value of a complex proposition at a given possible world is not a function of the truth values of its component at that world. In either case, there is no benefit to the third value, each proposition is either verified or not verified at a given world. If you want to insist on calling a situation where neither $p$ nor $\neg p$ is verified a third value, be my guest, but it adds nothing to the theory. – Pilcrow Feb 02 '22 at 12:48
  • @The_Sympathizer No, Heyting algebras are perfectly analogous to Boolean algebras. After all, you could equally well take the Boolean algebra of all subsets of $[0, 1]$ and then boggle at what it could possibly mean for a classical proposition to have the truth value $(0, 0.1) \cup (0.2, 0.9)$ in this Boolean algebra. What's happening here is probably that when you think about Boolean algebras, you really only think about the two-element Boolean algebra, whereas when you think about Heyting algebras, you're thinking about all of them. – Pilcrow Feb 02 '22 at 12:49
  • So then why can a Boolean algebra be understood as truth values in one case but the Heyting algebras can not in any case? – The_Sympathizer Feb 08 '22 at 03:32
  • @The_Sympathizer There are Heyting algebras that can be thought as algebras of truth values. Heck, any algebra can – if you can supply an intelligible story explaining what exactly makes it an algebra of truth values. An example is the interval $[0, 1]$ viewed as a Heyting algebra. This is a natural algebra of truth values extending ${ 0, 1 }$. It's just that the logic of this algebra is not intuitionistic logic, but something stronger (the Goedel–Dummett logic). – Pilcrow Feb 08 '22 at 10:39
  • So why does the space of topological open subsets not make sense as truth values for intuitionistic logic with the story given (i.e. that they represent "what parts of the total evidence base required to assert a proposition as true, that you actually have, via some suitable mapping")? – The_Sympathizer Feb 12 '22 at 23:23
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    @The_Sympathizer Well, if that makes sense to you, then that's fine with me. shrug But the question that you originally asked seems to imply that it doesn't (or at least didn't). In any case, I don't think we are getting anywhere in this discussion, so I am going to stop here. – Pilcrow Feb 13 '22 at 00:15
  • I just came back to this. It seems to make some more sense now on second reading, but it also seems to imply a virtually-complete shift away from the meaning of the valuation in classical logic. That is, that the valuation is now itself a proposition, as opposed to an assertion of the truth of a proposition. This seems to make it less natural to use such an interpretive framework to handle classical logic as a special case of intuitionistic logic (i.e. where we enforce LEM), and so what I think I was trying to go for was a framework that would not suffer from that apparent problem. – The_Sympathizer Jan 14 '24 at 18:50
  • That is to say, in classical logic, we might decide that atomic propositional variable, say $p$, means "the car is red" (here, $\lambda_\mathrm{refl} \in (650, 700)\ \mathrm{nm}$). Two truth values. But here, it seems $p$ now has been changed semantically to something like "what is known to be true about the color of the car", and not that specific proposition, given we can assign it any topologically open subset. And what I was after was an interpretation where we can retain something like the classical semantic assignment of the variable, just changing the semantics of the valuation. – The_Sympathizer Jan 14 '24 at 18:53
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Hence, what I wonder then is if we can come up with some algorithm - which need not be efficient - that:

  1. Takes as input an intuitionistic propositional formula,
  2. Takes as input an assigment of T/F/U values to each atomic variable,
  3. Evaluates the formula to the correct T/F/U value. Most importantly, that tautological formulas always evaluate to T and contradictory ones always to F.

Is such an algorithm possible? If not, why not? If so, what does it look like?

Let $p$ be the given formula. For every variable $x$ assigned T, let $p := x \to p$. For every variable $x$ assigned F, let $p := \neg x \to p$. The problem then reduces to checking whether $p$ is a tautology (T), $\neg p$ is a tautology (F), or neither (U). SEP says

The decidability of IPC [intuitionistic propositional logic] was first obtained by Gentzen in 1935.

referencing [1]. A tighter tesult is that this problem is PSPACE-complete [2]:

There is a polynomial time translation of intuitionistic logic into the modal logic S4 due to Tarski (see Fitting [3, p.43]). Ladner [4] shows that S4 can be decided in p-space, so the problem is p-space complete.

See this question for more details.


  1. Gentzen, G., 1934–5, “Untersuchungen Über das logische Schliessen,” Mathematische Zeitschrift, 39: 176–210, 405–431.

  2. Intuitionistic propositional logic is polynomial-space complete. Richard Statman. Theoretical Computer Science 9 (1):67-72 (1979).

  3. Intuitionistic Logic, Modal Theory and Forcing. Melvin Fitting. North-Holland, Amsterdam (1969).

  4. The computational complexity of provability in systems of modal propositional logic. Richard E. Ladner. SIAM Journal on Computing 6 (3):467–480 (1977).

user76284
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  • If it’s not too involved for you, Would you mind defining for me what exactly is indicated by , ⊩, and for this matter ≤, with respect to (﹖some structure defined on )model ?, at least as it pertains to formula ? And what is the relationship between and , with respect to a proposition describable under the model and ascertainment of its most precise valid determinable truth value? – user946772 Jan 13 '22 at 21:11
  • @11qq00 See page 2 of the linked PDF. – user76284 Jan 13 '22 at 22:34
  • Thanks. This actually was exactly what I ended up coming up with - except for the tautology algorithm - but didn't get around to posting it. It can be understood through the "truth as proof" philosophy in the intuitionistic logic. I am curious, though - how is it that it can even be done in finite time at all if $W$ is presumably an arbitrary set that could even be infinite and thus impossible to loop over? – The_Sympathizer Jan 14 '22 at 09:11
  • @The_Sympathizer The approach of reducing IPC to S4 and deciding S4 seems rather involved. I've linked to another question with simpler approaches (that might not be as efficient). – user76284 Jan 14 '22 at 20:55