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by Corry Shores
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[The following is summary. All boldface in quotations are mine unless otherwise noted. Bracketed commentary is my own, as are paragraph enumerations, which follow the paragraph divisions in the text. As proofreading is incomplete, you will find typos and other districting errors. I apologize in advance.]
Summary of
John Nolt
Logics
Part 5: Nonclassical Logics
Chapter 16: Radically Nonclassical Logics
16.2 Intuitionistic Logics
16.2.1–16.2.29
[Basic set-up of Kripkean intuitionistic semantics]
Brief summary:
Many logics base their notion or evaluation of truth on the correspondence of thoughts with the facts of a real world. Intuitionistic logic, however, holds that the “truth” or “falsity” of a statement is determined by whether or not the available data warrant us to assert it. We call this warrant, proof, or evidence confirmation. To assert a proposition means we have warrant to confirm it. And to assert its negation means we have warrant to say that it is not the case. In our formalized intuitionistic semantics, we assign to propositions one of two values: {1} confirmed, ‘C’, when it is warranted to assert it, and {2} unconfirmed, ‘U’, when it is not warranted to assert it. But there are two sorts of being unconfirmed for a proposition, namely, being either refuted (that is, confirming its negation) or being neither confirmed nor refuted (that is, the proposition is neither confirmed while its negation is not confirmed). One example of this “neither” status is the following. Consider the proposition P that there are seven consecutive sevens somewhere in the decimal expansion of the number π. Now suppose that no computer or human working on determining as many decimals of π as possible, after whatever extent of time, have discovered seven consecutive sevens. That means we do not have grounds to confirm P. But we also have no proof that there are not seven consecutive sevens, so we cannot confirm ~P. Another example has to do with historical statements which cannot be confirmed or denied, like “Napoleon ate breakfast on September 9, 1807.” Now, if we can neither confirm P nor confirm ~P, this means we cannot confirm P∨~P (which would be understood in intuitionistic logic to mean “‘P’ is either confirmed or refuted”). Thus the law of excluded middle does not hold in intuitionistic logic. We also cannot make double negation inferences. In other words, the intuitionistic value of ~~P is not necessarily the same as P. ~~P means that it is refuted that P is refuted. (But consider again the π example above. Here we can say that it is refuted that P is refuted, or ~~P, because we can know that no one has refuted it yet. So ~~P = C. But we also cannot confirm P, meaning that P = U. So we cannot derive P from ~~P in this instance by simply changing the value twice. Also it is possible to modify the example such that P = U and ~~P = U. The computer may have calculated the seven consecutive sevens, but on account of some technical problem, we cannot ever obtain that information from the computer’s processes. In other words, not only does negation not work by doubly switching the values, it also does not produce a consistent pattern of valuation in all cases). Now, in three-valued logics, we can always derive P from ~~P. So this is one way that distinguishes intuitionistic logic from three-valued (and also classical) logics. This in part leads us to regard the “neither” value (neither confirmed nor denied) as not being a third value but rather as a subspecies of non-confirmation. To more formally evaluate propositions in this bivalent way in intuitionistic logic, we need to consider the following concepts. One is “state of evidence” or “evidential state”, which is “the total evidence available to a person or culture at a given time” (431) and which are structurally similar to possible worlds in Kripkean model semantics. On the basis of evidential states, propositions are determined to be either confirmed or unconfirmed. A first evidential state is said to be consistent with (or have access from) a second one if all the confirmation assignments of the first are given as well in the second, even though the second may have additional confirmations not in the first. In intuitionistic semantics, it is stipulated that when something is confirmed in a first evidential state, it is confirmed in all others that the first is consistent with (those from which the first is accessible). This is the epistemic necessity of confirmation.
A Kripkean model in intuitionistic predicate logic is defined in the following way:
DEFINITION A Kripkean valuation or Kripkean model v for a formula or set of formulas of intuitionistic predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds or evidential states of v.
2. A reflexive, transitive relation ℛ, consisting of a set of pairs of worlds from Wv.
3. For each world w in Wv a nonempty set Dw of objects, called the domain of w, such that for any worlds x and y, if xℛy, then Dx is a subset of Dy.
4. An assignment to each name α in that formula or set of formulas of an extension v(α) that is a member of the domain of at least one world.
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5. An assignments to each predicate Φ and world w in that formula or set of formulas, an extension v(Φ, w) such that
i. If Φ is a zero-place predicate, v(Φ, w) is one (but not both) of the values C or U such that if v(Φ, w) = C, then for all u such that wℛu, v(Φ, u) = C.
ii. If Φ is a one-place predicate, v(Φ, w) is a set of members of Dw such that if d ∈ v(Φ, w), then for all u such that wℛu, d ∈ v(Φ, u).
iii. If Φ is an n-place predicate (n>1), v(Φ, w) is a set of ordered n-tuples of members of Dw such that if <d1, ..., dn> ∈ v(Φ, w), then for all u such that wℛu, <d1, ..., dn> ∈ v(Φ, u).
(432-433)
The following are the confirmation conditions for complex formulas in intuitionistic logic.
Let Φ and Ψ be any formulas and v any valuation of Φ and Ψ whose accessibility relation is ℛ. Then
1.
If Φ is a one-place predicate and α is a name, then v(Φα, w) = C iff v(α) ∈ v(Φ, w);
v(Φα, w) = U iff v(α) ∉ v(Φ, w).
2.
If Φ is an n-place predicate (n>1) and α1 ... , αn are names, then
v(Φα1, ... , αn, w) = C iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = U iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3.
If α and β are names, then v(α = β, w) = C iff v(α) = v (β);
v(α = β, w) = U iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = C iff for all u such that wℛu, v(Φ, u) ≠ C.
v(~Φ, w) = U iff for some u such that wℛu, v(Φ, u) = C.
5.
v(Φ & Ψ, w) = C iff both v(Φ, w) = C and v(Ψ, w) = C.
v(Φ & Ψ, w) = U iff either v(Φ, w) ≠ C or v(Ψ, w) ≠ C, or both.
6.
v(Φ ∨ Ψ, w) = C iff either v(Φ, w) = C or v(Ψ, w) = C, or both.v(Φ ∨ Ψ, w) = U iff both v(Φ, w) ≠ C and v(Ψ, w) ≠ C.
7.
v(Φ → Ψ, w) = C iff for all u such that wℛu, v(Φ, u) ≠ C or v(Ψ, u) = C, or both.
v(Φ → Ψ, w) = U iff for some u such that wℛu, v(Φ, u) = C and v(Ψ, u) ≠ C.
8.
v(Φ ↔ Ψ, w) = C iff for all u such that wℛu, v(Φ, u) = v(Ψ, u).
v(Φ ↔ Ψ, w) = C iff for some u such that wℛu, v(Φ, u) ≠ v(Ψ, u).
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9.
v(∀βΦ, w) = C iff for all worlds u such that wℛu and for all potential names α of all objects d in Du, v(α,d) (Φα/β , u) = C;
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v(∀βΦ, w) = U iff for some world u such that wℛu, and some potential name α of some object d in Du, v(α,d) (Φα/β , u) ≠ C.
10.
v(∃βΦ, w) = C iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = C;
v(∃βΦ, w) = U iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ C.
(Nolt 433-434)
Negation means refutation, which is more than non-confirmation in one evidential state. It means that there are no evidential states accessible from the first where the unrefuted formulation is confirmed. It is similar to impossibility in modal logic. Along with negation, the conditional, biconditional, and universal quantification require comparison with other evidential states accessible to the first. But conjunction, disjunction, and existential quantification need only the evidential state in question. A formula is intuitionistically valid if all evidential states confirm it. It is intuitionistically inconsistent if there is at least one evidential state that does not. A sequent is intuitionistically valid if there are no evidential states where the premises are confirmed while the conclusion is not.
Summary
16.2.1
[According to some, we do not have direct access to real things and situations but only to our impressions of them, and thus our criteria for evaluating truth should as well be based on such reception.]
We will now deal with a type of logic that is based on a very different sort of claim regarding our beliefs about how our statements can be said to be true. In all the other logics, we assumed that our statements could be said to correspond adequately or not to real situations in the world. But intuitionist philosophers “doubt that we can ever attain truth in this sense of” “a kind of correspondence between propositions or thoughts and reality” (427). Nolt gives an example of thinking soup is boiling, then going to see that it is. But we only ever access our perceptions of the soup, and not the soup itself. Because we do not have access to the reality to which our propositions correspond, we “can never know the relation between my thought and reality itself.” [Since truth lies in the adequate correspondence between thought and reality] we thus “can never know truth”. [His final sentence here is: “And if truth is something unknowable, then perhaps our semantics should be based on something more empirical” (428). I am not sure I grasp it entirely yet, but it might be the following. Recall the notion of semantics. Nolt writes in section 3.1 “Semantics is the study of meaning. The logical meaning of an expression is usually understood as its contribution to the truth or falsity of sentences in which it occurs that we therefore should not base our logical semantic” (39, boldface his). The semantics of a logic, recall, is often given as a set of rules for how to assign truth-values to propositions (see for example section 11.2.1 and section 12.1; or Agler Symbolic Logic section 6.4). His point might be that we should assign the truth-value of statements on the basis of certain empirical factors, like the evidence our senses give us.]
All the logics thus far considered have as their fundamental semantic concept the notion of truth. Truth is usually understood as a kind of correspondence between propositions or thoughts and reality. But many philosophers, believing that we have no access to reality as it is in itself, doubt that we can ever attain truth in this sense. I may think that the soup is boiling and then go to the stove and see that it is. In this sense I may confirm my thought that the soup is boiling. But my seeing or hearing (or even touching) the soup does not, so this line of reasoning goes, reveal the soup as it is in reality, but only the soup as I see or hear or feel it. I may, | in other words, compare my thought with the soup as perceived by me, but never with the soup as it is in itself. But if I can never know the relation between my thought and reality itself, then I can never know truth. And if truth is something unknowable, then perhaps our semantics should be based on something more empirical.
(427-428)
16.2.2
[Intuitionism bases truth not on the correspondence between thought and reality but rather between thought and evidence that gives proof, warrant, or confirmation for the thought.]
So instead of basing our evaluation of truth on the basis of a relation between thought and reality, we might instead base it on thought and evidence, as intuitionists suggest. We can see for example if the perceptions or intuitions we have warrant the claims we make.
One suggestion is to base our semantics on relations not between thought and reality, but between thought and evidence – relations such as proof, warrant, or confirmation. My perception of the soup is a form of evidence that proves, warrants, or confirms my thought or assertion that the soup is boiling. Thus thought or assertion, which I experience, is compared with evidence, which I also experience, rather than with reality or the world, which I allegedly never experience as it is in itself. Another word for such direct experience is ‘intuition’. Accordingly, the resolve to restrict semantics to entities that can be made evident to direct experience is called intuitionism.
(428, in the original, only ‘intuitionism’ is in bold. Here underlining and other boldface is added.)
16.2.3
[Intuitions may claim either that the real world is irrelevant to semantics or that there is no such thing.]
The traditional notion of truth supposes a world in itself. Intuitionists say that we never experience the world in itself, thus we must reject the traditional notion of truth. Intuitionists may claim either that the world in itself may exist but is irrelevant to semantics, or they may go further and say that there is no such world in itself.
Intuitionists reject the traditional notion of truth because it posits a world-as-it-is-in-itself – something we never experience as such – as that to which thought or assertion corresponds. For the intuitionist this world-in-itself is irrelevant to semantics. Having gone this far, one might even be tempted to conclude that there is no such thing as the world as it is in itself. Some prominent philosophers have drawn this conclusion.
(428)
16.2.4
[Intuitionists might claim that a certain kind of reality does not exist. Brouwer thought that mathematical objects are constructions.]
[I will not be able to state the next point with certainty, so read the quotation to follow. The idea is that for some intuitionist philosophers, there is no such thing as reality, but they mean a certain limited sort of reality. Brouwer holds such a position, but I am not sure how it works in this case. It might be that for him there is no mathematical reality but only our constructed mathematical objects. But I am not sure if he is saying there is a real world but that it does not really have mathematical objects in it.]
Others have accepted this conclusion only in regard to certain kinds of reality. The originator of intuitionism, L. E. J. Brouwer, was concerned primarily with mathematics, not with the world in general. He held that mathematical objects (numbers, functions, sets, etc.) exist only insofar as we construct them or define the means for their construction. The propositions of mathematics, then, are not true in the sense of corresponding to some independently existing reality, but rather merely confirmable, refutable, or neither, by the evidence of our calculations and proofs.
(428)
16.2.5
[Michael Dummett has a less subjectivistic intuitionist theory of meaning where the meaning of terms is a matter of the publically observable conditions that warrant its usage or “assertibility”.]
[I am again not certain I can accurately restate the next ideas. They might be the following. Michael Dummett thought that we learn the meanings of terms by seeing how they are used and associating the terms with the conditions that warrant their usage in such situations. These conditions must be publically observable, or else a linguistic community would be unable to learn them. This is supposed to be a less subjectivistic sort of intuitionism. (Where this is a little unclear for me is that publically observable conditions could suggest a real world that is perceived commonly by different people). Let me quote:]
Lately others – most notably Michael Dummett – have held that even the semantics of ordinary discourse is best understood as intuitionistic. Dummett’s view is less “subjectivistic” than that of earlier intuitionists. He holds that what constitutes the meanings of terms must be publicly observable; otherwise, our ability to learn language would be inexplicable. Thus we learn the meanings of terms by observing them in use and so associating them with publicly evident assertibility conditions. Meaning, then, is constituted, not by truth conditions (for truth, as we know, is not always publicly evident!), but by assertibility conditions. To know the meaning of a term is to know the publicly evident conditions under which it is appropriate to assert various sentences containing it.
(428)
16.2.6
[“Confirmation” is warrant or evidence for asserting a proposition. Also, we can assert the negation of a proposition if we are warranted to do so (by disproof or other evidence).]
So all types of intuitionism “Replace the traditional concept of truth with some notion of warrant or evidence” (428). Nolt says we will “use the general term confirmation to stand for all of these notions of warrant or evidence” (428, boldface in original). But this means that we are slightly changing the meaning of formulas and statements: “to assert a formula Φ is to say not that it corresponds to reality, but that it is evident, warranted, or (in our jargon) confirmed” (428). In mathematical contexts, to assert Φ means that either we have proof of Φ, “in more ordinary contexts” it means that we are warranted to assert it (429). This then also means that to assert ∼Φ “is to say that Φ is refutable” (429). In mathematical contexts, to assert ∼Φ “means that we have a disproof of Φ” which often involves deriving a contradiction from it; and in everyday contexts, it means that we are warranted to deny Φ (429).
16.2.7
[This intuitionist understanding of warrant or confirmation means that certain notions that are intuitive in classical logic, like the principle of excluded middle, do not hold in intuitionist logic. P∨~P means in intuitionism: ‘P’ is either confirmed or refuted. But we have not yet been able to calculate whether or not there are seven consecutive sevens in the number π. So we cannot confirm or deny the claim that there are, which means we cannot say that P∨~P holds necessarily.]
There is a problem with this idea, however. Not all propositions can be confirmed or refuted. Nolt has us consider one of Brouwer’s examples, a proposition that Nolt names “P”:
There are seven consecutive sevens in the decimal expansion of the number π, that is, 3.141....
(Nolt 429)
We now suppose that no human or computer has yet found seven consecutive sevens. [Since we do not have the seven consecutive sevens, that means we cannot confirm it, so we are not warranted to assert P. However, we also do not have a proof to show there cannot be seven consecutive sevens, that also means we are not warranted to assert ~P. Now consider P∨~P. Normally we would say this is valid, because it cannot be neither P nor ~P. Regardless of whatever value we assign to P, it will always come out true. But now we are using a different way to assign truth. The meaning of P∨~P here means: “ ‘P’ is either confirmed or refuted”. But it is neither. Thus we cannot have excluded middle as a certainty.]
But not every proposition is either confirmed or refuted; indeed, not every proposition can be either confirmed or refuted, even in principle. Consider, to use one of Brouwer’s examples, this proposition, which we shall symbolize as ‘P’:
There are seven consecutive sevens in the decimal expansion of the number π, that is, 3.141....If there are seven consecutive sevens, say, in the first million (or billion or trillion) digits of this decimal expansion, then we can know conclusively that there are simply by calculating that many digits. In fact, though I don't, someone may know this already, since π has in fact been calculated by computers to many millions of digits. But suppose that no matter now far we calculate, we never find seven consecutive sevens. Then, since neither we nor our computers (both being finite creatures) can calculate all the infinitely many digits of this decimal expansion, then (provided there is no noncalculational way to decide the question-and there does not seem to be) proposition P is in that case neither confirmable nor refutable, even in principle. There can, in other words, never, not even in principle, be sufficient evidence to confirm the disjunction ‘P∨~P’, whose intuitionistic meaning is that ‘P’ is either confirmed or refuted. Hence intuitionists reject, even as an ideal, the law of excluded middle, Φ∨∼Φ, which they read as asserting that Φ is either confirmed or refuted.
(429)
16.2.8
[Situations where we have warrant neither to assert nor deny a claim are found often in such non-mathematical situations like statements about historical facts that would be very difficult or impossible to know; for example: “Napoleon ate breakfast on September 9, 1807.”]
Nolt says that we encounter in non-mathematical situations these sorts of cases where we can neither confirm nor deny something. He gives the example of a statement about the past, “that Napoleon ate breakfast on September 9, 1807.” From the perspective of classical logic, this statement is either true or false (429). But intuitionists are not so much interested in this notion of its truth or falsity and instead are concerned with whether or not we have warrant to believe such a claim (429).
16.2.9
[Because we have three intuitionistic logical situations: confirm, refute, and neither, intuitionistic logic would seem at first to be a three-valued logic. But it is more subtle than that.]
We see from the above that we have three logical situations. Either we can confirm the proposition, we can refute it, or we can do neither. Thus, Nolt observes, it would seem to be something like a three-valued logic. Yet, he cautions, “it is more subtle than that” (429).
16.2.10
[One way that intuitionistic logic is different from classical and three-valued logic is that in the other logics double negation holds, but it does not hold in intuitionistic logic. For, ∼∼Φ means, “it is refuted that Φ is refuted”. But to refute a refutation leaves open the possibility that we do not have warrant to either refute or to affirm it.]
To see how intuitionistic logic is different from three-valued and classical logics, he has us consider the inference rule of double negation, which is valid in both classical and multivalued systems but is invalid in intuitionistic ones. We begin by noting that ∼Φ means “Φ is refuted”. The double negation would then mean “it is refuted that Φ is refuted” (429). But to refute that something is refuted, in this intuitionistic context, does not necessarily mean that it is confirmed. It could also simply mean that we do not know either way. We consider again the example proposition: “Napoleon ate breakfast on September 9, 1807.” [Suppose it is refuted. That would mean we have evidence to suggest it cannot have happened. Now furthermore suppose that we refute that it was refuted. Perhaps that is like saying that we have reason to believe that we lack warrant to deny it. But that does not mean we have necessarily gained information to affirm it. We could be back in the situation where we can neither confirm not deny it. But recall from section 15.2 the truth-table for negation in three valued logics:
Suppose the first case. P is true. That means ~P is false and ~~P is true, or the same value we started with for P. Suppose instead P is false. That means ~P is true and ~~P is false. Again it is the same value as we started with. And finally suppose P is I. That means ~P is I and ~~P is I. We again have the same value we began with, and thus double negation holds in three-valued logic but not in intuitionistic logic.]
We can begin to appreciate the subtlety by considering the inference rule of double negation, or ~E (from ∼∼Φ infer∼Φ), which is valid in both classical and multivalued systems; it is invalid in intuitionistic logic. For to an intuitionist, since negation signifies refutation, ∼∼Φ means “it is refuted that Φ is refuted.” But suppose we somehow refute the view that the proposition that Napoleon ate breakfast on September 9, 1807, is refuted. That is not tantamount to confirming this proposition; it may only demonstrate our ignorance. Hence from ∼∼Φ an intuitionist may not in general infer Φ. This is a drastic departure both from classical logic and from multivalued logics based on the notion of truth!
(429)
16.2.11
[But in intuitionistic logic, we cannot calculate the confirmation-values of complex formula on the basis of the values of subformulas, and thus it is not truth-functional. For example, if P has neither value, we cannot confirm ~P, for then P has the value refuted rather than its assumed neither-value. But again supposing P has neither value, we can either refute ~P, by knowing that there is no warrant to negate it, or we can say it has neither confirmation value, were we to lack knowledge as to whether it is confirmable or not.]
[I may get these next ideas wrong, so please consult the quotation to follow. As we saw above, in the three-valued logics, when the proposition has the third value, its negation has it too. So we can always derive the truth-value of ~Φ on the basis of Φ. But suppose we are using intuitionistic logic, and we can neither confirm nor deny Φ. What value could we assign ~Φ? Nolt will demonstrate now that ~Φ’s value is not consistently determined by the value of Φ, because in some cases Φ can be neither confirmed nor refuted but ~Φ is refuted, and in other cases Φ can be neither-valued and also ~Φ is too. First we should eliminate the possibility that Φ is neither confirmed nor refuted but ~Φ is confirmed. Were that so, then we are confirming that Φ is refuted. But that contradicts our original assumption that Φ is neither confirmed nor refuted. So ~Φ cannot be confirmed. But there are still these other two possibilities: {1} Φ is neither-valued and ~Φ is refuted. Above we saw that “Napoleon ate breakfast on September 9, 1807” was neither-valued, but we refuted that “Napoleon ate breakfast on September 9, 1807 is refuted”. So we cannot confirm or deny P and thus it is neither valued, but we can confirm ~~P. (I am not exactly sure how that might happen. He says that “That is not tantamount to confirming this proposition; it may only demonstrate our ignorance.” Does he mean that we have warrant to refute that P is refuted because we do not have enough knowledge to know if it is refuted? But that would not seem like enough to refute it but rather only to say it is not confirmed. To refute ~P, perhaps we need evidence to say that it was not refuted. Suppose there is a criminal investigation. Let us consider an accusation and call it ‘P’, something like ‘x committed murder’. Now, on the one hand, we do not know what evidence the investigators are finding, but on the other hand we somehow know with certainty that if the accusation was refuted by the available evidence, the charges would be dropped. We see that the charges have not been dropped, so we can refute that that the charge was refuted; that is to say, we can refute ~P. Yet at the same time, we see see that the case is not closed, so we cannot confirm that affirm that x did not commit the murder, because the investigators probably have not found enough evidence to know with certainty. So we cannot also confirm ~P. I am just guessing.) The other possibility is {1} Φ is neither-valued and ~Φ is neither-valued. (I again am not sure how to illustrate this. With our above example, suppose the situation is a little different. Suppose we know that the investigators would not immediately close the case if they found evidence to refute the accusation. Perhaps they need the suspect to be unsure about their fate for some reason, maybe to extract information from them. So from our perspective outside the investigation, we note that the charges have not been dropped. This neither confirms nor refutes that the charges have been refuted.) Nolt also seems to say that it is not simply in this case that we encounter these problems, but rather, in general, intuitionist logic is not truth-functional, because you cannot assign values of complex formulas on the basis of their parts. (I am not sure if he is saying we run into similar problems with the other operators, or if the problem of negation is enough to say there is no truth-functionality whatsoever, perhaps because of the prevalence of negation.) This means that we cannot form for intuitionistic logic a three-valued system to determine truth-values compositionally. Let me quote, as I might have this wrong:]
Moreover, if intuitionistic logic were three-valued in the way suggested above, then the intuitionistic operator ‘~’ would have a truth (or, rather, confirmation!) table that would tell how to calculate the value of ~Φ from the value of | Φ. But there can be no such table. For suppose the value of Φ were “neither,” that is, neither confirmed nor refuted. Then ~Φ could not have the value “confirmed,” since that would mean that it is confirmed that Φ is refuted and hence that Φ itself is refuted, rather than having the value “neither.” But the meaning assigned by the intuitionist to ‘~’ does not determine which of the other two values (“neither” or “refuted”) ~Φ ought to have. We saw in the preceding paragraph that ~Φ might be refuted though Φ is “neither.” But it might also be “neither” if Φ is “neither”; that is, both the proposition Φ that Napoleon ate breakfast on September 9, 1807, and its negation may be neither confirmed nor refuted. Thus when Φ is “neither,” ~Φ may be either refuted or “neither.” It follows that intuitionist propositional logic is non-truth-functional (or, more accurately, non-confirmation-functional). That is, the semantic values assigned to a complex formula in a given situation cannot in general be calculated merely from the values of its subformulas in that situation. Thus no confirmation-functional three-valued logic will suffice.
(429-430)
16.2.12
[The most natural formal semantics for intuitionistic logic is a bivalent semantics, with the values confirmed (C) and unconfirmed (U).]
Nolt says that rather than a three-valued semantics, “The most natural formal semantics for intuitionistic logic is in fact bivalent (two-valued)” (430). Those values are “confirmed” (abbreviated C) and “unconfirmed” (abbreviated U). [Later we will see how “refuted” and “neither” are understood under these two values.] Instead of ‘C’ or ‘U’, they might be written as ‘T’ or ‘F’ or as ‘1’ or ‘0’. (430)
16.2.13
[We will just use two values, confirmed and unconfirmed. The equivalent here for refuted is when a proposition is both unconfirmed and its negation is confirmed.]
[We notice that ‘refutation” is not specified. It is defined by how a proposition and its negation are valued. If a proposition is U and its negation is C, that is, if it unconfirmed while its negation is confirmed, then it is refuted. Now note again, we see that in this situation, the proposition is unconfirmed. It is still possible that as well, the negation is also not-confirmed, giving us Φ=U and ~Φ=U. In this case, the proposition is not refuted (and not confirmed).]
Refutation, on this semantics, is merely a subspecies of nonconfirmation. More specifically, Φ is refuted if and only if ~Φ is confirmed – in which case Φ has the value U and ~Φ the value C. Φ may, however, have the value U even if ~Φ also has the value U. Figure 16.1 illustrates the relations of these concepts.
(430)
| Confirmation value of Φ |
| / \ |
| Φ is confirmed | Φ is unconfirmed |
| / \ |
| Φ is refuted | Φ is neither confirmed |
(based on fig. 16.1, p.430)
16.2.14
[Our bivalent intuitionistic semantics comes from Saul Kripke]
So we will do a bivalent intuitionistic semantics, and Nolt has chosen one coming from Saul Kripke. (citing his “Semantical Analysis of Intuitionistic Logic I”). Given the similarities, Nolt suggests we review the material on S4 in section 12.1. (430-431)
16.2.15
[One difference between our Kripkean intuitionistic semantics and the Kripkean possible worlds semantics is that in intuitionistic semantics, we are dealing with epistemic possibility. We have (the equivalent of) possible worlds when “everything that is the case within them is consistent with the available evidence”.]
[Recall from section 11.2.1 that in Leibnizian modal semantics, we specify a set of possible worlds with their own sets of objects and facts (true and false formulas). We then noted in section 12.1 that one limitation with Leibnizian modal semantics is that it only models logical truths, but not physical ones that might vary without leading to logical contradictions. For example, it is not physically possible (in our world) for an object to accelerate faster than the speed of lights; but it is logically possible, since it is not contradictory. (It reminds me of Hume’s notion of matters of fact, which are arbitrary, logically speaking, because they cannot be deduced but instead must be discovered. See section 4.1 and section 4.2 of his Enquiry.) Kripke’s solution was to introduce the accessibility relation into this possible world modeling. In our example, we model our own world with its given laws of physics. But then we model another world with other laws. Suppose world 2 is exactly like our own, but in it objects can accelerate past the speed of light. Our world then does not have “access” to that world, because it has physical behavior that does not happen in ours. But now, take the perspective of world 2. In that world, objects can go both below and above the speed of light. But in world 1 (our world) things can only go no faster than the speed of light. Thus our world is accessible to world 2, because all things that can happen in our world can also happen in world 2.] [In Kripkean modal semantics, the issue was one of real possibility. But now we are dealing with epistemic possibility. In this context, when we use the notion of ‘worlds’, we say that worlds are possible when “everything that is the case within them is consistent with the available evidence” (431).]
A Kripke model V for intuitionistic logic, like Kripkean models for modal logics, specifies a set Wv of “possible worlds” and an accessibility relation ℛ on those worlds. However, the modality represented by the model is not alethic possibility, but a specific kind of epistemic possibility – that is, possibility relative to what has been confirmed. If we think of possible worlds as worlds, then their possibility consists in the fact that everything that is the case within them is consistent with the available evidence.
(431, boldface mine except for “epistemic possibility”)
16.2.16
[In intuitionistic semantics, we do not have worlds-in-themselves, thus we are not working really with “possible worlds”. Rather they are “states of evidence” (evidential states), which are “the total evidence available to a person or culture at a given time”. By means of them we may categorize propositions as either confirmed or unconfirmed. Other evidential states can be said to be consistent with our own (they are ones we have access to). In intuitionistic semantics, it is stipulated that when something is confirmed in one evidential state, it is thereby confirmed in all others consistent with the first. This is the epistemic necessity of confirmation.]
But recall that for intuitionism, there are no worlds-in-themselves. We only have available evidence. So we should think of a world as the body of available evidence at some time. This would be the “state of evidence” by which propositions are “categorized not as true or false, but as confirmed or unconfirmed” (431). [The next idea I might get wrong, so please consult the quotation below. It might be something like the following. The evidential state of one world can be said to be consistent with another. I am not sure what that means, but the impression I get is that we need to look at states sharing formulas. Suppose two states have all the same formulations with the same confirmation assignments. Formally speaking, there would be no difference between them. So our evidential states are not identical in some way. We seem to be working specifically with states where one has more confirmations than another for the same formulas. (Perhaps also the confirmations are the same for shared formulas, but the second has more formulas than the first, but I do not know. My guess is that we think of the formulas and objects as having some confirmation value in each state.) So suppose you have one world where formula P is confirmed and formula Q is unconfirmed, and suppose in another world you have both being confirmed. This would mean.
Evidential state 1:
P = C and Q = U
Evidential state 2:
P = C and Q = C
Here we would say that state 1 is a (proper) subset of 2, because all confirmations of 1 are found in 2 (but not all of 2 are in 1). So we say that 1 is consistent with 2 (but that 2 is not consistent with 1). We say that 1 is accessible from 2 (but that 2 is not accessible from 1). And thus we say that 2 has access to 1 (but 1 does not have access to 2). And as we see later, we write it 1ℛ2. I may have this terribly wrong, but I want to at least establish some specific way to understand the situation. The next point seems to be the following. We can on the one hand see the consistency relations from the sets of value assignments. But if we also stipulate that one world is consistent with a second one, and if we do not know any of the value assignments for the second one’s formulas, we can infer the confirmations in the second by matching them with the first. As he writes: “Confirmation is thus a kind of epistemic necessity”.]
But this is not the most illuminating way to think of possible worlds. Indeed, the central idea of intuitionism is to avoid positing such worlds-in-themselves and to formulate a semantics only in terms of the available evidence. It is better, then, to regard these “worlds” as representations of states of evidence-that is, of the total evidence available to a person or culture at a given time. In each such state, propositions are categorized not as true or false, but as confirmed or unconfirmed. Now a proposition which is confirmed in an evidential state w must also be confirmed in every evidential state that is epistemically possible with respect to w. For example, if I have confirmed (by calculation, say) that 472 + 389 = 861 , then in every evidential state that is consistent with my current evidence, 472 + 389 = 861 . It is, in other words, epistemically necessary for me that 472 + 389 = 861 . Confirmation is thus a kind of epistemic necessity.
(431, italics his)
16.2.17
[So when a proposition is confirmed in one evidential state, it must regarded as confirmed in all other ones accessible to (consistent with) the first.]
This means that when a proposition is confirmed in one evidential state, it must be represented as such in all other evidential states assessable from it (431).
Accordingly, the confirmation of a proposition Φ in a given evidential state w must be represented semantically not merely by the assignment of the value C to Φ at w, but by the assignment of the value C to Φ at w and at all “worlds” (evidential states) accessible from w.
(431)
16.2.18
[In classical modal semantics, a proposition can be true in one world but false in another accessible from the first, unlike how in this intuitionistic semantics being confirmed in one evidential state necessitates its confirmation in all others accessible from the first.]
So one important difference between Kripkean classical modal semantics and this intuitionistic semantics is that in the classical systems, a proposition can be true in one world but false in another accessible to the first. But here being confirmed in one evidential state means that necessarily it is confirmed in all others accessible to the first. (431)
16.2.19
[So if we go to other evidential states accessible from the first one, we will have all the same confirmed propositions (never any less) and we may have additional ones as well.]
[I do not understand everything about the next part. The idea might be that from one evidential state we might move to others accessible from the first, and those others must have exactly the same confirmed propositions, and it may have more. But what I do not understand is how we would be following “accessibility relations out from a given world”. In practice for example, if we find ourselves in one evidential state, what does it mean for us to follow out of it to another one? Is it simply to learn new things? But later he says that the new states cannot be future ones. So I have no concrete conception of this yet. Let me quote:]
Consequently, as we follow accessibility relations out from a given world, the set of confirmed propositions can only get larger or, at minimum, remain constant; it never decreases.
(431)
16.2.20
[The accessibility relation cannot take us to future evidential states, because they might contradict our current one, and this contradiction is not permissible.]
Nolt says that we should not be tempted to think of the accessibility relation here as us learning more things over time. [The problem with this future interpretation is that we can later learn things that contradict our current ones. I do not fully understand this problem, so I will guess. Perhaps the problem is that the proposed action of moving from an evidential state 1 to a state 2 could only happen were 1 consistent with 2. I cannot figure out a reason why that might be so, but just suppose it for the moment. Now, if we learn something new later in the second evidential state whereby a proposition P that was confirmed before is now not confirmed, that would mean world 1 is not consistent with 2. If our above reasoning was right, this cannot happen, because you can only move to a new state consistent with the first. But that is not the way things work in life, is it? Do we not always find ourselves learning that something previously confirmed is now unconfirmed or refuted? Maybe the idea is the following. Suppose we learn something new that unconfirms a proposition that in the prior evidential state was confirmed. Perhaps the response here would be that we really were not warranted previously to have confirmed it (because for example we did not sufficiently examine the situation). But these are wild guesses. Nolt says in a footnote that we could understand the accessibility relation as taking us to a future evidential state if we incorporate tense operators. So I suppose it would be like in evidential state 1, P is C at t1, and in evidential state two, P is C at t1 and also P is U at t2. (Maybe in state 1, P is U at t2, even though in the situation we are modelling, we did not know that yet.) ]
Some accounts of intuitionistic logic suggest that ℛ be understood temporally so that accessible evidential states are those we might arrive at through time as we learn more. But this interpretation is misleading, particularly for nonmathematical applications. What we confirm in the future may contradict what we confirm in the present, if only because (from a classical standpoint) the world has changed in the meantime. In reality it may also happen that evidence we obtain in | the future contradicts our present evidence, even if the relevant facts haven’t changed, because our present evidence is mistaken. (What I think is the boiling of the soup, for example, may turn out on closer inspection to be only the play of shadow from a nearby fan.) Kripkean semantics, however, never allows anything confirmed in an evidential state accessible from w to conflict with anything confirmed in w. Therefore ℛ is best understood as taking us not to possible future states of evidence, but to states that, consistent with the evidence we have currently, we might now enjoy if we could now have more evidence.6
(431-432, boldface mine)
6 It is, of course, possible to provide explicitly for such future (and also past) states of evidence by adding tense operators to Kripkean intuitionistic logic.
(432)
16.2.21
[The accessibility relation must be reflexive: a state of evidence must be consistent with itself.]
Nolt now discusses the structure of the accessibility (consistency) relation ℛ between evidential states. The first is that an evidential state must be consistent with itself, thus ℛ is reflexive.
Because ℛ expresses this specific kind of epistemic possibility (namely, consistency with current evidence), ℛ must have a specific structure. It must, first of all, be reflexive – at least if we suppose (as Kripke does) that evidential states are noncontradictory, that is, self-consistent.
(432)
16.2.22
[ℛ is also transitive. What is confirmed in one evidential state is confirmed in all others consistent with the first. If the first is consistent with the second, and the second with the third, then the first is consistent with the third.]
Also, ℛ is transitive, meaning that if an evidential state 1 is consistent with 2, and 2 with 3, then 1 is consistent with 3. [Note, I may have my terminology confused, but based on how I currently grasp it, for the evidential states mentioned below in the first instance, w3 is a subset of w2 , which is a subset of w1. Then next it seems to order them in the opposite direction, as we normally do. If they are ordered in the same direction, then I am confused about the terminology of ‘consistent with/accessible from’, which I have so far have understood to order the states from subset to larger set.]
Moreover, ℛ must be transitive. That is, if an evidential state w2 is consistent with (accessible from) another state w1, and a third state w3 is consistent with w2 , then w3 is consistent with w1. This follows from the assumption noted above that every proposition confirmed in a particular evidential state w is also confirmed in all evidential states accessible from w. For then if w1ℛw2 and w2ℛw3, all the propositions confirmed in w1 are also confirmed in w2 and all of those confirmed in w2 are confirmed in w3. Hence all of those confirmed in w1 are confirmed in w3. But this means that w3 is consistent with the evidential state w1 so that (since ℛ represents consistency with current evidence) w1ℛw3.
(432)
16.2.23
[We define Kripkean valuation for formulas in intuitionistic predicate logic similarly to Kripkean modal logic, except we think of the worlds as evidential states, and instead of T or F we have C or U. Also, negation, conditional, biconditional, and universal quantification differ.]
For a predicate intuitionistic logic, we need to “assign a domain to each evidential state” (432). Nolt says that “Intuitively, this domain represents the objects whose existence we have confirmed” (432). Now, since objects can exist without their existence being confirmed, as we follow accessibility relations out to more informed evidential states, there could be more objects in those domains, if their existence has been confirmed. But since whatever is confirmed in one domain must also be confirmed in those others it has access too, the epistemically related domains cannot diminish as we move outward through them. [Below, after the quotation this draws from, Nolt will provide the definition for a Kripkean model in intuitionistic predicate logic, followed by the semantics of its confirmation conditions]
Finally, with respect to predicate logic, we must assign a domain to each evidential state. Intuitively, this domain represents the objects whose existence we have confirmed. Since it is epistemically possible for there to be objects whose existence we have not confirmed, we allow that as we move out from a given evidential state along a trail of accessibility, the domain may grow. But it may not diminish, for if the existence of an object is confirmed, then it is not epistemically possible for that object not to exist. That is, if w1ℛw2, then the domain of w2 must include all the objects in the domain of w1 and maybe some additional objects as well. With these principles in mind we make the following definition:
DEFINITION A Kripkean valuation or Kripkean model v for a formula or set of formulas of intuitionistic predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds or evidential states of v.
2. A reflexive, transitive relation ℛ, consisting of a set of pairs of worlds from Wv.
3. For each world w in Wv a nonempty set Dw of objects, called the domain of w, such that for any worlds x and y, if xℛy, then Dx is a subset of Dy.
4. An assignment to each name α in that formula or set of formulas of an extension v(α) that is a member of the domain of at least one world.
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5. An assignments to each predicate Φ and world w in that formula or set of formulas, an extension v(Φ, w) such that
i. If Φ is a zero-place predicate, v(Φ, w) is one (but not both) of the values C or U such that if v(Φ, w) = C, then for all u such that wℛu, v(Φ, u) = C.
ii. If Φ is a one-place predicate, v(Φ, w) is a set of members of Dw such that if d ∈ v(Φ, w), then for all u such that wℛu, d ∈ v(Φ, u).
iii. If Φ is an n-place predicate (n>1), v(Φ, w) is a set of ordered n-tuples of members of Dw such that if <d1, ..., dn> ∈ v(Φ, w), then for all u such that wℛu, <d1, ..., dn> ∈ v(Φ, u).
(432-433, boldface in the original)
Now, rather than the truth conditions for complex formulas, we give what we might call their confirmation conditions. Let Φ and Ψ be any formulas and v any valuation of Φ and Ψ whose accessibility relation is ℛ. Then
1.
If Φ is a one-place predicate and α is a name, then v(Φα, w) = C iff v(α) ∈ v(Φ, w);
v(Φα, w) = U iff v(α) ∉ v(Φ, w).
2.
If Φ is an n-place predicate (n>1) and α1 ... , αn are names, then
v(Φα1, ... , αn, w) = C iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = U iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3.
If α and β are names, then v(α = β, w) = C iff v(α) = v (β);
v(α = β, w) = U iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = C iff for all u such that wℛu, v(Φ, u) ≠ C.
v(~Φ, w) = U iff for some u such that wℛu, v(Φ, u) = C.
5.
v(Φ & Ψ, w) = C iff both v(Φ, w) = C and v(Ψ, w) = C.
v(Φ & Ψ, w) = U iff either v(Φ, w) ≠ C or v(Ψ, w) ≠ C, or both.
6.
v(Φ ∨ Ψ, w) = C iff either v(Φ, w) = C or v(Ψ, w) = C, or both.v(Φ ∨ Ψ, w) = U iff both v(Φ, w) ≠ C and v(Ψ, w) ≠ C.
7.
v(Φ → Ψ, w) = C iff for all u such that wℛu, v(Φ, u) ≠ C or v(Ψ, u) = C, or both.
v(Φ → Ψ, w) = U iff for some u such that wℛu, v(Φ, u) = C and v(Ψ, u) ≠ C.
8.
v(Φ ↔ Ψ, w) = C iff for all u such that wℛu, v(Φ, u) = v(Ψ, u).
v(Φ ↔ Ψ, w) = C iff for some u such that wℛu, v(Φ, u) ≠ v(Ψ, u).
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9.
v(∀βΦ, w) = C iff for all worlds u such that wℛu and for all potential names α of all objects d in Du, v(α,d) (Φα/β , u) = C;
|
v(∀βΦ, w) = U iff for some world u such that wℛu, and some potential name α of some object d in Du, v(α,d) (Φα/β , u) ≠ C.
10.
v(∃βΦ, w) = C iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = C;
v(∃βΦ, w) = U iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ C.
(Nolt 433-434, boldface in the original)
16.2.24
[Negation means refutation, which is more than non-confirmation in one evidential state. It means that there are no evidential states accessible from the first where the unrefuted formulation is confirmed. It is similar to impossibility in modal logic.]
Nolt will now explain how negation, conditional, biconditional, and universal quantification differ in intuitionistic semantics, beginning with negation. [I may get this wrong, so please consult the quotation below. It seems we need to distinguish ~P being confirmed (that is, P being refuted), and P being unconfirmed. So we cannot infer one from the other, I suppose unlike how in a normal bivalent truth situation we can say that P being false is the same as ~P being true. But I am not sure about that. At any rate, confirming ~P, which is refuting P, means that it is impossible to confirm P (rather than just it not being confirmed in some world). It seems that the idea of possibility (or necessity) of confirmation is like that in modal logic, where if in another world something is the case, it is possible in the first world, and if it is so in all worlds, than it is necessarily the case in the first world. Here the idea is that ~P is confirmed in some evidential state if there is no other state accessible from the first where P is confirmed. Perhaps there is the following intuition here. We suppose that as we go outward in states, we go to increasingly better informed ones. If none of the better informed states can confirm P, then in a lesser informed state we can confirm its negation. It is maybe like saying, “even with all the best possible evidence we cannot confirm P, so we must confirm ~P.”)
The negation operator expresses not merely nonconfirmation, but refutation. Refutation is a kind of epistemic impossibility. Intuitively, ~Φ is confirmed (i.e., Φ is refuted) iff current evidence precludes any possibility of the confirmation of Φ. Formally, ~Φ is confirmed in a given evidential state w iff no state in which Φ is confirmed is compatible with (accessible from) w. Thus negation is an epistemic impossibility operator; ‘~’ has the same semantics in intuitionistic logic as ‘□~’ – or, equivalently, ‘~◊’ – in Kripkean modal logic.
(433)
16.2.25
[In intuitionistic semantics, the conditional is evaluated similarly to classical logic, except here we look to another evidential state to see if the antecedent is not confirmed or the consequent is confirmed.]
[The conditional works much like in classical logic, except the confirmation must be made in another evidential state.]
To assert Φ → Ψ is to say that any epistemically possible state that confirms Φ also confirms Ψ. Formally, Φ → Ψ is confirmed in an evidential state w iff in each evidential state compatible with (accessible from) w, either Φ is not confirmed or Ψ is confirmed. Except for the replacement of truth by confirmation and falsehood by nonconfirmation, the semantics for the intuitionistic conditional is the same as that for the classical strict conditional (see Section 12.3).
(434)
16.2.26
[A biconditional is confirmed if there are no evidential states where the two terms take different values.]
[The biconditional is evaluated as confirmed by seeing if there is any evidential state that distinguishes the values for the two terms.]
The biconditional is in effect a conjunction of strict conditionals. To assert Φ ↔ Ψ is to say that no epistemically possible state differentiates the two.
(434)
16.2.27
[Universal quantification applies to all objects whose existence is compatible with our current evidential state.]
[I may not get the next concept right. I will guess it is the following, but consult the quotation. The universal quantifier ∀ ranges over all designated objects not just in one evidential state but as well to all such designated objects in other evidential states accessible from the first.]
The universal quantifier ‘∀’ means “for all epistemically possible objects,” rather than just “for all objects whose existence has been confirmed.” It has the same semantics as ‘□∀’ in Kripkean modal logic. To assert ∀βΦ in intuitionistic logic, then, is not just to say that Φ is confirmed to apply to all objects whose existence has been confirmed, but to assert that Φ has been confirmed to apply to all objects whose existence is compatible with our current evidential state.
(434)
16.2.28
[The other operations, namely, conjunction, disjunction, and existential quantification do not involve consideration of other evidential states.]
[While the operations above may have been similar to classical logic, what we notice is that they involve considering the other evidential states. The idea he next makes regarding conjunction, disjunction, and existential quantification is that they may only require our current evidential state and therefore are more analogous to their classical counterparts.]
The intuitionistic meanings of conjunctions, disjunctions, and existential statements are, by contrast, more direct analogues of their classical meanings. To assert Φ & Ψ is to assert that both Φ and Ψ are confirmed. To assert Φ ∨ Ψ is to assert that at least one of these disjuncts is confirmed. And to assert ∃βΦ is to claim confirmation of the existence of an object to which Φ applies.
(434)
16.2.29
[A formula is intuitionistically valid if all evidential states confirm it. It is intuitionistically inconsistent if there is at least one evidential state that does not. A sequent is intuitionistically valid if there are no evidential states where the premises are confirmed while the conclusion is not.]
Nolt next defines validity and consistency. [A formula is intuitionistically valid if in all models all the evidential states confirm it. (It would seem that it is like there is no possible evidence that could contradict it / designate it as nonconfirmed). And a formula is intuitionistically inconsistent if there are no models where the evidential states confirm it. A sequent is intuitionistically valid if there are no models where there are evidential states in which the premises are confirmed but the conclusion is not.]
This new semantics requires new definitions for the fundamental semantic concepts. We shall call a sequent intuitionistically valid iff there is no intuitionistic Kripke model containing some evidential state in which the sequent’s premises are confirmed and its conclusion is not confirmed. Intuitively, this means that any evidence that confirmed the premises would also confirm the conclusion. A formula is intuitionistically valid if it is confirmed in all evidential states in all intuitionistic Kripke models. A formula is intuitionistically inconsistent iff there is no evidential state in any intuitionistic Kripke model in which it is confirmed. And so on.
(435)
Nolt then begins some formal demonstrations of the ideas in this section. The first is the invalidity of the principle of excluded middle. [I will first try to offer some reasoning for it. But you should skip to the quotation for the exact formulation. So we want to prove that P∨~P is not intuitionistically valid. Recall that a formula is intuitionistically valid if it is confirmed in all evidential states in all worlds. What we will do is derive the formulation saying that it is not confirmed in one evidential state, thus determining it as invalid. We begin with worlds 1 and 2. Here, 1 is accessible from 1; 2 is accessible from 2; and 1 is accessible from 2. (or however the following should be worded: 1ℛ1, 2ℛ2, and 1ℛ2.) The value of P in 1 is U and in 2 it is C. Nolt’s proof begins by observing that That v(‘P’, 2) = C and 1ℛ2. It then seems he substitutes a variable in for 2 to this to get: v(‘P’, u) = C and 1ℛu. Now recall valuation rule 4:
4.
v(~Φ, w) = C iff for all u such that wℛu, v(Φ, u) ≠ C.
v(~Φ, w) = U iff for some u such that wℛu, v(Φ, u) = C.
Nolt will say that “Thus by valuation rule 4, v(‘~P’, 1) ≠ C.” So world 1 is consistent with 2, meaning that its confirmations are a subset of world 2. For whatever formula they share, their valuations must be identical. Now what does the second line of rule 4 say? The negation of a formula in one world is unconfirmed if in another world from which the first is accessible that formula is confirmed. (In other worlds, if in a larger state of evidence consistent with the first a formula is confirmed, then its negation in the first world is unconfirmed.) So in our proof, we can say that ~P is unconfirmed in state 1, because P is confirmed in state 2. We next appeal to valuation rule 6.
6 .
v(Φ ∨ Ψ, w) = C iff either v(Φ, w) = C or v(Ψ, w) = C, or both.v(Φ ∨ Ψ, w) = U iff both v(Φ, w) ≠ C and v(Ψ, w) ≠ C.
We note that in evidential state 1, we have that P is not confirmed (from the original assignment of it as U) and we also have ~P is not confirmed (from the fact that P is confirmed in evidential state 2, and state 1 is consistent with 2). The second line of rule 6 says that if both terms in the disjunction are not confirmed, then the whole disjunction is not confirmed. Here we have those conditions for P∨~P, thus it is unconfirmed (and hence not confirmed) in state 1. Recall again that a formula is intuitionistically valid if it is confirmed in all evidential states in all worlds. But here P∨~P is not confirmed in state 1. Thus it is not intuitionistically valid. That ends the proof. Now recall what constitutes refutation, namely, “Φ is refuted if and only if ~Φ is confirmed – in which case Φ has the value U and ~Φ the value C.” But here in state 1 we stipulated that P is unconfirmed and we derived that ~P is unconfirmed. Thus P is neither confirmed nor refuted. Nonetheless, it is still possible that P be confirmed in state 1, because it is so in state 2.]
METATHEOREM: The formula ‘P∨~P’ is not intuitionistically valid.
PROOF: Consider the valuation v whose set Wv of worlds is {1, 2} and whose relation ℛ is {<1, 1>, <2, 2>, <1, 2>} such that
v(‘P’, 1) = U
v(‘P’, 2) = C
(Clearly ℛ is reflexive and transitive and v meets the conditions on confirmation value assignments imposed by the definition of a valuation.) Since v(‘P’, 2) = C and 1ℛ2, there is some world u (namely, 2) such that 1ℛu and v(‘P’, u) = C. Thus by valuation rule 4, v(‘~P’, 1) ≠ C. So, since v(‘P’, 1 ) ≠ C, by rule 6 v(‘P ∨ ~P’, 1) ≠ C. Hence ‘P ∨ ~P’ is not intuitionistically valid. QED
The model here represents an evidential state (world 1) in which ‘P’ is neither confirmed nor refuted, but relative to which it is epistemically possible that ‘P’ be confirmed (world 2).
(435)
[Let me note something about the value assignments and world relations in this proof. We have that
v(‘P’, 1) = U
v(‘P’, 2) = C
and that 1ℛ2. We might wonder now how the one and only mentioned formula, P, does not have the same valuation in both related worlds. Whatever is confirmed in state 1 must be confirmed in state 2. But not everything confirmed in state 2 must be confirmed in state 1. For, state 2 in a sense knows more that state 1.]
[sections 16.2.30–16.2.36 are excluded from this post.]
From:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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