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
16.3
Relevance Logics
16.3.1-16.3.21
[basic set-up of dialethic, relevant logic semantics]
Brief summary:
In many logics, the following sequent is valid: P, ~P ⊢ Q. For, no matter which values we assign the formulas, none will make the premises true and the conclusion not true, as all the premises can never be made true anyway. This raises the concern that you can validly draw an inference that is completely irrelevant to the premises, which goes against our intuitions regarding how an inference from premises should work. (There are a number of irrelevant sequents that do not involve deriving irrelevant conclusion from contradictions, for example: Abe Lincoln was truthful. ∴ Nothing is both alive and not alive; Ta ⊢ ~∃x(Ax & ~Ax). Another is: P ⊢ Q → Q.) Now, inconsistencies can appear in many humanly constructed realms, like in statutory law, games, fictions, and even semantics as with the liar paradox. There are logics designed to prevent such irrelevant sequents, called relevance or relevant logics. Most relevant logics hold the following three things: {1} inconsistent premises do not imply every propositions but only those relevant to them; {2} valid formulas only follow validly from related premises; and {3} there is a sort of conditional that can only be true when its antecedent and consequent are relevant to one another. Paraconsistent logics do not allow for all propositions to be derived from inconsistent premises. Relevance logics do allow certain conclusions to be drawn from contradictory premises, but only relevant ones. Since not all propositions can be derived from contradictions in relevance logics, they can be considered paraconsistent. When a sequent lacks a counterexample, it is valid, and thus in classical logic P, ~P ⊢ Q is valid. (That is to say, no possible value assignment, when we are restricted to just true and just false, can make the premises true and the conclusion not true.) There are two sorts of strategies for ensuring that irrelevant sequents have counterexamples. {1} We add criteria of relevance to the definition of validity, or {2} we keep this definition of validity, but we expand our notion of counterexample to include ones that invalidate irrelevant sequents. The semantics we outline here, a “dialethic” logic, takes the second approach. It rejects bivalence and has the following four value assignments: {1} just true, {2} just false, {3} both true and false, and {4} neither true nor false. For technical reasons we exclude the conditional operator, and we think of the values in terms of sets: {T}, {F}, {T, F} or { }. The valuation rules and truth tables are:
1.
T ∈ v(~Φ) iff F ∈ v(Φ).
F ∈ v(~Φ) iff T ∈ v(Φ).
2.
T ∈ v(Φ & Ψ) iff T ∈ v(Φ) and T ∈ v(Ψ).
F ∈ v(Φ & Ψ) iff F ∈ v(Φ) or F ∈ v(Ψ), or both.
3.
T ∈ v(Φ ∨ Ψ) iff T ∈ v(Φ) or T ∈ v(Ψ), or both.
F ∈ v(Φ ∨ Ψ) iff F ∈ v(Φ) and F ∈ v(Ψ).
Truth Table for Negation
Truth Table for Conjunction
Truth Table for Disjunction
As we can see, the value assignments follow classical logic as much as possible. (When a formula’s component value is neither, but that value would not change a classical valuation anyway, then it takes the classical valuation. When part or all of a formula has both values, we determine all the possible classical valuations taking each singular value independently and include all the results in the value-set for the whole complex formula.) What is important in our validity evaluations of sequents is that the premises be at least true (even if they are also false) and the conclusion be not at all true (so if the conclusion is both true and false, then we cannot say it is not true. But if it has neither value or is just false, then we can say it is not true.) Consider the evaluation for P, ~P ⊢ Q:
As we can see, in the 10th and 12th lines, the premises are (at least true) and the conclusion is not (at all) true. (In line 13, for example, the premises are at least true, and the conclusion is at least false, but the conclusion is also at least true and thus not not-true. So that line’s valuation, v(P)={T,F} and v(Q)={T,F}, does not create a counterexample.) This dialethicist semantics does not overapply and create counterexamples for sequents that our intuition tells us should be valid; so for example it does not invalidate: ‘P & Q ⊢ P’. However, from contradictions we can derive relevant consequences, as in: ‘P & ~P ⊢ P’. One unfortunate exception, however, is that disjunctive syllogism, ‘P ∨ Q, ~P ⊢ Q’, is invalid in this dialethic semantics.
Summary
16.3.1
[In a relevance logic, only inferences where the premises are relative to the conclusion are valid. Thus P, ~P ⊢ Q would not be valid in a relevance logic.]
[The inference rules of the logics we have considered so far are meant to reflect our insights about how reasoning should work. But there is one insight they do not reflect, namely, that the premises of an argument should be relevant to the conclusion. Graham Priest, in chapter 2 of Logic: A Very Short Introduction shows the problem of irrelevance of inferences of the form q, ¬q / p, for example, “The Queen is rich,” “The Queen is not rich,” therefore “Pigs can fly”. ]
Relevance logic ( also called relevant logic) is a form of logic that does not count an inference valid unless its premises are relevant to its conclusion. All the logics we have considered until now validate irrelevant inferences. In particular, the sequent ‘P, ~P ⊢ Q’ is valid in every system we have surveyed. And though most of the nonclassical logics we have considered lack some of the valid formulas of classical logic, still the formulas which are valid in those systems validly follow from any set of premises, whether relevant or not.
(439)
16.3.2
[Most relevant logics hold the following three things: {1} inconsistent premises do not imply every propositions but only those relevant to them; {2} valid formulas only follow validly from related premises; and {3} there is a sort of conditional that can only be true when its antecedent and consequent are relevant to one another.]
There is dispute over what constitutes the relevance connection between the premises and conclusion. But most relevantists agree on the following three claims [quoting]:
1. Inconsistent premises do not imply every proposition, but only propositions relevantly related to them.
2. A valid formula does not validly follow from every set of premises, but only from premises relevant to it.
3. There is a kind of conditional that is true only if its antecedent and consequent are relevantly connected.
(439)
Nolt will now look at justification for these claims.
16.3.3
[Inconsistencies can appear in many humanly constructed realms, like in statutory law, games, fictions, and even semantics as with the liar paradox.]
Nolt then explains: “Advocates of classical logic often argue that there is no problem in allowing inconsistent premises to imply any conclusion; since inconsistent premises cannot all be true, arguments which employ them are always unsound and hence always negligible” (439). [Nolt discusses different senses for “soundness”. See footnote 11 on p.138. (He also discusses soundness and completeness with regard to proofs and semantics on p.83: “This will enable us to see that our proof technique is sound – that is, that if we start with assumptions true on some valuation, we shall always, no matter how many times we apply these rules, arrive at conclusions that are likewise true on that valuation. Thus a proof establishes that there are no counterexamples to the sequent of which it is a proof; it is a third formal method (in addition to truth tables and trees) for showing that a sequent is valid. In Section 5.10 we shall show that the entire system of rules introduced here is not only sound but also complete – that is, capable of providing a proof for every valid sequent of propositional logic.”) It would seem that by ‘sound’ here he means that each of the premises is true. (From footnote 11 on p.138: “... a sound argument – that is, a valid argument with true premises”.) If we assume that only a proposition or its negation can be true, but not both, then one would say that an argument from inconsistent premises cannot be sound. The interesting point here is that classically minded logicians would say that it is all well and good that you can derive irrelevant conclusions from inconsistent premises, because the argument is unsound anyway. It seems their thinking is that we must disqualify such arguments from the beginning, so it does not matter whether they derive relevant or irrelevant conclusions. I find this odd. On the one hand they want to say that the form of the inference is valid. But on the other hand, part of that form is a contradiction; and what disqualifies it is necessitated by this formal element (the contradiction makes the premises in part untrue), even though the cause of the disqualification is claimed to be a non-formal matter (the truth of the formulas rather than the structure of the sequent). So they want to keep the form as valid while neglecting the fact that the form disqualifies certain sequents.] Nolt then notes how in statutory laws, a mistake can occur where for example one law written at one time would imply that a corporation is liable while at another written at another time would imply it is not liable. There can be a period of time before the legal contradiction is resolved (439). And “Similar contradictions may arise in other humanly constructed realms, such as games and fiction – and perhaps even in semantics itself, in the case of such paradoxical sentences as ‘This sentence is not true’ (see Section 15.2)” (439).
16.3.4
[Classical logic has the principle of explosion, which means that from contradictory premises you can infer all propositions, many of which being irrelevant to the premises. Paraconsistent logics do not admit of this principle, and thus with them you cannot derive all propositions from inconsistent premises. Relevance logics do allow conclusions to be drawn from contradictory premises, but only relevant ones. Since not all propositions can be derived from contradictions here, relevance logics are paraconsistent logics.]
[Classical logic has the principle of explosion, so]
To grant, on the basis of such examples, that inconsistencies are sometimes the case while retaining classical logic is disastrous. If the law contains a true contradiction, for example, then using classical logic we may soundly infer that | everyone is guilty of embezzlement, that bologna is blue, and infinitely many other absurdities.
(439)
For these humanly constructed realms that admit of contradictions, we might want “a logic which only allows only relevant conclusions to be validly derived from these contradictions.” Paraconsistent logics [do not admit of the principle of explosion and thus] do not allow all propositions to follow from a contradiction. Since relevance logics only allow from contradictory premises just relevant conclusions, they do not allow all propositions to follow and thus are paraconsistent logics.
It would be useful, then, to have for the domain of law and for other domains that may admit contradictions a logic which allows only relevant conclusions to be validly derived from these contradictions. Unlike most other forms of logic, this new logic would isolate the consequences of contradictions, preventing them from “infecting” irrelevant areas of knowledge. Logics according to which contradictions do not imply all propositions are said to be paraconsistent. Relevance logics are paraconsistent logics.
(440, boldface in original)
16.3.5
[Even though hypothetical reasoning is thought to be in some way at greater liberty, it would still be problematic in hypothetical reasoning to derive irrelevant conclusions from contradictory premises.]
[I am not certain about Nolt’s next point, so please check the quotation below. Maybe it is that some might think it is ok to have contradictions in hypothetical thinking, but really that is quite problematic.]
But we need not hold that there actually are true contradictions to have reservations about such reasoning. Contradictions are frequently encountered in hypothetical reasoning, and even there it seems odd, if not worse, to reach a contradiction and then infer something wholly irrelevant.
(440)
16.3.6
[It is also odd to prove a valid formula from irrelevant premises.]
Nolt also says that we might also probably find it problematic to prove a valid formula from irrelevant premises. He gives the example (quoting):
Abe Lincoln was truthful.
∴ Nothing is both alive and not alive.
(In symbols this might be ‘Ta ⊢ ~∃x(Ax & ~Ax)’
(440)
Since we find this is odd, and yet it is fine in classical logic, “We seem, then, to have an intuitive, nonclassical notion of validity. Relevantists hope to formalize that notion” (440).
16.3.7
[Relativists argue for a “natural” conditional that cannot be true when the antecedent and the consequent are not relevantly related, as in the case: ‘Snow is white ⇒ Rome is in Italy.’ All the other conditionals we have seen so far allow for irrelevant inferences.]
[Relevantists also note how the material conditional can be problematic in cases of irrelevance, and so] “relevantists hold that there is a ‘natural’ conditional that is true only if its antecedent and consequent are relevantly connected” (440). Nolt has us use the ⇒ symbol for this sort of relevance conditional, and he gives the following example of a conditional where the irrelevance of the antecedent and consequent is enough to qualify it as false:
Snow is white ⇒ Rome is in Italy
(440)
[So even though both antecedent and consequent are true, and even though the truth-table would normally evaluate this as true, it is false because of its irrelevance.] [Nolt next explains that other sorts of conditionals can produce inferences that go against our intuitions, on account of relevance. The first one is about Lewis conditions. But I have not summarized that section yet, so I cannot say much about it. I will mention some things, but please consult the text itself (pp.351-356). In that section, we learn that for Lewis, “if ... then” statements have the following meaning:
‘If kangaroos had no tails, they would topple over’ seems to me to mean something like this: In any possible state of affairs in which kangaroos have no tails, and which resembles our actual state of affairs as much as kangaroos having no tails permits it to, the kangaroos topple over.
(Nolt 351, citing Lewis Counterfactuals p.1)
More generally, we may say:
If Φ then Ψ is true in a world w iff in all the worlds most like w in which Φ is true, Ψ is also true.
(351)
We are to consider worlds that our most like ours where the antecedent is true, and see if the consequent is as well. Consider ‘If Socrates is a rock, then Socrates is a chihuahua’ (352). With the more conventional way of evaluating this, we would say that there are no (practically) possible worlds where the antecedent is true, then the consequent becomes (trivially) true. But, it is odd to think that we can infer that Socrates is a chihuahua from his being a rock. So in this Lewis approach, we continue to think of worlds more remote from ours, going out from practically possible ones to just logically possible ones. In such worlds where it is conceivable for Socrates to be a rock, it is not also true that he is a chihuahua, making the sentence false. Lewis’ conception as we can see involves understanding possible worlds as admitting of degrees of closeness or similarity. In Nolt’s explanation, we are to think then of the ℛ relation as have a degree falling under a scale ranging from 0 (complete lack of relative possibility) to 1 (highest degree of relative possibility). We then would designate the ℛ values as triples, with the second member being a world possible relative to the first, and the third member is the degree of that the second member is possible relative to the first. He writes:
ℛ = {<1, 1, 1>, <1, 2, 0.7>, <2, 1, 0, <2, 2, 1>}
This means that worlds 1 and 2 are each fully possible relative to themselves, world 2 is possible relative to world 1 with a degree of 0.7, and world 1 is not at all possible to world 2. Rather that writing this all out in English, let’s use the notation ℛ(1, 2) = 0.7 to mean that the degree to which world 2 is possible relative to world 1 is 0.7.
(353)
Nolt then gives his reformulation of Lewis conditional ‘□→’
v(Φ □→Ψ, w) = T iff there is some world u such that v(Φ, u) = T, and there is no world z such that ℛ(w, z) ≥ ℛ(w, u), v(Φ, z) = T, and v(Ψ, z) ≠ T.
(354)
(I will guess at the meaning. A formulation like ‘if Φ then Ψ’ is true if {a} it is true in a second world, and {b} there is no third world that {i}is more similar to the first than the second is and {ii} in which ‘if Φ then Ψ’ is not true. So let us look at the sorts of conditionals that allow for inferences to be drawn irrelevantly.]
In particular, where ‘A’ and ‘B’ express unrelated propositions, relevantists object to inferences such as
A, B ⊢ A □→ B
which is valid for the Lewis conditional,
(440)
[Perhaps the idea is that we can find another similar world where the conclusion must be true when the premises are true. Here, we see that this could be the case even if the premises are unrelated.]
B ⊢ A → B
~A ⊢ A → B
which are valid for the material and intuitionistic conditionals,
[For the material conditional’s definition, where the conditional is false only when the antecedent is true and the consequent false, we see that these are valid. (The first makes the consequent true and the second makes the antecedent false). For intuitionistic conditionals, recall their confirmation condition from section 16.2.23.
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.
(433)
The first conditional in question is:
B ⊢ A → B
So suppose in our evidential state we are confirming B. That means we must confirm it in all accessible evidential states. That means in these other states, the consequent is confirmed, and so this would seem to be valid. The second conditional in question is:
~A ⊢ A → B
Here A is refuted (and that ~A is affirmed). This is the confirmation condition for negation:
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.
That means A is at least unconfirmed in all other accessible states. This makes the antecedent unconfirmed and thus it is valid. I may not have this right, so please trust the text instead of my explanation.]
and
□B ⊢ A → B
~◊A ⊢ A → B
which are valid for all the conditionals we have so far studied, except for the Lewis conditional and the conditional of Bochvar’s multivalued logic.
(440)
[Let us begin with
□B ⊢ A → B
and let us see if we can determine why it would be invalid for the Lewis conditional and for Bochvar’s multivalued logic conditional. The valuation for necessity in section 11.2.1 is:
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u in Wv, v(Φ, u) ≠ T;
(316)
Perhaps the idea here is that □B means that in all other worlds no matter how remotely possible the consequent will be true and thus inference valid.
v(Φ □→Ψ, w) = T iff there is some world u such that v(Φ, u) = T, and there is no world z such that ℛ(w, z) ≥ ℛ(w, u), v(Φ, z) = T, and v(Ψ, z) ≠ T.
But it is supposed to be invalid. Maybe the idea is that □B only guarantees the truth of B in worlds that are exactly alike, but I am not sure. Keeping with Lewis conditionals, consider:
~◊A ⊢ A → B
with the evaluation:
v(◊Φ, w) = T iff for some world u in Wv, v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T.
(316)
Suppose ~◊A means that in no other world is A true. That means the antecedent is false and the inference valid. But again, it is supposed to be invalid for Lewis conditionals, and I cannot figure out why. Perhaps ~◊A means that in no other world with a possibility relation of 1 is A true. That is a bad guess, however. Let us now recall from section 15.2 the truth tables for Bochvar’s multivalued logic:
And consider again:
□B ⊢ A → B
Because B is true in all worlds, we need to see if there is any instance where the premises are true but the conclusion is not true. We see in row 8 that B would be true but the conditional I. That perhaps makes it invalid. And for:
~◊A ⊢ A → B
Suppose this means the antecedent is false. Given that we do not list the negation of A (or Φ) in the above chart on the right, let us look for the rows where Φ is F, because that means the ~A of the premise would be true. We see in row 6 that the premise would be true but the conclusion I, which perhaps qualifies as not true, and is thus invalid there.] But, Nolt adds, “For the conditionals of relevance logic, none of these sequents are valid” (440).
16.3.8
[In one relevance semantics uses a notion of relevant conjunction, called fusion, but we will not examine it here.]
Nolt turns now to articulating a semantics for relevance logics. Nolt says one option uses “a non-truth functional form of relevant conjunction, called fusion, which seems to have no straightforward natural language equivalent” (441). But, Nolt explains, it is beyond the scope of this book to provide a treatment of fusion (441).
16.3.9
[We examine instead a relevantist semantics that excludes conditionals, because they are not easily included.]
The sort of relevantist semantics that Nolt will present is a truth-functional one without conditionals, because the inclusion of conditionals is not easily done. [Nolt says in a footnote that the best effort for this is an article by Graham Priest and Richard Sylvan, “Simplified Semantics for Basic Relevant Logics”.] (441)
16.3.10
[Classical logic says a sequent is valid if no counterexample can be found, that is to say, if no value assignment can make the premises true and the conclusion false. This allows certain irrelevant inferences to made validly. One relevantist solution is to require relevance somehow. The semantics we examine here, however, is bivalent, and it loosens restrictions on counterexamples to allow ones that would invalidate irrelevant inferences.]
Nolt explains that the relevantist semantics we will examine “is a radical rejection of bivalence” (441) [So it will reject the limitation to just two logical values.] He then notes the issue of validity. In classical logic we would say that “a sequent is valid iff it lacks a counterexample” (441). Nolt gives some examples of sequents that are valid in classical logic [but that draw irrelevant conclusions]:
P, ~P ⊢ Q
P ⊢ Q → Q
(441)
[As we will see below, finding a counterexample means assigning truth values such that the premises are true and the conclusion false. On pp.14-15 Nolt discusses how there cannot be a counterexample when there are inconsistent premises. This is because we cannot make a value assignment where the premises are true and the conclusion false, for the specific reason that there is no way to make both P and ~P true in classical logic, given its semantics for negation. For the second formula, perhaps there is no negation, because regardless of whether Q is true or false, the conditional will be true, thus we cannot possibly make the conclusion false no matter what. Graham Priest, in chapter 2 of Logic: A Very Short Introduction discusses this notion of vacuous validity.] [Because this notion of validity leads to irrelevant inferences,] “Relevance logicians find this definition too permissive” (441). Nolt then describes two ways to “tighten up” the definition of validity to suit the concerns of relevantists (441). In the first approach, we add criteria calling for relevance in order for there to be validity. In the second approach, this classical notion of validity is maintained, but we loosen the restrictions on what constitutes as a counterexample, so that others can be given to invalidate irrelevant inferences. Nolt will examine a semantics that takes this second approach (441).
16.3.11
[A counterexample to a sequent provides a valuation making the premises true but the conclusion false.]
Nolt now states that “A counterexample is a valuation which makes the premises, but not the conclusion, of a sequent true” (441). He then asks what valuation could possibly give a counterexample for such sequents as we saw above? (441)
16.3.12
[Three valued logics can make certain irrelevant sequents invalid. We wonder now how they might make inconsistent premises both be true.]
[Recall from section 15.2 the three valued semantics for the conditional of Bochvar
and Kleene
Now consider P ⊢ Q → Q. Suppose that Q is I. In both semantics, that means Q → Q has the value I. But I is not true. So we can have the premises as as true but the conclusion as not true, when v(P) = T and v(Q) = I. But what about P, ~P ⊢ Q? How could we make a proposition and its negation both be true?
16.3.13
[Dialethic logic allows for us to assign both true and false to a formula. In cases of inconsistent premises, if they both have the value true and false, then they are both at least true, and thus their irrelevant sequent can be invalidated.]
One way to get inconsistent premises to be true is to say that the unnegated form is both true and false, making the negated form both true and false, thereby allowing for the contradictory premises to be at least true. Dialethic semantics allows for such two-valued assignments. [I normally see it spelled “dialetheic”, as in Graham Priest’s writings.]
Both premises could, perhaps, be true if ‘P’ were both true and false. For in that case ‘~P’ would also, presumably, be both true and false. Hence both premises would be true, and both would also be false. Semantics which permit the assignment of both values to a proposition are called dialethic (literally, “two-truth”). To see how such a semantics might have some practical application in the field of law, consider this instance of ‘P, ~P ⊢ Q’:
Corporation X is liable. Corporation X is not liable.
∴ Bologna is blue.
To this argument we can imagine the following counterexample. Suppose that the legislature has enacted contradictory laws which make it both true and false that Corporation X is liable. Then since it is true that Corporation X is liable, the first | premise is true. And since it is false that Corporation X is liable, the second premise is true. But the conclusion, let us agree, is not true. Of course the premises are both false as well. But if we define a counterexample as a situation in which the premises are true and the conclusion is not true, this situation fits that definition.
(441-442)
16.3.14
[Dialethicists claim that inconsistent propositions can be coherently conceived.]
Nolt says that in chapter 1, he qualified that in informal counterexample must be “a coherently conceivable situation in which the premises are true and the conclusion untrue” (442). In our example, it is conceivable that contradictory laws can be made. But that does not mean that the two inconsistent propositions can be conceived together coherently. “Dialethicists in effect propose a liberalization of our notion of coherence so that we can coherently conceive such contradictions in certain situations” (442). [Note, I often see it spelled, “dialetheists”.]
16.3.16
[Dialethic counterexamples do not overapply to sequents that we intuitively think to be valid but rather apply just to irrelevant sequents.]
The dialethicist counterexamples will only apply “to sequents that we would generally recognize as irrelevant”, and it does not overapply so to invalidate many sequents we would want to say are valid. So it does not invalidate for example ‘P & Q ⊢ P’, because even with its extra values, we cannot produce a counterexample for it. [Let us consider the following valuation: v(P) = T,F and v(Q) = T. (As we will see with the truth tables below,) this would mean that the premise is both true and false, and the conclusion is true and false. That furthermore means that the premise is at least true and the conclusion is at least false. But we need to stick to the precise wording for validity: the premises are true and the conclusion is not true. The conclusion is at least true also, even though it is also false. So the criterial for invalidity does not hold here.]
We might fear that their proposal would generate counterexamples everywhere, leading us to a wholesale denial of validity. In fact, however, the new counterexamples it produces (classical counterexamples still stand) apply only to sequents that we would generally recognize as irrelevant. Dialethicism does not invalidate, for example, the sequent ‘P & Q ⊢ P’ (simplification). Any truth-value assignment (whether dialethic or not) that makes the premise true must also make the conclusion true; there is no counterexample. Of course if ‘P’ is both true and false and ‘Q’ is true, then the premise is both true and false and so is the conclusion. Hence the premise is true and the conclusion false. But this is still not a counterexample, for we have defined a (formal) counterexample as a valuation on which the premises are all true and the conclusion is not true. The conclusion of this inference might be false when the premise is true – and false as well –but in any case this conclusion cannot fail to be true when the premise is true. Thus simplification remains valid on a dialethic semantics.
(442)
16.3.14
[For example, the relevant sequent P & ~P ⊢ P is valid in this semantics, but the irrelevant sequent P & ~P ⊢ Q is not.]
[Now consider: P & ~P ⊢ P and its possible valuations.
As we can see, the only line that could potentially invalidate the sequent is the third one. But there, although the premises are at least true and the conclusion at least false, the conclusion is also at least true as well. So we cannot say that there is a valuation that makes the premises true and the conclusion not true. Now consider: P & ~P ⊢ Q and its possible valuations.
Here we can see that when v(P) = T,F and v(Q) = F, then the premises can be true and the conclusion not true.]
For the same reason ‘P & ~P ⊢ P’ is valid, though ‘P & ~P ⊢ Q’ is not. Contradictions thus have consequences – but only relevant consequences.
(442)
16.3.15
[The dialethic semantics we will use assigns one of four value-sets to proposition letters: {T}, {F}, {T, F} or { }.]
[The next idea is that the semantics we will use has for values, true, false, both, and neither. (Nolt says this is dialethic, but as I understand, dialetheic would mean gaps and so there are three values, namely, just true, just false, and both true and false. An analetheic logic would be one with: just true, just false, and neither true nor false. Nolt’s semantics has all four options, which I have seen in something called first degree entailment. See section 8.4 of Graham Priest’s An Introduction to Non-Classical Logic.) Nolt says that we will think of these values as sets of values. I am not sure why exactly. My best guess is that the neither-value is for some reason best understood as an empty set, and the fact we have a two-value option suggests that we should think of it as a set of values. (Suppose instead we had value ‘B’ for ‘both values’ and ‘N’ for ‘neither-value’. Here the truth tables could still be worked out, but were we to use sets instead, we can see more directly how B takes both values and N, neither.)]
Before going on, we ought to be more explicit about the semantics we are using. Recall that we are considering only the nonconditional fragment of propositional logic. Since this logic is purely propositional, a valuation can be merely an assignment of truth values to sentence letters. But since propositions may receive either value, neither, or both, it is convenient to think of what is assigned to a sentence letter as a set of truth values. Any of these four sets may be assigned:
{T} {F} {T, F} { }
More precisely:
DEFINITION A dialethic valuation or dialethic model for a formula or set of formulas of propositional logic is an assignment of one, but not more than one, of the sets {T}, {F}, {T, F} or { } to each sentence letter in that formula or set of formulas.
(442)
16.3.16
[This dialethic semantics will stick to classical valuations as much as possible. When a formula’s component value is neither, but that value would not change a classical valuation anyway, then it takes the classical valuation. When part or all of a formula has both values, we determine all the possible classical valuations taking each singular value independently.]
The valuation rules for our dialethic semantics will “mimic the classical rules as closely as possible” (442). Suppose we have a complex formula. First now consider if one part of it is missing a truth value, but it is not needed to make a classical valuation. In that case, it gets the classical valuation anyway. [There is no illustration for this, but let us consider one: P&Q where v(P)={F} and v(Q)={ }. Here it would not matter for a classical valuation, because one false conjunct is enough to make the conjunction false.] Second, consider if one or all parts of it have both values. That means we do a classical evaluation for each possibility in two values, which can potentially assign a double value for the whole formula. [This case is illustrated in the following paragraphs.]
16.3.17
[Consider: v(‘P’)={T,F} and v(‘Q’)=F”. What is: v(‘P&Q’)? Even though one part of the complex formula has two values, the complex formula itself only has the value false. For, regardless of whether ‘P’ is taken to be just true or just false, its classical valuation would still make the complex formula just false.]
We begin with an example to illustrate the second case where part or all of a formula has both values, and we thus do a classical valuation for all possibilities of individual values. Here we “suppose that v(‘P’)={T,F} and v(‘Q’)=F” (443). We wonder, what is: v(‘P&Q’)? Since ‘’Q’ is false, then regardless of whether ‘P’ is seen as just true or just false, in both cases ‘P&Q’ will be false under a classical evaluation. So it is just {F}, even though one of its components has both values.
16.3.18
[Consider for contrast: v(‘P’)={T,F} and v(‘Q’)={T}. Then, v(‘P&Q’)={T,F}, because taking both values of P independently for classical valuations, in one case P&Q is true and in the other it is false.]
But now consider: v(‘P’)={T,F} and v(‘Q’)={T}. What then is: v(‘P&Q’)? With P being at least true, and Q being true, then under a classical valuation, one value of P&Q is {T}. But with P also being at least false and Q still being true, then under a classical valuation, another value of P&Q is false. Thus v(‘P&Q’)={T,F}. (473)
16.3.19
[Our dialethic semantics can be formalized (see rules and tables below).]
Nolt then gives the valuation rules for the operators in our semantics:
1.
T ∈ v(~Φ) iff F ∈ v(Φ).
F ∈ v(~Φ) iff T ∈ v(Φ).
2.
T ∈ v(Φ & Ψ) iff T ∈ v(Φ) and T ∈ v(Ψ).
F ∈ v(Φ & Ψ) iff F ∈ v(Φ) or F ∈ v(Ψ), or both.
3.
T ∈ v(Φ ∨ Ψ) iff T ∈ v(Φ) or T ∈ v(Ψ), or both.
F ∈ v(Φ ∨ Ψ) iff F ∈ v(Φ) and F ∈ v(Ψ).
These rules may also be represented, though less compactly, as four-valued truth tables:
Truth Table for Negation
Truth Table for Conjunction
Truth Table for Disjunction
Notice that where Φ and Ψ have exactly one truth value each, these are just the classical truth tables, and that in other cases these tables retain as much as possible of the classical valuation rules.
(443-444, boldface in the original. pages divide at the 11th line of the conjunction table)
16.3.20
[We can determine validity using truth tables.]
We can determine the validity of sequents by making a truth table. If there is a line where the premises are at least true and the conclusion is not at all true, then the sequent is invalid, and it is valid otherwise. One version of DeMorgan’s laws can be shown valid this way:
(444-445, page divides at line 8 of the table)
Nolt also notes that this table shows ‘P ∨ Q’ and ‘~(~P & ~Q)’ to be logically equivalent (445).
16.3.21
[Disjunctive syllogism is invalid in dialethic logic.]
But, disjunctive syllogism is not valid in dialethic logic. We can see in line 10 (below) that the premises are at least true and the conclusion false. The 12th line is also a counterexample.
(modified from 445)
[Sections 16.3.22-16.3.28 are omitted from this summary.]
From:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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