Priest (7.5) An Introduction to Non-Classical Logic, ‘Many-valued Logics and Conditionals,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[Priest, Introduction to Non-Classical Logic, entry directory]

 

[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]

 

 

 

 

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

Part II:

Quantification and Identity

 

7

Many-valued Logics

 

 

7.5

Many-valued Logics and Conditionals

 

 

 

 

Brief summary:

(7.5.1) We will now examine the conditional operator in many-valued logics. (7.5.2) We will assess whether or not some problematic inferences using conditionals are valid in K₃, Ł₃, LP₃, and RM₃, by making a table. (In the table below, a ‘✓’ means the inference or formula is valid in the given system, and an ‘×’ means it is not valid.)

 

		K3	Ł3	LP	RM3
1	q ⊨ p ⊃ q	✔	✔	✔	×
2	¬p ⊨ p ⊃ q	✔	✔	✔	×
3	(p ∧ q) ⊃ r ⊨ (p ⊃ r) ∨ (q ⊃ r)	✔	✔	✔	✔
4	(p ⊃ q) ∧ (r ⊃ s) ⊨ (p ⊃ s) ∨ (r ⊃ q)	✔	✔	✔	✔
5	¬(p ⊃ q) ⊨ p	✔	✔	✔	✔
6	p ⊃ r ⊨ (p ∧ q) ⊃ r	✔	✔	✔	✔
7	p ⊃ q, q ⊃ r ⊨ p ⊃ r	✔	✔	×	✔
8	p ⊃ q ⊨ ¬q ⊃ ¬p	✔	✔	✔	✔
9	⊨ p ⊃ (q ∨ ¬q)	×	×	✔	×
10	⊨ (p ∧ ¬p) ⊃ q	×	×	✔	×

 

(7.5.3) Generally speaking, the many-valued logics still validate many of the problematic inferences using the conditional. (7.5.4) We have the intuitions that in finitely many-valued logics, the following two things should hold:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Only in K₃ does (ii) not hold. (7.5.5) Given these two rules, suppose we have a many-valued logic with one more formula than there are truth-values; that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated. (7.5.6) But there are counter-examples to this claim (and in these counter-examples, the intuitive sense of the sentences does not allow for any true biconditional combinations of two different sentences, even though technically they should evaluate as true). For instance, “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true” (127). (So suppose we have a three-valued logic, and John has 1 hair on his head. That means “John has 2 hairs on his head if and only if John has 3 hairs on his head” is true (or at least true, or ‘designated’ whatever way), on account of both sides being false, even though the intuitive sense of the formulation would make the biconditional false (or at least senseless); for, John’s having x number of hairs on his head should not be conditional on his having x ± 1 hairs on his head. Thus, finitely many-valued logics will always be potentially vulnerable to the following problem: because the disjunction of all biconditionals should be true, then at least one must be true, meaning that in the case of propositions like “John has x number of hairs”, there must be at least one true one that reads “John has x number of hairs on his head only if John has x + 1 number of hairs on his head.” But that is senseless even though it would be evaluated as true.)

 

 

 

 

 

 

Contents

 

7.5.1

[Turning to the Conditional in Many-Valued Logics]

 

7.5.2

[A Table of Problematic Conditional Inferences in Many-Valued Logics]

 

7.5.3

[Evaluating the Many-Valued Logics]

 

7.5.4

[Some Disjunction and Biconditional Rules for Many-Valued Logics to Show Why Many Problematic Conditional Inferences are Inevitable in Them]

 

7.5.5

[The Disjunction of all Biconditionals as a Logical Truth]

 

7.5.6

[Counter-Examples to This Claim]

 

 

 

 

 

 

Summary

 

7.5.1

[Turning to the Conditional in Many-Valued Logics]

 

[We will now examine the conditional operator in many-valued logics.]

 

[(ditto)]

Further details of the properties of ∧, ∨ and ¬ in the logics we have just met will emerge in the next chapter. For the present, let us concentrate on the conditional.

(125)

[contents]

 

 

 

 

 

 

7.5.2

[A Table of Problematic Conditional Inferences in Many-Valued Logics]

 

[We will assess whether or not some problematic inferences using conditionals are valid in K₃, Ł₃, LP₃, and RM₃, by making a table.]

 

[The issue of problematic inferences using the conditional is something we have explored quite a bit in previous sections. See sections 1.6, 1.7, 1.8, 1.9, 1.10, 4.5, 4.6, 4.8, 4.9, and 5.2. Priest now summarizes many of these problematic inferences that use the conditional in a table where we can see whether or not they are are valid in K₃, Ł₃, LP₃, and RM₃. Recall that part of this evaluation involves the designated value, which is the truth-preserving value (like 1 in classical logic and i and 1 in LP) and which is not the same in all of these systems. In the table below, a ‘✓’ means the inference or formula is valid in the given system, and an ‘×’ means it is not valid.]

In past chapters, we have met a number of problematic inferences concerning conditionals. The following table summarises whether or not they hold in the various logics we have looked at. (A tick means yes; a cross means no.)

|

		K3	Ł3	LP	RM3
1	q ⊨ p ⊃ q	✔	✔	✔	×
2	¬p ⊨ p ⊃ q	✔	✔	✔	×
3	(p ∧ q) ⊃ r ⊨ (p ⊃ r) ∨ (q ⊃ r)	✔	✔	✔	✔
4	(p ⊃ q) ∧ (r ⊃ s) ⊨ (p ⊃ s) ∨ (r ⊃ q)	✔	✔	✔	✔
5	¬(p ⊃ q) ⊨ p	✔	✔	✔	✔
6	p ⊃ r ⊨ (p ∧ q) ⊃ r	✔	✔	✔	✔
7	p ⊃ q, q ⊃ r ⊨ p ⊃ r	✔	✔	×	✔
8	p ⊃ q ⊨ ¬q ⊃ ¬p	✔	✔	✔	✔
9	⊨ p ⊃ (q ∨ ¬q)	×	×	✔	×
10	⊨ (p ∧ ¬p) ⊃ q	×	×	✔	×

 

(1) and (2) we met in 1.7, and (3)–(5) we met in 1.9, all in connection with the material conditional. (6)–(8) we met in 5.2, in connection with conditional logics. (9) and (10) we met in 4.6, in connection with the strict conditional. The checking of the details is left as a (quite lengthy) exercise. For K₃, a generally good strategy is to start by assuming that the premises take the value 1 (the only designated value), and recall that, in K₃, if a conditional takes the value 1, then either its antecedent takes the value 0 or the consequent takes the value 1. For L₃, it is similar, except that a conditional with value 1 may also have antecedent and consequent with value i. For LP, a generally good strategy is to start by assuming that the conclusion takes the value 0 (the only undesignated value), and recall that, in LP, if a conditional takes the value 0, then the antecedent takes the value 1 and the consequent takes the value 0. For RM₃, it is similar, except that if a conditional has value 0, the antecedent and consequent may also take the values 1 and i, or i and 0, respectively. And recall that classical inputs (1 or 0) always give the classical outputs.

(125-126)

[contents]

 

 

 

 

 

 

7.5.3

[Evaluating the Many-Valued Logics]

 

[Generally speaking, the many-valued logics still validate many of the problematic inferences using the conditional.]

 

[If we look at the table, we can see that there are many check-marks, which means that these problematic inferences are commonly valid in the many-valued logics. If if we ignore conditionals with an enthymematic ceteris paribus clause (see section 5.2), all these many-valued logics still “suffer from some of the same problems as the material conditional” (126). Priest then writes: “K₃ and Ł₃ also suffer from some of the problems that the strict conditional does. In particular, even though (10) tells us that (p ∧ ¬p) ⊃ q is not valid in these logics, contradictions still entail everything, since p ∧ ¬p can never assume a designated value. By contrast, this is not | true of LP (as we saw in 7.4.4) ...” (126-127). I am not sure I get that, but maybe the ideas are the following. One of the problems of the strict conditional is the explosion of contradictions, where contradictions entail everything (see especially section 4.8). But, as we can see from row 10, this is not valid in K₃ and Ł₃. Let us consider the evaluation of ‘⊨ (p ∧ ¬p) ⊃ q’ in K₃ and also ‘p ∧ ¬p ⊨ q’ in K₃, because I am guessing that is the distinction here. Recall from section 7.3.2 that the evaluation for negation, conjunction, and the conditional are the following.

 

f¬
1	0
i	i
0	1

 

f∧	1	i	0
1	1	i	0
i	i	i	0
0	0	0	0

 

f⊃	1	i	0
1	1	i	0
i	1	i	i
0	1	1	1

(122)

 

And the designated value is 1. So we need to see if there is any instance in the evaluation where the value of the whole conditional is not 1. If there is such an instance, it is invalid, and if there is no such instance, it is valid.

 

p  q	(p	∧	¬p)	⊃	q
1  1	1	0	0	1	1
1  i	1	0	0	1	i
1  0	1	0	0	1	0
i  1	i	i	i	1	1
i  i	i	i	i	i	i
i  0	i	i	i	i	0
0  1	0	0	1	1	1
0  i	0	0	1	1	i
0  0	0	0	1	1	0

 

As we can see, ‘⊨ (p ∧ ¬p) ⊃ q’ is not valid in K₃, because when p is i and q is either i or 0, then the conditional is i. Now let us evaluate p ∧ ¬p ⊨ q’ in K₃. If there are any cases where the premises are 1 and the conclusion is not 1, then it is valid. But note that if there are no cases where the premises are 1 to begin with, then it will still be valid, although “vacuously valid” (see Priest’s Logic: A Very Short Introduction, section 2.)

 

p  q	(p	∧	¬p)	⊨	q
1  1	1	0	0		1
1  i	1	0	0		i
1  0	1	0	0		0
i  1	i	i	i		1
i  i	i	i	i		i
i  0	i	i	i		0
0  1	0	0	1		1
0  i	0	0	1		i
0  0	0	0	1		0

 

As we can see, there are no cases where the premises are 1, so it is vacuously valid and thus explosion holds in K₃, and it also holds in Ł₃. But, since in LP, i is a designated value, and since when p is i and q is 0 the premises are i but the conclusion 0, that means it is not valid in LP (see section 7.4.4). Yet Priest continues, “but this is so only because modus ponens is invalid, since (p ∧ ¬p) ⊃ q is valid, as (10) shows.” I am not sure why this formula shows that modus ponens is invalid. I will need to come back to this. But for now I will note that under the exact same evaluation there is the same row as the counter-example for modus ponens, as we saw in section 7.4.5:

 

p  q	p	p ⊃ q	⊨ q
1  1	1	1	1
1  i	1	i	i
1  0	1	0	0
i  1	i	1	1
i  i	i	i	i
i  0	i	i	0
0  1	0	1	1
0  i	0	1	i
0  0	0	1	0

 

In all, the system that has the fewest valid problematic inferences using the conditional is RM₃.]

As can be seen from the number of ticks, the conditionals do not fare very well. If one’s concern is with the ordinary conditional, and not with conditionals with an enthymematic ceteris paribus clause, then one may ignore lines (6)–(8). But all the logics suffer from some of the same problems as the material conditional. K₃ and Ł₃ also suffer from some of the problems that the strict conditional does. In particular, even though (10) tells us that (p ∧ ¬p) ⊃ q is not valid in these logics, contradictions still entail everything, since p ∧ ¬p can never assume a designated value. By contrast, this is not | true of LP (as we saw in 7.4.4), but this is so only because modus ponens is invalid, since (p ∧ ¬p) ⊃ q is valid, as (10) shows. (Modus ponens is valid for the other logics, as may easily be checked.) About the best of the bunch is RM₃.

(126-127)

[contents]

 

 

 

 

 

 

7.5.4

[Some Disjunction and Biconditional Rules for Many-Valued Logics to Show Why Many Problematic Conditional Inferences are Inevitable in Them]

 

[We have the intuitions that in finitely many-valued logics, the following two things should hold:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Only in K₃ does (ii) not hold.]

[Priest will now explain why the conditional in finitely many-valued logics will probably be problematic. In this section we make the first step, so the full reasoning will not here be given. But we need to note a few important things here. First consider how disjunction works. If we can affirm either just A on the one hand or alternatively just B on the other hand, then we should be able to affirm A ∨ B, because we know one of the disjuncts would be true, which is enough for the whole disjunction to be true. Thus:

(i) if A (or B) is designated, so is A ∨ B

(127)

The next observation is that A ≡ A should be a logical truth, even in a many-valued logic. Why? The reasoning is a little tricky, but it seems to be the following. Suppose A is valued at i. Let us make that concrete with an example. We have a moving object, and for A we have, “the moving object is at point 1 at time 1.” Suppose we think that should be both true and false. Regardless, would we not still think that the following should at least also be both true and false: “If the moving object is at point 1 at time 1, then the moving object is at point 1 at time 1”? So, we are claiming that it is reasonable to say that ‘if A then A’ has designated value whenever A itself has one. Now, since A is A, then that implies the inversion of the ‘if A then A’ also holds, and so A iff A should also hold as well. Now, in this case, A will have to have the same value as itself, so both sides of the biconditional will always have the same value, and regardless, the whole biconditional will have a designated value. The next idea seems to be that we suppose we have B, but we assign it the same value as A. And then we say that the same thing should hold for the biconditional here, namely, that:

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

I may have the reasoning wrong there, but this is my guess. Priest ends by saying that this holds in all the finitely many-valued logics we have seen, but for K₃, (ii) fails. ]

But there are quite general reasons as to why the conditional of any finitely many-valued logic is bound to be problematic. For a start, if disjunction is to behave in a natural way, the inference from A (or B) to A ∨ B must be valid. Hence, we must have:

(i) if A (or B) is designated, so is A ∨ B

Also, A ≡ A ought to be a logical truth. (Even if A is neither true nor false, for example, it would still seem to be the case that if A then A, and so, that A iff A.) Hence:

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Note that both of these conditions hold for all the logics that we have looked at, with the exception of K₃, for which (ii) fails.

(127)

[contents]

 

 

 

 

 

 

7.5.5

[The Disjunction of all Biconditionals as a Logical Truth]

 

[Given these two rules, suppose we have a many-valued logic with one more formula than there are truth-values; that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated.]

 

[The argumentation continues to get tricky. I may have the reasoning wrong, so see the quotation below. I am guessing we are doing the following. We are dealing with finitely many-valued logics. So that means regardless of the number of values, they will have a total number of values, that we can use the variable n to signify. So first we will consider any n-valued logic that satisfies

(i) if A (or B) is designated, so is A ∨ B

and

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Now we will consider n+1 propositional parameters, p₁, p₂, . . . , p_n+1. Recall from section 1.2 that a propositional parameter is something like a formula which in our texts above we are writing as A, B, or C. The point is that we have one more formula than the total number of possible truth-values for that system. So at least two formulas will need to have the same truth-value. Now, given what (ii) says, that means for these two same-valued formulas, their biconditional must have a designated value. And, since we now have at least one designated biconditional, that would make it that the disjunction of all biconditionals will have to be designated, by rule (i). So in sum, suppose we have a many-valued logic with one more formula than there are truth-values, that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated.]

Now, take any n-valued logic that satisfies (i) and (ii), and consider n+1 propositional parameters, p₁, p₂, . . . , p_n+1. Since there are only n truth values, in any interpretation, two of these must receive the same value. Hence, by (ii), for some j and k, p_j ≡ p_k must be designated. But then the disjunction of all biconditionals of this form must also be designated, by (i). Hence, this disjunction is logically valid.

[contents]

 

 

 

 

 

 

7.5.6

[Counter-Examples to This Claim]

 

[But there are counter-examples to this claim (and in these counter-examples, the intuitive sense of the sentences does not allow for any true biconditional combinations of two different sentences, even though technically they should evaluate as true). For instance, “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true” (127). (So suppose we have a three-valued logic, and John has 1 hair on his head. That means “John has 2 hairs on his head if and only if John has 3 hairs on his head” is true (or at least true, or ‘designated’ whatever way), on account of both sides being false, even though the intuitive sense of the formulation would make the biconditional false (or at least senseless); for, John’s having x number of hairs on his head should not be conditional on his having x ± 1 hairs on his head. Thus, finitely many-valued logics will always be potentially vulnerable to the following problem: because the disjunction of all biconditionals should be true, then at least one must be true, meaning that in the case of propositions like “John has x number of hairs”, there must be at least one true one that reads “John has x number of hairs on his head only if John has x + 1 number of hairs on his head.” But that is senseless even though it would be evaluated as true.)]

 

[So as we saw in section 7.5.5, if we accept the following two intuitive notions:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Then we should conclude that for any finitely many-valued logical system with n truth-values and n+1 propositional parameters, that the disjunction of all its biconditionals will necessarily be true. Priest then gives a counter-example. “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’.” Priest says that any biconditional of these formulas would seem to be false, and thus the disjunction of all of them will not have any true one in it. I am not following this well, and I also do not see very how this shows us why the conditional for any finitely many-valued logic is bound to be problematic. Let me go through this as best I can. The idea originally was that in such a set of propositions, at least two would have to have the same value. So here, I would think that there would be two with the value false, and thus at least one biconditional that is true. Suppose we are using a three-valued logic with the values 0, 1, and i. And suppose John has 1 hair on his head. That means we have the following four propositions:

(1) John has 1 hair on his head.

(2) John has 2 hairs on his head.

(3) John has 3 hairs on his head.

(4) John has 4 hairs on his head.

Now, 3 and 4 are false, so their biconditional is true. So I do not see yet why “Any biconditional relating a pair of these would appear to be false.” But this is my failing. I am just not sure where I go wrong in the above explication. My best guess at the moment is that the problems come from the intuitive sense of these biconditional formulations: One of them will be, “John has 1 hair on his head if and only if John has 2 hairs on his head.” Now, as we know, this is senseless. John does in fact have 1 hair on his head, but this cannot be biconditional with him having 2 hairs on his head. By extension, even though technically “John has 3 hairs on his head if and only if John has 4 hairs on his head” is true, it is also for the same reason senseless. In other words, John’s not having some number of hairs on his head should not be biconditional on him having some other false number of hairs on his head. In other words, because every biconditional will biconditionally equate statements saying John has a different number of hairs on his head would seem on the level of its sense to necessarily always be false, even though technically they might evaluate as true whenever both sides of the biconditional are false. So on the level of sense, John’s having x number of hairs cannot be conditioned on him having x+1 number of hairs, and vice versa. That is my best guess at the moment. Still, even supposing this to be the case, I am not entirely sure why that would show there to be something problematic with conditionals in finitely many-valued logics. Is it because the biconditional is composed of conditionals, so if there is a paradox with the biconditionals there is a problem with conditionals in general? In other words, since the biconditionals imply that we will have  conditionals of the form “John has x number of hairs if John has x+1 number of hairs” where at least one biconditional combination of them will have to be true, even though we know that cannot be so in any case, given the intuitive sense of the constituent conditional sentences, that we will always have this problem with conditionals in finitely many-valued logics. Let me quote, as I am guessing very wildly here.]

But this seems entirely wrong. Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true.

[contents]

 

 

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

 

 

Priest (9.2) An Introduction to Non-Classical Logic, ‘Introduction [to ch.9, “Logics with Gaps, Gluts and Worlds”],’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[Priest, Introduction to Non-Classical Logic, entry directory]

 

[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]

 

 

 

 

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

Part I:

Propositional Logic

 

9.

Logics with Gaps, Gluts and Worlds

 

9.2

Adding →

 

 

 

 

Brief summary:

(9.2.1) In order to introduce a well-functioning conditional into FDE, we could build a possible world semantics upon it. “To effect this, let us add a new binary connective, →, to the language of FDE to represent the conditional. By analogy with Kυ, a relational | interpretation for such a language is a pair ⟨W, ρ⟩, where W is a set of worlds, and for every w ∈ W, ρ_w is a relation between propositional parameters and the values 1 and 0” (163-164). (9.2.2) We will use the symbol → for the conditional operator in our possible worlds FDE semantics. We still use the ρ relation to assign truth-values. But we also will specify the worlds in which that value holds. (9.2.3) The evaluation rules for ∧, ∨ and ¬ and just like those for FDE, only now with worlds specified.

A ∧ Bρ_w1 iff Aρ_w1 and Bρ_w1

A ∧ Bρ_w0 iff Aρ_w0 or Bρ_w0

(164)

 

A ∨ Bρ_w1 iff Aρ_w1 or Bρ_w1

A ∨ Bρ_w0 iff Aρ_w0 and Bρ_w0

 

¬Aρ_w1 iff Aρ_w0

¬Aρ_w0 iff Aρ_w1

(not in the text)

(9.2.4) In our possible worlds FDE, a conditional is true if in all worlds, whenever the antecedent is true, so is the consequent. And it is false if there is at least one world where the antecedent is true and the consequent false.

A → Bρ_w1 iff for all w′ ∈ W such that Aρ_w′1, Bρ_w′1

A → Bρ_w0 iff for some w′ ∈ W, Aρ_w′1 and Bρ_w′0

(9.2.5) In our possible worlds FDE, “semantic consequence is defined in terms of truth preservation at all worlds of all interpretations:

Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w ∈ W: if Bρ_w1 for all B ∈ Σ, Aρ_w1

(164)

(9.2.6) “A natural name for this logic would be Kυ₄. We will call it, more simply, K₄” (164).

 

 

 

 

 

 

Contents

 

9.2.1

[Introducing the Conditional into FDE with Possible Worlds Semantics]

 

9.2.2

[Notational Conventions: →, ρ, etc.]

 

9.2.3

[The Evaluation Rules for ∧, ∨ and ¬.]

 

9.2.4

[The Evaluation Rule for the Conditional →]

 

9.2.5

[Semantic Consequence as Truth Preservation at All Worlds]

 

9.2.6

[Naming this Logic Kυ₄ or K₄]

 

 

 

 

 

Summary

 

9.2.1

[Introducing the Conditional into FDE with Possible Worlds Semantics]

 

[In order to introduce a well-functioning conditional into FDE, we could build a possible world semantics upon it.]

 

[Let us review some things about First Degree Entailment. First recall from section 1.3.1 the notion of interpretation in classical logic:

An interpretation of the language is a function, v, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as v(p) = 1 and v(q) = 0.

(5)

In section 8.1 we learned how in FDE our interpretations – rather than being functions that assign values  as in the other cases –are instead formulated as relations between formulas and standard truth values. In section 8.2, we noted the following in our brief summary:

In our semantics for First Degree Entailment (FDE), our only connectives are ∧, ∨ and ¬ (with A ⊃ B being defined as ¬A ∨ B.) FDE uses relations rather than functions to evaluate truth. So a truth-valuing interpretation in FDE is a relation ρ between propositional parameters and the values 1 and 0. We write pρ1 for p relates to 1, and pρ0 for p relates to 0. This allows a formula to have one of the following four value-assignment situations: just true (1, e.g.: pρ1), just false (0, e.g.: pρ0), both true and false (1 and 0, e.g.: pρ1, pρo), and neither true nor false (no such valuing formulations). In FDE, being false (that is, relating to 0) does not automatically mean being untrue (that is, not relating to 1), because it can still be related to 1 along with 0. For formulas built up with connectives, we use the same criteria as in classical logic to evaluate them, only here we can have formulas taking both values.

(from our brief summary of section 8.2)

As we can see, there is no conditional operator in FDE. Now recall from section 8.6.5 that modus ponens fails for the conditional operator in FDE (this has to do with the fact that disjunctive syllogism fails in FDE.) Priest next notes that “In any case, as we have seen, using possible-world semantics provides a much more promising approach to the logic of conditional operators.” I am not certain, but perhaps he is referring to the strict conditional. (That is a guess, because we have found problems with the strict conditional, like explosion. See section 4.8.) So, to better incorporate the conditional into FDE, we might combine FDE with possible-world semantics.]

9.2.1 FDE has no conditional operator. The material conditional, A ⊃ B, does not even satisfy modus ponens, as we saw in 8.6.5. In any case, as we have seen, using possible-world semantics provides a much more promising approach to the logic of conditional operators. Thus, an obvious thing to do is to build a possible-world semantics on top of the relational semantics of FDE.

(163)

[contents]

 

 

 

 

9.2.2

[Notational Conventions: →, ρ, etc.]

 

[We will use the symbol → for the conditional operator in our possible worlds FDE semantics. We still use the ρ relation to assign truth-values. But we also will specify the worlds in which that value holds.]

 

[We will now use → for the conditional in our possible worlds FDE. It is a binary connective (connecting the antecedent to the consequent). Since we are dealing with possible worlds, that means a conditional can have a different truth value depending on which world it is said to hold (or not hold) in. So suppose we have an A → B formula, and it is true in world 1 but false in world 2. Recall that ρ is our truth-assigning relation. So we would have A → Bρ_w11 and A → Bρ_w20.]

To effect this, let us add a new binary connective, →, to the language of FDE to represent the conditional. By analogy with Kυ, a relational | interpretation for such a language is a pair ⟨W, ρ⟩, where W is a set of worlds, and for every w ∈ W, ρ_w is a relation between propositional parameters and the values 1 and 0.

(163-164)

[contents]

 

 

 

 

9.2.3

[The Evaluation Rules for ∧, ∨ and ¬.]

 

[The evaluation rules for ∧, ∨ and ¬ and just like those for FDE, only now with worlds specified.]

 

[Recall from section 8.2.6 the evaluation rules for the connectives ∧, ∨ and ¬. Now we will relativize them for worlds. Priest gives the one for conjunction, and I will guess the formulations for disjunction and conjunction.

A ∧ Bρ_w1 iff Aρ_w1 and Bρ_w1

A ∧ Bρ_w0 iff Aρ_w0 or Bρ_w0

(164)

 

A ∨ Bρ_w1 iff Aρ_w1 or Bρ_w1

A ∨ Bρ_w0 iff Aρ_w0 and Bρ_w0

 

¬Aρ_w1 iff Aρ_w0

¬Aρ_w0 iff Aρ_w1

(not in the text)

]

The truth and falsity conditions for the extensional connectives (∧, ∨ and ¬) are exactly those of 8.2.6, except that they are relativised to each world, w. Thus, for example, the truth and falsity conditions for conjunction are:

A ∧ Bρ_w1 iff Aρ_w1 and Bρ_w1

A ∧ Bρ_w0 iff Aρ_w0 or Bρ_w0

(164)

[contents]

 

 

 

 

9.2.4

[The Evaluation Rule for the Conditional →]

 

[In our possible worlds FDE, a conditional is true if in all worlds, whenever the antecedent is true, so is the consequent. And it is false if there is at least one world where the antecedent is true and the consequent false.]

 

[Recall from section 4.5.4 and 5.2.8 that I tried to formulate the rule for evaluating the strict conditional. We now get the correct formulation for the strict conditional:

v_w(A⥽B) = 1 if for all w′ such that v_w′ (A) = 1, v_w′ (B) = 1;

v_w(A⥽B) = 0 if for some w′, v_w′ (A) = 1 and v_w′ (B) = 0.

(164)

The formulation for → will be similar, only now using the ρ relation. A conditional is true if in all worlds, whenever the antecedent is true, so is the consequent. And it is false if there is at least one world where the antecedent is true ant the consequent false.]

For the truth and falsity conditions for →, recall that the truth and falsity conditions for ⥽ in Kυ come to this:

v_w(A⥽B) = 1 if for all w′ such that v_w′ (A) = 1, v_w′ (B) = 1; and v_w(A⥽B) = 0 if for some w′, v_w′ (A) = 1 and v_w′ (B) = 0. Making the obvious generalisation:

A → Bρ_w1 iff for all w′ ∈ W such that Aρ_w′1, Bρ_w′1

A → Bρ_w0 iff for some w′ ∈ W, Aρ_w′1 and Bρ_w′0

(164)

[contents]

 

 

 

 

9.2.5

[Semantic Consequence as Truth Preservation at All Worlds]

 

[In our possible worlds FDE, “semantic consequence is defined in terms of truth preservation at all worlds of all interpretations: Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w ∈ W: if Bρ_w1 for all B ∈ Σ, Aρ_w1” (164).]

 

[Recall from section 8.2.8 that semantic consequence in FDE is defined as:

Σ ⊨ A iff for every interpretation, ρ, if Bρ1 for all B ∈ Σ then Aρ1

(p. 144, section 8.2.8)

and for modal logics (section 2.3.11):

Σ ⊨ A iff for all interpretations ⟨W, R, v⟩ and all w ∈ W: if v_w(B) = 1 for all B ∈ Σ, then v_w(A) = 1.

(p.23, section 2.3.11)

We combine them for our definition of semantic validity in possible worlds FDE.]

Semantic consequence is defined in terms of truth preservation at all worlds of all interpretations:

Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w ∈ W: if Bρ_w1 for all B ∈ Σ, Aρ_w1

(164)

[contents]

 

 

 

 

9.2.6

[Naming this Logic Kυ₄ or K₄]

 

[“A natural name for this logic would be Kυ₄. We will call it, more simply, K₄” (164).]

 

[Priest will now say that “A natural name for this logic would be Kυ₄. We will call it, more simply, K₄.” I do not understand the naming conventions, so should not comment. K is the name for normal modal logics (section 2.1.2). We can place constraints on the accessibility relation R like:

ρ (rho), reflexivity: for all w, wRw.

σ (sigma), symmetry: for all w₁, w₂, if w₁Rw₂, then w₂Rw₁.

τ (tau), transitivity: for all w₁, w₂, w₃, if w₁Rw₂ and w₂Rw₃, then w₁Rw₃.

(p.36, section 3.2.3)

to get more versions of K, like Kρ or Kρσ.  Another restriction is υ: “let an υ-interpretation – ‘υ’ (upsilon) for universal – be an interpretation in which R satisfies the following condition: for all w₁ and w₂, w₁Rw₂ – everything relates to everything” (p.45, section  3.5.1). In section  3.5.4, Priest explains that Kρστ and Kυ are equivalent logical systems. So we have already a sense for Kυ. Perhaps the idea is that our possible worlds FDE will be (so far) a normal modal logic with the universal constraint, meaning that every world has an accessibility relation to every other world (and thus also they have reflexivity, symmetry, and transitivity), but I am guessing. Yet, what about the subscript “3”? I will guess further. Recall from section 7.3 strong Kleene three-valued logic, written as K₃. Just as a guess, I wonder if the subscript there means three-valued, and so here Priest calls our possible worlds 4-value situation semantics Kυ₄ and more simply, K₄.]

A natural name for this logic would be Kυ₄. We will call it, more simply, K₄.

(164)

[contents]

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

.