Pirates & Revolutionaries: strict conditional

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[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

Conditional Logics

5.2

Some More Problematic Inferences

Brief summary:

(5.2.1) There are three inferences involving the conditional that are valid in classical logic and for the strict conditional, but as we will see in the next section, they are problematic. They are: {1} Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B; {2} Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C; and Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A. (5.2.2) Here are the problematic counter-example illustrations. {1} Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B; “If it does not rain tomorrow we will go to the cricket. Hence, if it does not rain tomorrow and I am killed in a car accident tonight then we will go to the cricket.” {2} Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C; “If the other candidates pull out, John will get the job. If John gets the job, the other candidates will be disappointed. Hence, if the other candidates pull out, they will be disappointed.” {3} Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A; “If we take the car then it won’t break down en route. Hence, if the car does break down en route, we didn’t take it.” (5.2.3) One might reply to the above objections by saying that they are enthymemes and thus would be valid were we to supply the right relevant information among the premises. (5.2.4) When we supply additional relevant material to the premises of these counter-example illustrations, they show their validity. (5.2.5) But since in such illustration counter-examples we cannot explicitly list all circumstances in the premises that are needed for the argument to be purely non-enthymemic, then this objection does not work absolutely sufficiently yet. (5.2.6) But in fact we can capture all of these infinitely many needed additional de-enthymemizing clauses by simply saying for all of them, “other things being equal,” which is called a ceteris paribus clause. (5.2.7) Ceteris paribus clauses {1} are conditioned by the other antecedent term they are conjoined with, because that term might require particular clauses be implied while others be excluded and {2} are context-dependent. (5.2.8) ‘A > B’ means a conditional with a ceteris paribus clause. And, “A > B is true (at a world) if B is true at every (accessible) world at which A ∧ C_A is true” (84).

5.2.1

[Three Problematic Conditional Inferences]

5.2.2

[Counter-Example Illustrations of These Problematic Conditional Inferences]

5.2.3

[A Possible Defense: The Counter-Examples are Enthymemes]

5.2.4

[Defusing the Counter-Examples by De-Enthymemizing Them]

5.2.5

[The Insufficiency of the Enthymemic Defense: It is Impossible to Completely De-Enthymemize Them]

5.2.6

[Completely De-Enthymemizing the Arguments by Adding a Ceteris paribus Clause (“other things being equal”).]

5.2.7

[Ceteris paribus Clauses as Being Conditioned by the Antecedent that They are Conjoined to and as Being Context-Dependent]

5.2.8

[A Conditional with a Ceteris paribus Clause as ‘A > B’. Defining It Like a Strict Conditional.]

Summary

5.2.1

[Three Problematic Conditional Inferences]

[There are three inferences involving the conditional that are valid in classical logic and for the strict conditional, but as we will see in the next section, they are problematic. They are: {1} Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B; {2} Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C; and Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A.]

[Priest will start now with the problematic inferences involving the conditional, and he says they can be shown to be valid in classical logic. I will try to make the tableaux for them, but probably I will do them wrong.

Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B

A ⊃ B ⊢ (A ∧ C) ⊃ B

A ⊃ B

↓

¬((A ∧ C) ⊃ B)

↓

A ∧ C

↓

¬B

↓

¬C

↙ ↘

¬A B

× ×

2¬⊃

3∧

1⊃

Valid

(7×5)

(7×4)

(This tableau is not in the text and is probably mistaken)

Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C

A ⊃ B, B ⊃ C ⊢ A ⊃ C

A ⊃ B

↓

B ⊃ C

↓

¬(A ⊃ C)

↓

¬C

↙ ↘

¬A B

↙ ↓ ↓ ↘

¬B C ¬B C

× × × ×

3¬⊃

1⊃

2⊃

Valid

(6×4)

(7×6)

(7×5)

(This tableau is not in the text and is probably mistaken)

Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A

A ⊃ B ⊢ ¬B ⊃ ¬A

A ⊃ B

↓

¬(¬B ⊃ ¬A)

↓

¬B

↓

¬¬A

↓

↙ ↘

¬A B

× ×

2¬⊃

4¬¬

1⊃

Valid

(6×5)

(6×3)

(This tableau is not in the text and is probably mistaken)

Priest also says that this holds when ‘⊃’ is replaced by ‘⥽’, but I will not try to make the tableaux.]

Let us start with the inferences. It is easy enough to check that the following are all valid in classical logic:

Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B

Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C

Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A

It is also easy to check that the same is true if ‘⊃’ is replaced by ‘⥽’. (The inferences all hold in L, and so in all modal systems.)

(82)

5.2.2

[Counter-Example Illustrations of These Problematic Conditional Inferences]

[Here are the problematic counter-example illustrations. {1} Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B; “If it does not rain tomorrow we will go to the cricket. Hence, if it does not rain tomorrow and I am killed in a car accident tonight then we will go to the cricket.” {2} Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C; “If the other candidates pull out, John will get the job. If John gets the job, the other candidates will be disappointed. Hence, if the other candidates pull out, they will be disappointed.” {3} Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A; “If we take the car then it won’t break down en route. Hence, if the car does break down en route, we didn’t take it.”]

[Priest next gives some inferences based on the three forms in section 5.2.1, and they would be valid in classical logic, even though these specific examples appear to be intuitively invalid. I will pair them off:

Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B

(1) If it does not rain tomorrow we will go to the cricket. Hence, if it does not rain tomorrow and I am killed in a car accident tonight then we will go to the cricket.

With regard to the formulation, A ⊃ B ⊨ (A ∧ C) ⊃ B, we might note that if C is true, then the antecedent of the conclusion is true as is the consequent. But if C is false then the antecedent is false and the conclusion true. Either way, the conclusion is true. But in the example, when we add another C proposition, it does matter whether it is true or false, because here its being true makes the consequent impossible.

Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C

(2) If the other candidates pull out, John will get the job. If John gets the job, the other candidates will be disappointed. Hence, if the other candidates pull out, they will be disappointed.

In this one it is less obvious to me how to grasp the problem. In the illustration, the other candidates get disappointed if John gets the job, but if they pull out of the competition, John will get the job. So we should normally conclude that if they other candidates pull out, they will be disappointed. My only guess at the moment is that the other candidates do not know that their actions lead to John getting the job, or, maybe the issue is that the simple act of pulling out is an individual decision, and they cannot know if all the other candidates will likewise pull out. In either case, it seems, the candidates will not be disappointed when they pull out, because in that moment of decision they do not realize the consequence will be that John will win, even though later they will be disappointed. So I am not really sure I grasp the problem with that one.

Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A

(3) If we take the car then it won’t break down en route. Hence, if the car does break down en route, we didn’t take it.

This example is also hard for me to grasp. It seems relatively harmless at first glance, even if it is a little paradoxical. The main idea seems to be that it is impossible for the car to break down if we take it, so suppose a situation where it breaks down while we are driving it, well, that is impossible, so it also must be the case that we never took the car out in the first place. Maybe the problem is that we should not suppose that the car breaks down under a mode of imagining it or considering it hypothetically but rather maybe we should think that indeed it did break down while we in fact were driving it, and thus we were not driving it while we were driving it. At any rate, in each case it is reasonable to think that the premises are true but the conclusion false.]

But now consider the three following arguments of the same respective forms:

(1) If it does not rain tomorrow we will go to the cricket. Hence, if it does not rain tomorrow and I am killed in a car accident tonight then we will go to the cricket.

(2) If the other candidates pull out, John will get the job. If John gets the job, the other candidates will be disappointed. Hence, if the other candidates pull out, they will be disappointed. |

(3) If we take the car then it won’t break down en route. Hence, if the car does break down en route, we didn’t take it.

If the conditional were either material or strict, then these inferences would be valid, which they certainly do not appear to be, since they may have true premises and a false conclusion. Hence, we have a new set of objections against the conditional being either material or strict. (And since the conditionals are indicative, they tell just as much against one who claims only that English indicative conditionals are material.)

(83-84)

5.2.3

[A Possible Defense: The Counter-Examples are Enthymemes]

[One might reply to the above objections by saying that they are enthymemes and thus would be valid were we to supply the right relevant information among the premises.]

Priest says that one reply to these counter-example objections from section 5.2.2 is to say they are enthymemes and thus that in their form the inference is valid when we include the relevant additional information. As he says, “suppose that I say: if this plane lands in Rome, it lands in Italy. Strictly speaking, one may say, the conditional is false. It is an enthymeme of the true conditional: if this plane lands in Rome, and Rome is in Italy, then this plane lands in Italy.”

What is one to say about these objections? It is often the case that, when one gives an argument, one does not mention explicitly some of the premises, perhaps because they are pretty obvious. Thus, I might say: this plane lands in Rome; therefore, this plane lands in Italy. Here I omit the fact that Rome is in Italy. Arguments where premises are omitted in this way are traditionally called enthymemes. Just as arguments can be enthymematic, so can conditionals. Thus, suppose that I say: if this plane lands in Rome, it lands in Italy. Strictly speaking, one may say, the conditional is false. It is an enthymeme of the true conditional: if this plane lands in Rome, and Rome is in Italy, then this plane lands in Italy.

5.2.4

[Defusing the Counter-Examples by De-Enthymemizing Them]

[When we supply additional relevant material to the premises of these counter-example illustrations, they show their validity.]

[As we noted above in section 5.2.3, one might object that these counter-examples are really enthymemes and thus are valid when the proper additional information is given among the premises. So for example with the first one, which was “If it does not rain tomorrow we will go to the cricket. Hence, if it does not rain tomorrow and I am killed in a car accident tonight then we will go to the cricket,” supposing it is an enthymeme, we would fill it out by making the following inclusion in the premises: “if it does not rain tomorrow and I am not killed in a car accident tonight, then we will go to the cricket tomorrow. Hence, if it does not rain tomorrow and I am killed in a car accident tonight then we will go to the cricket” Or consider the second one, “If the other candidates pull out, John will get the job. If John gets the job, the other candidates will be disappointed. Hence, if the other candidates pull out, they will be disappointed.” Here, the second premise, which reads, “If John gets the job, the other candidates will be disappointed,” should really be, “if John gets the job and the other candidates do not pull out, they will be disappointed.”]

Now consider the first argument of 5.2.2 . A natural thing to say is that the inference is valid. It is just that the premise is not, strictly speaking, true. What we are assenting to, when we assent to the premise, is really the conditional: if it does not rain tomorrow and I am not killed in a car accident tonight, then we will go to the cricket tomorrow. The premise is an enthymematic form of that. Similar comments can be made about the other arguments of 5.2.2. Thus, the second premise of the second argument is, strictly speaking, false. What is true is that if John gets the job and the other candidates do not pull out, they will be disappointed. Thus, one may defuse these counter-examples.

(83)

5.2.5

[The Insufficiency of the Enthymemic Defense: It is Impossible to Completely De-Enthymemize Them]

[But since in such illustration counter-examples we cannot explicitly list all circumstances in the premises that are needed for the argument to be purely non-enthymemic, then this objection does not work absolutely sufficiently yet.]

[Priest now considers how the above solutions in section 5.2.4 are still not adequate. Suppose for the first one we say for the premises, as we suggested above: “if it does not rain tomorrow and I am not killed in a car accident tonight, we will go to the cricket.” For the same reason we object to the premise being just “if it does not rain tomorrow, we will go to the cricket” being an enthymeme (namely, that it assumes that it is possible for it to rain and that also we get killed in a car accident and yet still somehow go to the game) holds also for “if it does not rain tomorrow and I am not killed in a car accident tonight, we will go to the cricket” (namely, it assumes that it is possible for it to rain tomorrow, that we do not get killed in a car accident, but also that we could instead get killed in a domestic incident or in some other way, and still somehow we go to the game.) As Priest notes, “The list of conditions is, arguably, open-ended and indefinite. So no conditional of this kind that we could formulate explicitly is true!”)

This move is essentially right, but it is a bit too swift, though. Come back to the premise of the first argument. If the conditional ‘if it does not rain tomorrow, we will go to the cricket’ is not true, then neither is the conditional ‘if it does not rain tomorrow and I am not killed in a car accident tonight, we will go to the cricket’. I might be killed in a domestic accident, all means of transport may break down tomorrow, we might be invaded by Martians, etc. The list of conditions is, arguably, open-ended and indefinite. So no conditional of this kind that we could formulate explicitly is true!

(84)

5.2.6

[Completely De-Enthymemizing the Arguments by Adding a Ceteris paribus Clause (“other things being equal”).]

[But in fact we can capture all of these infinitely many needed additional de-enthymemizing clauses by simply saying for all of them, “other things being equal,” which is called a ceteris paribus clause.]

[In section 5.2.5, we noted that the defense for the counter-examples was not so obviously sufficient, because we can always think of yet another proposition that is needed among the premises to ensure the validity of the argument. Priest notes now that there is a sh0rt-hand for capturing these infinitely many additions: we may add “other things being equal,” and this is called a ceteris paribus clause. So whenever we say that these counter-examples should be valid, we are really regarding them as having ceteris paribus clauses.]

Fortunately, though, we can capture all the open-ended conditions in a catch-all clause. We can say: ‘if it does not rain tomorrow then, other things being equal, we will go to the cricket’ or ‘if it does not rain tomorrow and everything else relevant remains unchanged, we will go to the cricket’. The Latin for ‘other things being equal’ is ceteris paribus, so we can call this a ceteris paribus clause. It is the conditional with the ceteris paribus clause that we are really assenting to when we assent to the premise of the first argument. Similarly for the other arguments.

(85)

5.2.7

[Ceteris paribus Clauses as Being Conditioned by the Antecedent that They are Conjoined to and as Being Context-Dependent]

[Ceteris paribus clauses {1} are conditioned by the other antecedent term they are conjoined with, because that term might require particular clauses be implied while others be excluded and {2} are context-dependent.]

[We can formulate an argument with a ceteris paribus clause in the premise conditional in the following way: “if A and C_Athen B,” where C_Ais the ceteris paribus clause. We can say two things about the ceteris paribus clause. {1} It depends on the content of A. For example, if A is, “it does not rain tomorrow,” then the ceteris paribus clause might need to include “we are not invaded by Martians.” However, if A is “flying saucers arrive from Mars,” then of course C_Acannot include “we are not invaded by Martians,” for obviously it is assumed by A that we are. {2} The ceteris paribus clause is context-dependent. (I did not grasp this example very well, so please do not trust my summarization and rather check the quotation below. In Priest’s illustration, he gives a situation with two observers, one having more information than the other. Here, Priest is the driver of a car, and he is stuck behind a truck, but he can see that another car is coming the other direction in the next lane. So Graham’s inference will involve a conditional whose C_Awill include the following: “If I pass the truck now, and there is a car coming the other way, there will be an accident.” But the passenger cannot see the car coming the other way. So their conditional will contain a C_Athat includes, “If Graham passes the truck now, and Graham is a good driver, then there will not be an accident.” The reason I am confused is that surely the passenger would also say, “If Graham passes the truck now, and if there is a car coming the other way, even if Graham is a good driver, there will be an accident.” Maybe the idea is that in this situation, it would never cross the passenger’s mind to include that clause (and hence the passenger’s inference concludes there will be no accident.) But surely the passenger would agree that this addition about incoming traffic should be included, if we mention it to them. So please check the quotation.]

A conditional of this kind is of the form ‘if A and C_Athen B’, where C_Ais the ceteris paribus clause. How does this clause function? It is no ordinary conjunct. For a start, as we have seen, it captures an open-ended set of conditions. It also depends very much on A. (That is what the subscript A is there to remind you of.) If A is ‘it does not rain tomorrow’, then C_Aincludes the condition that we are not invaded by Martians. If A is ‘flying saucers arrive from Mars’, it does not.

Finally, it is context-dependent. For example, suppose that I am driving, and am stuck behind a truck. A is ‘I overtake now’. From where I sit, I can see that there is a car coming the other way. This is part of my C_A. Hence, I can truly assert ‘If I overtake now, there will be an accident.’ You, on the other hand, are sitting in the passenger seat and cannot see the oncoming traffic. You do know, however, that I am a safe driver. That is part of your C_A. Hence you can truly assert ‘If Graham overtakes now, there will not be an accident’.

(84)

5.2.8

[A Conditional with a Ceteris paribus Clause as ‘A > B’. Defining It Like a Strict Conditional.]

[‘A > B’ means a conditional with a ceteris paribus clause. And, “A > B is true (at a world) if B is true at every (accessible) world at which A ∧ C_A is true” (84).]

[Priest says that we will write A > B to mean a conditional with a ceteris paribus clause. Now recall from section 4.5.2 and section 4.5.3 that we define the strict conditional as: □(A ⊃ B). And recall from section 2.3.5 that we define the necessity operator in the following way.

v_w(□A) = 1 if, for all w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

(Priest p.22, section 2.3.5)

(22)

And recall also from section 4.5.4 how I am not able to find where there is a statement for how to evaluate the conditional in modal logic, but I proposed the following:

v_w(A ⊃ B) = 1 if v_w(A) = 0 or v_w(B) = 1, and 0 otherwise.

(not in Priest that I know of or where, yet. See section 4.5.4)

Thus I wonder if we can say that:

v_w□(A ⊃ B) = 1 if, for all w′ ∈ W such that wRw′, v_w′(A) = 0 or v_w′(B) = 1, and 0 otherwise.

(not in Priest and probably wrong)

At any rate, given the semantics of the strict conditional, we would also say that: “A > B is true (at a world) if B is true at every (accessible) world at which A ∧ C_A is true.” He then tells us we will spell this out more precisely.]

Let us write A > B for a conditional with a ceteris paribus clause. Suppose one accepts a strict account of the conditional. Then a conditional A ⥽ B is true (at a world) if A ⊃ B is true at every (accessible) world; that is, if B is true at every (accessible) world at which A is true. Thus, the conditional A > B is true (at a world) if B is true at every (accessible) world at which A ∧ C_A is true. How do we spell out this idea more precisely?

(84)

From:

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

24 May 2018

Priest (4.9) An Introduction to Non-Classical Logic, ‘Lewis’ Argument for Explosion,’ summary

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

Summary of

Graham Priest

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

Part I:

Propositional Logic

Non-Normal Modal Logics; Strict Conditionals

4.9

Lewis’ Argument for Explosion

Brief summary:

(4.9.1) Strict conditionals do not require relevance, as we see for example with: ⊨ (A ∧ ¬A) ⥽ B. So we might object to them on this basis. (4.9.2) C.I. Lewis argues that (A ∧ ¬A) ⥽ B is intuitively valid, because from A ∧ ¬A it is intuitively valid to infer A and ¬A; from ¬A it is intuitively valid to infer ¬A ∨ B, and from A and ¬A ∨ B it is intuitively valid, by disjunctive syllogism, to derive B. [Now, if each step has a connection on the basis of its intuitive validity, that means the final conclusion B should have a connection, by extension, to A ∧ ¬A on the basis of the intuitively valid steps leading from the premise to the final conclusion. So despite objections to the contrary, there is a connection between the antecedent and consequent in (A ∧ ¬A) ⥽ B, according to Lewis. (4.9.3) C.I. Lewis also formulates an argument for the connection between antecedent and conclusion for A ⥽ (B ∨ ¬B), but this argument is a bit less convincing than the one for (A ∧ ¬A) ⥽ B.

4.9.1

[Strict Conditionals as Lacking Relevance]

4.9.2

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in (A ∧ ¬A) ⥽ B by Means of Disjunctive Syllogism]

4.9.3

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in A ⥽ (B ∨ ¬B)]

Summary

4.9.1

[Strict Conditionals as Lacking Relevance]

[Strict conditionals do not require relevance, as we see for example with: ⊨ (A ∧ ¬A) ⥽ B. So we might object to them on this basis.]

[Recall from section 4.5.2 and section 4.5.3 that the strict conditional A⥽B is defined as □(A ⊃ B). In previous sections – see for example section 4.6 and section 4.8 – Priest has considered objections for the strict conditional ⥽ as providing a correct account of the conditional. Priest will now consider a final objection to the this claim about the correctness of the strict conditional. He notes that we have the intuition that this definition is inadequate, because we expect in a conditional that there is some kind of connection between the antecedent and the consequent (for otherwise, what is the sense of the conditionality of their relation?). But strict conditionals do not require any such connection. For example, there is no connection between A ∧ ¬A and B, (even though, as we saw in section 4.6.3: ⊨ (A ∧ ¬A) ⥽ B.)]

Let us end by considering a final objection to ⥽ as providing a correct account of the conditional. It is natural to object that this account cannot be correct, since a conditional requires some kind of connection between antecedent and consequent; yet a strict conditional requires no such connection. There is no connection in general, for example, between A ∧ ¬A and B.

(76)

4.9.2

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in (A ∧ ¬A) ⥽ B by Means of Disjunctive Syllogism]

[C.I. Lewis argues that (A ∧ ¬A) ⥽ B is intuitively valid, because from A ∧ ¬A it is intuitively valid to infer A and ¬A; from ¬A it is intuitively valid to infer ¬A ∨ B, and from A and ¬A ∨ B it is intuitively valid, by disjunctive syllogism, to derive B. (Now, if each step has a connection on the basis of its intuitive validity, that means the final conclusion B should have a connection, by extension, to A ∧ ¬A on the basis of the intuitively valid steps leading from the premise to the final conclusion. So despite objections to the contrary, there is a connection between the antecedent and consequent in (A ∧ ¬A) ⥽ B, according to Lewis.)]

[Despite what we said about relevance above in section 4.9.1, C.I. Lewis does see a connection in the strict conditional even in explosive formulas like ⊨ (A ∧ ¬A) ⥽ B. (On explosion and the strict conditional, see section 4.8). Only, the connection here is one obtained by a series of inferences, each of which is presumably intuitively valid. (So if each inference is intuitively valid, then they have a logical connection. And so ultimately the explosive inference is intuitively valid). We begin with a premise that is a contradiction: A ∧ ¬A. We then infer the conjects from this conjunction, ¬A and A. From ¬A we infer the disjunction ¬A ∨ B, which with A and by disjunctive syllogism, we infer B. (The idea might be the following, but I am just guessing here. By modus ponens, from A, A ⥽ B we can infer B. And, A⥽B is equivalent ¬A ∨ B. And as we see, by disjunctive syllogism from A, ¬A ∨ B we can infer B. Furthermore, maybe another idea here is that when there are premises validly making some other formula true, then you can make the premises be the antecedents and the conclusion the consequent in another formula that will be valid, but I am guessing. So because A ⥽ B is equivalent to ¬A ∨ B, and because the inference from A ∧ ¬A to B is shown to be valid using disjunctive syllogism on premises validly derived from A ∧ ¬A, that means (A ∧ ¬A) ⥽ B should be intuitively valid. Again, these are guesses. See the quotation below.]

C.I. Lewis, who did accept as an adequate account of the conditional, thought that there was a connection, at least in this case. The connection is shown in the following argument:

xxxxxxxxxxxxxA∧¬A

xxxxxxxxxxxx______

xxxxxA∧¬Axxxxx¬A

xxxxx____xxxxx___

xxxxxxxAxxxxx¬A∨B

xxxxx____________

xxxxxxxxxxxB

Premises are above lines; conclusions are below. The only ultimate premise is A∧¬A; the only ultimate conclusion is B. The inferences that the argument uses are: inferring a conjunct from a conjunction; inferring a disjunction from a disjunct; and the disjunctive syllogism: A, ¬A ∨ B ⊢ B. Of course, all these are valid in the modal logics we have looked at. If contradictions do not entail everything, then one of these must be wrong. We will return to this point in a later chapter.

(76)

4.9.3

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in A ⥽ (B ∨ ¬B)]

[C.I. Lewis also formulates an argument for the connection between antecedent and conclusion for A ⥽ (B ∨ ¬B), but this argument is a bit less convincing than the one for (A ∧ ¬A) ⥽ B.]

[Priest then notes that “Lewis also argued that there is a connection in the case of the conditional A ⥽ (B ∨ ¬B) as well,” using the following argument. We begin with A. From this we infer (A ∧ B) ∨ (A ∧ ¬B) (I am not exactly sure how, but maybe the reasoning is something like the following. Either B or ¬B holds, on account of excluded middle. Since we have affirmed A, then either (A ∧ B) or (A ∧ ¬B) holds.) From this we infer A ∧ (B ∨ ¬B) (I am not sure how again, but it seems like we extract the A as being the common affirmed formula in both, leaving (B ∨ ¬B).) And from this we infer (B ∨ ¬B) by pulling it out as one of the conjuncts. So by beginning with A, we can validly infer (B ∨ ¬B), and thus A ⥽ (B ∨ ¬B).) Priest says this argument is less convincing than the prior one, because “the first step seems evidently to smuggle in the conclusion” (77). (But I am not sure how that works other than the fact that the (B ∨ ¬B) that we want to derive is built into (A ∧ B) ∨ (A ∧ ¬B) by a sort of distribution.) Please see the quotation below, as I do not know the precise reasoning for each step.]

Lewis also argued that there is a connection in the case of the conditional A ⥽ (B ∨ ¬B) as well. The connection is provided by the | following argument:

xxxxxxxxxxxxxA

xxxxx_________________

xxxxx(A ∧ B) ∨ (A ∧ ¬B)

xxxxx_________________

xxxxxxxxA ∧ (B ∨ ¬B)

xxxxxxx______________

xxxxxxxxx(B ∨ ¬B)

This argument is less convincing than that of 4.9.2, however, since the first step seems evidently to smuggle in the conclusion.

(76-77)

From:

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

22 May 2018

Priest (4.8) An Introduction to Non-Classical Logic, ‘The Explosion of Contradictions,’ summary

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

Summary of

Graham Priest

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

Part I:

Propositional Logic

Non-Normal Modal Logics; Strict Conditionals

4.8

The Explosion of Contradictions

Brief summary:

(4.8.1) One of the paradoxes of the strict conditional is: ⊨ (A ∧ ¬A) ⥽ B. By modus ponens we derive: (A∧¬A)⊨B. In other words, contradictions entail everything (any arbitrary formula whatsoever). But this is counter-intuitive, and there are counter-examples that we will consider. (4.8.2) The first counter-example: Bohr knowingly combined inconsistent assumptions in his model of the atom, but on that account the model functioned well. However, explosion does not hold here, because we cannot on the basis of the contradiction infer everything else, like electronic orbits being rectangles. (4.8.3) The second counter-example: we can have inconsistent laws without their contradiction entailing everything. (4.8.4) The third counter-example: there are perceptual illusions that give us inconsistent impressions without giving us all impressions. For example, the waterfall illusion gives us the impression of something moving and not moving, but it does not thereby also give us every other impression whatsoever. The fourth counter-example: there can be fictional situations where contradictions hold but that thereby not all things hold as well.

4.8.1

[The Strict Conditional Involves the Explosion of Contradictions]

4.8.2

[Counter-Example 1: The Bohr Model’s Contradictory Assumptions as Non-Explosive]

4.8.3

[Counter-Example 2: Inconsistent Legislation]

4.8.4

[Counter-Example 3: Perceptual Illusions. Counter-Example 4: Fictional Situations]

Summary

4.8.1

[The Strict Conditional Involves the Explosion of Contradictions]

[One of the paradoxes of the strict conditional is: ⊨ (A ∧ ¬A) ⥽ B. By modus ponens we derive: (A∧¬A)⊨B. In other words, contradictions entail everything (any arbitrary formula whatsoever). But this is counter-intuitive, and there are counter-examples that we will consider.]

[Let us first recall some notions regarding the strict conditional. In section 4.5.2 and section 4.5.3 we learned that the strict conditional is defined as “□(A ⊃ B),” and it is symbolized as A⥽B. In section 4.6.2 and section 4.6.3, we learned that modal systems that can handle conditionality should be systems where modus ponens holds: A, A⥽B ⊨ B. (I did not know why exactly this is necessary, but I guessed it was for the following reason. Suppose modus ponens does not hold. That would mean by affirming the antecedent, we could not obtain the consequent. But were that the case, then we have lost a basic intuition we have about conditionality, namely, that the consequent will follow necessarily from the antecedent.) We learned in section 4.6.2 that for modus ponens to hold in a modal system, it needs the ρ-constraint (reflexivity). (Recall it from section 3.2.3: “ρ (rho), reflexivity: for all w, wRw” p.36.) But we then learned in section 4.6.3 that no matter how many other constraints we add to ρ, we will always obtain the paradoxes of strict implication, with one being: ‘⊨ (A ∧ ¬A) ⥽ B’. Now in our current section, Priest says that by modus ponens, from ⊨ (A ∧ ¬A) ⥽ B we can derive (A∧¬A)⊨B. (I do not know exactly how that works, however. I guess the idea is that if we establish the conditional, and if we have modus ponens, then that means simply from the antecedent being affirmed we can infer the consequent as a semantic consequence. The important philosophical point here is that) the strict conditional in any modal system that can handle conditionality leads us to being able to derive any arbitrary formula whatsoever from a contradiction. As Priest puts it: “Contradictions would entail everything.” But this is counter-intuitive. Priest will now give three counter-examples of situations or theories that are inconsistent but also where we should not be able thereby to infer that everything whatsoever holds.]

The toughest objections to a strict conditional, at least as an account of the indicative conditional, come from the fact that ⊨(A∧¬A)⥽B. If this were the case, then, by modus ponens, we would have (A∧¬A)⊨B. Contradictions would entail everything. Not only is this highly counterintuitive, | there would seem to be definite counter-examples to it. There appear to be a number of situations or theories which are inconsistent, yet in which it is manifestly incorrect to infer that everything holds. Here are three very different examples.

(74-75)

4.8.2

[Counter-Example 1: The Bohr Model’s Contradictory Assumptions as Non-Explosive]

[The first counter-example: Bohr knowingly combined inconsistent assumptions in his model of the atom, but on that account the model functioned well. However, explosion does not hold here, because we cannot on the basis of the contradiction infer everything else, like electronic orbits being rectangles.]

[I do not know much about the first example, so please see the quotation below. The basic idea is that Bohr knowingly combined two inconsistent assumptions in his model of the atom, namely, he assumes “the standard Maxwell electromagnetic equations” but also “that energy could come only in discrete packets (quanta).” Yet, despite its obvious inconsistency, both assumptions were needed for the model to work and “many of its observable predictions were spectacularly verified.” Priest’s philosophical point here is that on the basis of this contradiction, we cannot infer everything else. “Bohr did not infer, for example, that electronic orbits are rectangles” (75).]

The first is a theory in the history of science: Bohr’s theory of the atom (the ‘solar system’ model). This was internally inconsistent. To determine the behaviour of the atom, Bohr assumed the standard Maxwell electromagnetic equations. But he also assumed that energy could come only in discrete packets (quanta). These two things are inconsistent (as Bohr knew); yet both were integrally required for the account to work. The account was therefore essentially inconsistent. Yet many of its observable predictions were spectacularly verified. It is clear though that not everything was taken to follow from the account. Bohr did not infer, for example, that electronic orbits are rectangles.

(75)

4.8.3

[Counter-Example 2: Inconsistent Legislation]

[The second counter-example: we can have inconsistent laws without their contradiction entailing everything.]

[In Priest’s second counter-example, we have two laws that together function together non-problematically in most cases, but in a particular situation they come into contradiction. Priest then says that on the basis of this contradiction, “it would be stupid to infer from this that, for example, the traffic laws are consistent” (75). (I did not quite get how that works. Are we saying that we can consider our two inconsistent laws as presenting a structure like A∧¬A, and “the traffic laws are consistent” is some arbitrary B that we try to derive from it? At any rate, surely at least we might say that from this contradiction we cannot derive any other traffic law we want.)]

Another example: pieces of legislation are often inconsistent. To avoid irrelevant historical details, here is an hypothetical example. Suppose that an (absent-minded) state legislator passes the following traffic laws. At an unmarked junction, the priority regulations are:

(1) Any woman has priority over any man.

(2) Any older person has priority over any younger person.

(We may suppose that clause 2 was meant to resolve the case where two men or two women arrive together, but the legislator forgot to make it subordinate to clause 1.) The legislation will work perfectly happily in three out of four combinations of sex and age. But suppose that Ms X, of age 30, approaches the junction at the same time as Mr Y, of age 40. Ms X has priority (by 1), but has not got priority (by 2 and the meaning of ‘priority’). Hence, the situation is inconsistent. But, again, it would be stupid to infer from this that, for example, the traffic laws are consistent.

(75)

4.8.4

[Counter-Example 3: Perceptual Illusions. Counter-Example 4: Fictional Situations]

[The third counter-example: there are perceptual illusions that give us inconsistent impressions without giving us all impressions. For example, the waterfall illusion gives us the impression of something moving and not moving, but it does not thereby also give us every other impression whatsoever. The fourth counter-example: there can be fictional situations where contradictions hold but that thereby not all things hold as well.]

[The third example is that there are perceptual illusions that can give us inconsistent impressions. For example, the waterfall illusion causes us to see something both in motion and not in motion. But thereby we do not perceive everything else, like for example that everything is red all over. The fourth example is that in fictional situations where there are contradictions, that does not entail that everything holds in that fictional situation. (For some reason the fourth one is placed in a footnote, despite being an excellent and convincing counter-example.)]

Third example: it is possible to have visual illusions where things appear contradictory. For example, in the ‘waterfall effect’, one’s visual system is conditioned by constant motion of a certain kind, say a rotating spiral. If one then looks at a stationary situation, say a white wall, it appears to move in the opposite direction. But, a point in the visual field, | say at the top, does not appear to move, for example, to revolve around to the bottom. Thus, things appear to move without changing place: the perceived situation is inconsistent. But not everything perceivable holds in this situation. For example, it is not the case that the situation is red all over.⁵

(75-76)

5. A fourth kind of example is provided by certain fictional situations, in which contradictory states of affairs hold. This may well be the case without everything holding in the fictional situation.

(76)

From:

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

2 May 2018

Priest (4.6) An Introduction to Non-Classical Logic, ‘The Paradoxes of Strict Implication,’ summary

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

Summary of

Graham Priest

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

Part I:

Propositional Logic

Non-Normal Modal Logics; Strict Conditionals

4.6

The Paradoxes of Strict Implication

Brief summary:

(4.6.1) We wonder if the definition of the strict conditional – A⥽B is defined as □(A⊃B) – is adequate. But first we need to address the matter of its variance under different systems of modal logic. (4.6.2) To model conditionality in general and the strict conditional in particular, we need modus ponens to hold, as it is a basic inferential principle that should hold when the conditional has its normal semantics. But in systems without the ρ-constraint (reflexivity), modus ponens will fail. Thus our system at least needs the ρ-constraint . (4.6.3) We need not narrow our systems down any further than systems with the ρ-constraint, because no matter what, they will all lead to the paradoxes of strict implication: ‘□B ⊨ A ⥽ B’, ‘¬◊A ⊨ A ⥽ B’; and also ‘⊨ A ⥽ (B ∨ ¬B) ’, ‘⊨ (A ∧ ¬A) ⥽ B’.

4.6.1

[The Question of the Adequacy of the Strict Conditional ⥽]

4.6.2

[The Need for the ρ-Constraint]

4.6.3

[The Paradoxes of Strict Implication]

Summary

4.6.1

[The Question of the Adequacy of the Strict Conditional ⥽]

[We wonder if the definition of the strict conditional – A⥽B is defined as □(A⊃B) – is adequate. But first we need to address the matter of its variance under different systems of modal logic.]

[Recall from section 4.5 the notion of the strict conditional. In section 4.5.3 we learn that the strict conditional, symbolized as ⥽, is defined in the following way: A⥽B is defined as □(A⊃B). (p.72, section 4.5.3). We now ask if this definition of the conditional is adequate. But as the properties of the strict conditional will vary according to the modal logic system at hand, we need to say more on this matter.]

Does it provide an adequate account of the conditional? Each system of modal logic gives ⥽ different properties. Hence, before we can answer that question, we need to address the question of which system of modal logic it is that is at issue. Let me make two comments on this.

(72)

4.6.2

[The Need for the ρ-Constraint]

[To model conditionality in general and the strict conditional in particular, we need modus ponens to hold, as it is a basic inferential principle that should hold when the conditional has its normal semantics. But in systems without the ρ-constraint (reflexivity), modus ponens will fail. Thus our system at least needs the ρ-constraint .]

[Priest first notes that modus ponens fails in systems without the ρ-constraint (the reflexivity constraint; see section 3.2.3). It seems that for conditionality we would want modus ponens to hold: A, A⥽B ⊨ B. I do not know the exact reason why, but it would seem that conditionality should allow us to infer the consequent from an affirmation of the antecedent. For otherwise, what is the sense of the conditional without that also holding? But, Priest says, in systems without the reflexivity constraint, modus ponens will not hold. Thus we at least need the reflexivity constraint.]

First, it is natural to suppose that any notion of necessity that is to be employed in defining a notion of conditionality must be at least as strong as Kρ (or Lρ if one is countenancing non-normal systems). This is because, without ρ, modus ponens fails: A, A⥽B ⊭ B. With it, it holds, as simple tableau tests verify.

(73)

4.6.3

[The Paradoxes of Strict Implication]

[We need not narrow our systems down any further than systems with the ρ-constraint, because no matter what, they will all lead to the paradoxes of strict implication: ‘□B ⊨ A ⥽ B’, ‘¬◊A ⊨ A ⥽ B’; and also ‘⊨ A ⥽ (B ∨ ¬B) ’, ‘⊨ (A ∧ ¬A) ⥽ B’.]

[Priest’s next point I might not summarize properly, but I am guessing it is the following. So far we specified that for the strict conditional we need systems with the reflexivity constraint. But we learn now that we need not narrow our systems down any further, because no matter how constrained we make them, all systems will lead to certain paradoxes. And it is these paradoxes that lead us to question the notion that the strict conditional models the English conditional.]

Second, a further determination of this question is not very important for what follows. This is because the major objections to the claim that English conditionals are strict hinge on a feature that the strict conditional possesses in all systems of modal logic. In all systems of modal logic the following hold:

□B ⊨ A ⥽ B

¬◊A ⊨ A ⥽ B

These facts are sometimes called the ‘paradoxes of strict implication’. A tableau test verifies that these hold in L, and so in all the normal and non-normal systems that we have looked at. Since, in all systems, we also have ⊨□(B∨¬B) and ⊨¬◊(A∧¬A), this gives us as special cases:

⊨ A ⥽ (B ∨ ¬B)

⊨ (A ∧ ¬A) ⥽ B

(73)