by Corry Shores

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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

5.

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).

[Three Problematic Conditional Inferences]

[Counter-Example Illustrations of These Problematic Conditional Inferences]

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

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

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

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

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

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

Summary

[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 | ||

1. .
2. .
3. .
4. .
5. .
6. .
7. | A ⊃ B ↓ ¬((A ∧ C) ⊃ B) ↓ A ∧ C ↓ ¬B ↓ A ↓ ¬C ↙ ↘ ¬A B × ×
| P .
P .
2¬⊃ .
2¬⊃ .
3∧ .
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 | ||

1. .
2. .
3. .
4. .
5. .
6. .
7. | A ⊃ B ↓ B ⊃ C ↓ ¬(A ⊃ C) ↓ A ↓ ¬C ↙ ↘ ¬A B ↙ ↓ ↓ ↘ ¬B C ¬B C × × × ×
| P .
P .
P .
3¬⊃ .
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 | ||

1. .
2. .
3. .
4. .
5. .
6. .
| A ⊃ B ↓ ¬(¬B ⊃ ¬A) ↓ ¬B ↓ ¬¬A ↓ A ↙ ↘ ¬A B × ×
| P .
P .
2¬⊃ .
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) ⊃BTransitivity:

A⊃B,B⊃C⊨A⊃ CContraposition:

A⊃B⊨ ¬B⊃ ¬AIt 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)

[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 downen 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 downen 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)

[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.

[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)

[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)

[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 aceteris paribusclause. It is the conditional with theceteris paribusclause that we are really assenting to when we assent to the premise of the first argument. Similarly for the other arguments.(85)

[*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 _{A }*then

*B*,” where

*C*is the

_{A }*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*cannot include “we are not invaded by Martians,” for obviously it is assumed by

_{A }*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*will 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

_{A }*C*that 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 }A conditional of this kind is of the form ‘if

AandCthen_{A }B’, whereCis the_{A }ceteris paribusclause. 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 onA. (That is what the subscriptAis there to remind you of.) If A is ‘it does not rain tomorrow’, thenCincludes the condition that we are not invaded by Martians. If_{A }Ais ‘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.

Ais ‘I overtake now’. From where I sit, I can see that there is a car coming the other way. This is part of myC. 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_{A}C. Hence you can truly assert ‘If Graham overtakes now, there will not be an accident’._{A}(84)

[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 allw′ ∈Wsuch thatwRw′,(A) = 1; and 0 otherwise.v_{w′}(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 allw′ ∈Wsuch thatwRw′,(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>Bfor a conditional with aceteris paribusclause. Suppose one accepts a strict account of the conditional. Then a conditionalA⥽Bis true (at a world) ifA⊃Bis true at every (accessible) world; that is, ifBis true at every (accessible) world at whichAis true. Thus, the conditionalA> B is true (at a world) ifBis true at every (accessible) world at whichA∧Cis true. How do we spell out this idea more precisely?_{A}(84)

From:

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

.

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