1. NEW FOUNDATIONS FOR IMPERATIVE LOGIC III: A General Definition of Argument Validity
Peter B. M. Vranas
University of Wisconsin-Madison
Talk at the University of Warsaw, 17 May 2012

2. A TAXONOMY OF ARGUMENTS
Pure declarative arguments
You sinned shamelessly.
So: You sinned.
Pure imperative arguments
Repent quickly.
So: Repent.
Mixed-premise declarative
If you sinned, repent.
You will not repent.
So: You did not sin.
Mixed-premise imperative
You sinned.
Cross-species declarative
Repent.
So: You can repent.
Cross-species imperative
You must repent.
Consider this argument: “If you sinned, repent; you sinned; so, repent.” The first premise, “If you sinned, repent”, is an imperative, or a prescription, as I prefer to call it. The second premise, “You sinned”, is a proposition. The conclusion, “Repent”, is again a prescription. To many people, this argument looks valid. But if imperatives cannot be true or false, then to say that this argument is valid is not to say that the truth of its premises guarantees the truth of its conclusion. So we need a different account of validity for such arguments. In this talk I propose such an account.
To be general, define an argument as an ordered pair whose first member is a nonempty set of propositions or prescriptions or both, namely the premises of the argument, and whose second member is either a proposition of a prescription, namely the conclusion of the argument. Say that an argument is declarative if its conclusion is a proposition, as in the three arguments in the left column of the table. Say that an argument is imperative if its conclusion is a prescription, as in the three arguments in the right column of the table. Say that an argument is pure if its premises and its conclusion are either all propositions or all prescriptions, as in the two arguments in the top row of the table. Say that an argument is mixed-premise if its premises include both a proposition and a prescription, as in the two arguments in the middle row of the table. Finally, say that an argument is cross-species if either its premises are all propositions and its conclusion is a prescription or the other way around, as in the two arguments in the bottom row of the table. On this classification, we have six kinds of arguments, which you can see on the table with examples. I will propose a general definition of argument validity: a definition that applies to all six kinds of arguments.

3. RELATED RESEARCH
New foundations for imperative logic I: Logical connectives, consistency, and quantifiers. Noûs (2008) 42:
In defense of imperative inference. Journal of Philosophical Logic (2010) 39:
New foundations for imperative logic II: Pure imperative inference. Mind (2011) 120:
New foundations for imperative logic IV: Soundness and completeness. In preparation.
The present work continues a series of papers on which I have been working for a while. The first paper was “New foundations for imperative logic I”, in which I proposed an account of prescriptions as ordered pairs of logically incompatible propositions, namely satisfaction propositions and violation propositions, and also among other things I defined the conjunction of prescriptions. In another paper, “In defense of imperative inference”, I replied to common objections to the possibility or the usefulness of imperative inference. In a third, recently published paper, I proposed an account of validity for pure imperative arguments. Let me clarify that to understand the present talk you don’t need to know anything about the previous papers. As another preliminary point, in all these papers, including the present one, I don’t introduce syntactic considerations: I don’t introduce any formal language. I plan to do this in the next paper in the series.

4. OVERVIEW Part 1: THE GENERAL DEFINITION Part 2:
CROSS-SPECIES ARGUMENTS
Part 3:
MIXED-PREMISE ARGUMENTS
My talk today has three main parts. In the first part, I propose and defend a general definition of argument validity. In the other two parts, I examine some implications of this definition for cross-species and mixed-premise arguments.

5. PRELIMINARIES
A prescription is an ordered pair of incompatible propositions: the satisfaction proposition and the violation proposition of the prescription. Their disjunction is the context, and its negation is the avoidance proposition of the prescription.
A prescription is unconditional if its context is necessary (e.g., “Repent”) and is conditional otherwise (e.g., “If you sinned, repent”).
One need only consider single-premise pure and cross-species arguments, and two-premise mixed-premise arguments.
As I said, I understand a prescription as an ordered pair of logically incompatible propositions. The first member of the pair is the satisfaction proposition of the prescription, and the second member of the pair is the violation proposition of the prescription. Take, for example, the prescription expressed by the imperative sentence “Repent”. Its satisfaction proposition is the proposition that you repent, and its violation proposition is the proposition that you do not repent. Call the disjunction of the satisfaction and the violation proposition the context of the prescription. For example, the context of the prescription “Repent” is the proposition that you repent or you do not repent. The context is necessary, so say that the prescription is unconditional. By contrast, the context of a conditional prescription is not necessary. Take, for example, the prescription “If you sinned, repent”. Its satisfaction proposition is the proposition that you sinned and you repent, its violation proposition is the proposition that you sinned and you do not repent, and its context is the proposition that you sinned. The context is not necessary, so the prescription is conditional. Say that a prescription is avoided if and only if it is neither satisfied nor violated. So the prescription “If you sinned, repent” is avoided if and only if you did not sin.
As another preliminary point, note that for the purpose of defining argument validity one need only consider single-premise pure and cross-species arguments and two-premise mixed-premise arguments. This is because, if an argument has multiple propositions or multiple prescriptions as premises, then one should be able to replace the multiple propositions with a single proposition, namely their conjunction, and also replace the multiple prescriptions with a single prescription, namely their conjunction, without affecting validity.

6. MERITING ENDORSEMENT
A typical reason for adducing a valid argument is to convince people that its conclusion is true (if it is a proposition) or that it is supported by reasons (if it is a prescription).
A proposition merits endorsement iff it is true, and a prescription merits endorsement iff it is supported by reasons (facts).
An argument is valid only if, necessarily, if its premises merit endorsement, then its conclusion also merits endorsement.
Let’s reflect on how we use arguments in everyday discourse. A typical reason for adducing a valid declarative argument is to convince people that its conclusion, which is a proposition, is true. Similarly, a typical reason for adducing a valid imperative argument is to convince people that its conclusion, which is a prescription, is supported by reasons. For example, suppose I give you an argument whose conclusion is “Repent”. Typically, I do so in order to convince you that there are reasons for you to repent, or in order words that the prescription “Repent” is supported by reasons. To have a uniform terminology for propositions and prescriptions, say that a proposition merits endorsement if and only if it is true, and that a prescription merits endorsement if and only if it is supported by reasons, which I take to be facts. On this terminology, a typical reason to adduce any valid argument, declarative or imperative, is to convince people that its conclusion merits endorsement. This suggests that any useful definition of argument validity will have the following consequence: An argument is valid only if, necessarily, if its premises merit endorsement, then its conclusion also merits endorsement. In other words, the property of meriting endorsement is transmitted from the premises to the conclusion of a valid argument.

7. GUARANTEEING/SUSTAINING
A fact guarantees a proposition P iff, necessarily, if the fact exists, then P is true.
A proposition merits endorsement (i.e., is true) iff it is guaranteed by some fact (e.g., the fact that the proposition is true). Similarly, a prescription merits endorsement iff it is supported by some fact.
Uniform terminology: a fact sustains a proposition P iff it guarantees P, and a fact sustains a prescription I iff it supports I.
The next step is to realize that there is a relation between facts and propositions, which I call the relation of guaranteeing, which is in an important respect analogous to the relation of supporting between facts and prescriptions. Here is a definition: a fact guarantees a proposition if and only if, necessarily, if the fact exists, then the proposition is true. For example, the fact that Berlin is in Germany and Paris is in France guarantees the proposition that Paris is in France. If you accept a traditional definition of truthmaking, then you can say that a fact guarantees a proposition if and only if the fact is a truthmaker for the proposition. However, that definition of truthmaking is controversial, so I don’t want to assume it and I don’t say anything more about truthmakers in what follows.
The next step is to realize that a proposition merits endorsement, in other words is true, if and only if it is guaranteed by some fact. Why is that? Well, if a proposition is true, then it is a fact that it is true, and that fact guarantees it. Conversely, if a proposition is guaranteed by some fact, in other words if there exists some fact such that, necessarily, if the fact exists then the proposition is true, then the proposition is true; so the biconditional holds. Here is then the analogy between guaranteeing and supporting: just as a prescription merits endorsement if and only if it is supported by some fact, a proposition merits endorsement if and only if it is guaranteed by some fact. To have a uniform terminology for propositions and prescriptions, say that a fact sustains a proposition if and only if it guarantees the proposition, and that a fact sustains a prescription if and only if it supports the prescription. On this terminology, a proposition or a prescription merits endorsement if and only if it is sustained by some fact.

8. PRO TANTO/ALL-THINGS- CONSIDERED ENDORSEMENT
Meriting pro tanto endorsement: being sustained by some fact. Too weak.
Meriting all-things-considered endorsement: being undefeatedly sustained by some fact.
(DP) An argument is valid only if, necessarily, if its premises are sustained by some fact, then its conclusion is also sustained by some fact.
(DA) An argument is valid only if, necessarily, if its premises are undefeatedly sustained by some fact, then its conclusion is also undefeatedly sustained by some fact.
What I just said corresponds to what may be called meriting pro tanto endorsement, namely being sustained by some fact. This should be distinguished from meriting all-things-considered endorsement, namely being undefeatedly sustained by some fact, in the sense of being sustained not only by that fact, but also by the conjunction of that fact with any other fact. Arguably, meriting pro tanto endorsement is too weak. Suppose I convince you that there is some reason for you to smother a crying baby with a pillow; the reason being that, if you did so, you would eliminate the annoyance of the baby’s cries. Clearly, this is not enough to convince you to smother the crying baby with a pillow. So one might say that a more typical reason for adducing a valid argument is to convince people that the conclusion merits all-things-considered endorsement, not just pro tanto endorsement. This distinction suggests that the previously mentioned desirable consequence of any useful definition of validity must be split into two consequences. The first desirable consequence is that an argument is valid only if, necessarily, if its premises are sustained by some fact, then its conclusion is also sustained by some fact; in other words, if its premises merit pro tanto endorsement, then its conclusion also merits pro tanto endorsement. The second desirable consequence is that an argument is valid only if, necessarily, if its premises are undefeatedly sustained by some fact, then its conclusion is also undefeatedly sustained by some fact; in other words, if its premises merit all-things-considered endorsement, then its conclusion also merits all-things-considered endorsement. Neither of these two desirable consequences entails the other, but fortunately we don’t need to choose: the definition I am about to propose has both as consequences.

9. THE GENERAL DEFINITION
General Definition of Argument Validity: An argument is valid iff, necessarily, every fact that sustains every premise of the argument also sustains the conclusion of the argument.
In support of the General Definition (GD): GD
(1) entails both DP and DA, and (2) yields as special cases:
 Definition 1: P entails P iff, necessarily, every fact that guarantees P also guarantees P.
 Definition 2: I entails I iff, necessarily, every fact that supports I also supports I .
Here is at last the General Definition of Argument Validity: an argument is valid if and only if, necessarily, every fact that sustains every premise of the argument also sustains the conclusion of the argument. Clearly, this definition applies to all six kinds of arguments I distinguished above, so it is general. A first point in its favor is that, as I show in the paper on which this talk is based, it has both of the previously mentioned desirable consequences. A second point in its favor is that it yields as special cases both the standard definition of validity for pure declarative arguments and my previously defended definition of validity for pure imperative arguments. Indeed, take a pure declarative argument from a proposition P to a proposition P. According to the General Definition, this argument is valid if and only if, necessarily, every fact that sustains the premise, in other words guarantees the premise, since the premise is a proposition, also sustains, namely guarantees, the conclusion. In the paper I show that this is equivalent to the standard definition, which says that the argument is valid if and only if, necessarily, if P is true then P is true. This equivalence should not be surprising once you recall that, as I explained, a proposition is true if and only if it is guaranteed by some fact. Similarly, consider a pure imperative argument from a prescription or imperative I to a prescription or imperative I. According to the General Definition, this argument is valid if and only if, necessarily, every fact that sustains the premise, namely supports the premise, since the premise is a prescription, also sustains, namely supports, the conclusion. This is in effect my previously proposed definition of validity for pure imperative arguments. So, in conjunction with the detailed defense of this definition that I provided in my previous publication, “New Foundations II”, the fact that the General Definition has this as a consequence is another point in favor of the General Definition.

10. CROSS-SPECIES ARGUMENTS
PART 2
Part 1:
THE GENERAL DEFINITION
Part 2:
CROSS-SPECIES ARGUMENTS
Part 3:
MIXED-PREMISE ARGUMENTS
Let’s take stock. So far we have a General Definition, and we have some weighty considerations in favor of the General Definition. But an important question remains: is the definition usable? The definition may look a bit too abstract: all this talk about facts and sustaining may smack of metaphysical speculation. So the question is: can we actually use the definition in order to determine whether any specific argument is or not valid? In fact we can: I have proven a series of theorems which render the definition eminently usable. In the second part of my talk, I explain how to use the definition to find out whether a cross-species argument is or not valid.

11. CROSS-SPECIES IMPERATIVE ARGUMENTS
Definition 3 (follows from GD): P entails I iff, necessarily, every fact that guarantees P supports I.
Equivalence Theorem 1: P entails I iff P entails that some fact whose existence follows from P undefeatedly supports I.
E.g., the following argument is valid:
The fact that you have sworn to tell the truth is an undefeated reason for you to tell the truth.
So: Tell the truth.
Cf. Hume’s thesis and Poincaré’s Principle.
Take any cross-species imperative argument, namely an argument from a proposition P to a prescription I. According to the General Definition, this argument is valid if and only if, necessarily, every fact that sustains the premise, namely guarantees the premise, since the premise is a proposition, also sustains the conclusion, namely supports the conclusion, since the conclusion is a prescription. In the paper I prove this theorem, Equivalence Theorem 1: P entails I if and only if P entails that some fact whose existence follows from P undefeatedly supports I. This may look, and in fact is, a bit complicated, but let’s not lose track of the general structure of the theorem. The theorem says that a cross-species imperative argument is equivalent to a pure declarative argument, in the sense that the first argument is valid if and only if the second argument is valid. The second argument may be a bit complicated, but since it is a pure declarative argument, we can use the tools of standard declarative logic to find out whether it is or not valid. That’s how this theorem renders the definition usable. As you can imagine, I was very excited when I discovered this theorem: it was not easy to find a pure declarative argument whose validity is both necessary and sufficient for the validity of a cross-species imperative argument.
Let’s take an example. Consider the following argument: The fact that you have sworn to tell the truth is an undefeated reason for you to tell the truththat’s the premise. Conclusion: Tell the truth. By applying the theorem, we can see that this argument is valid. Why? Because the theorem says that the proposition P which is the premise entails the prescription “Tell the truth” if and only if P entails that some fact whose existence follows from P undefeatedly supports “Tell the truth”. Indeed, this is so because, necessarily, if the premise is true, then some fact whose existence follows from the premise, namely the fact that you have sworn to tell the truth, undefeatedly supports “Tell the truth”because that’s what the premise says, and if the premise is true what it says holds.
The validity of this argument may seem to conflict with what is known in the literature as Poincaré’s Principle, which basically says that, apart from certain trivial cases, one cannot derive a prescription from a proposition. This principle is inspired by Hume’s is/ought thesis, which says that, apart from certain trivial cases, one cannot derive an ought from an is, or one cannot derive a normative proposition from a non-normative one. However, if Poincaré’s Principle is inspired by Hume’s thesis, then strictly speaking the principle should not say that one cannot derive a prescription from any proposition; it should say instead that one cannot derive a prescription from any non-normative proposition. So understood, the principle does not conflict with the validity of the argument in my example because the premise of the argument is a normative proposition: it entails the existence of a normative reason.

12. FURTHER CONSEQUENCES OF EQUIVALENCE THEOREM 1
P entails I only if P entails that some fact undefeatedly supports I.
So the following arguments are not valid:
(1) You will tell the truth.
So: Tell the truth.
(2) There is a reason for you to tell the truth.
(3) Jupiter is the largest planet.
So: Either go to Jupiter or don’t go to the largest planet.
We have seen that certain prescriptions follow from certain normative propositions. But can a prescription follow from any non-normative proposition? The answer is “no”. It is a consequence of the theorem that a proposition entails a prescription only if the proposition entails that some fact undefeatedly supports the prescription. So the proposition must be normative: it must entail the existence of a normative reason. By applying this necessary condition for validity, we can see that the following arguments are not valid. First argument: “You will tell the truth; so, tell the truth”. Clearly, the premise does not entail that there is any reason for you to tell the truth; so the necessary condition for validity is violated, and this argument is not valid. I trust you will agree that this is the intuitively correct result. Second argument: “There is a reason for you to tell the truth; so, tell the truth”. The premise, “There is a reason for you to tell the truth”, does not entail that there is an undefeated reason for you to tell the truth; so the necessary condition for validity is again violated, and the argument is not valid. Initially, this argument might look valid to some people, but if you think for a moment you will realize that it is not valid. Go back to the example with the crying baby: clearly, from the premise that you have some reason to smother a crying baby with a pillow the prescription does not follow “Smother the crying baby with a pillow”you need an undefeated reason, not just some reason, for the conclusion to follow. Third argument: “Jupiter is the largest planet; so, either go to Jupiter or don’t go to the largest planet”. A similar argument has been proposed by Peter Geach as a counterexample to Poincaré’s Principle; somehow, Geach thinks the argument is valid. But the premise, “Jupiter is the largest planet”, does not entail the existence of any normative reason; so again the necessary condition for validity is violated, and the argument is not valid.

13. CROSS-SPECIES DECLARATIVE ARGUMENTS
Definition 4 (follows from GD): I entails P iff, necessarily, every fact that supports I guarantees P.
Equivalence Theorem 2: I entails P iff P follows from the proposition that there is a fact which possibly supports I.
E.g., the following are equivalent (and valid):
(1) Marry me. So: Possibly, there is a reason
for you to marry me.
(2) There is a fact which is possibly a reason
for you to marry me. So: Possibly, there is
a reason for you to marry me.
Take any cross-species declarative argument, namely an argument from a prescription I to a proposition P. According to the General Definition, this argument is valid if and only if, necessarily, every fact that sustains, namely supports, the premise, also sustains, namely guarantees, the conclusion. In the paper, I prove this theorem, Equivalence Theorem 2: I entails P if and only if P follows from the proposition that there is a fact which possibly supports I. Again, this theorem tells us that a cross-species declarative argument is equivalent to a pure declarative argument, in the sense that the first argument is valid if and only if the second argument is valid.
Take an example: Marry me; so, possibly, there is a reason for you to marry me. According to the theorem, this cross-species declarative argument is valid if and only if the following pure declarative argument is valid: There is a fact which is possibly a reason for you to marry me; so, possibly, there is a reason for you to marry me. Argument (2) is valid, as one can easily see by using standard quantified modal logic. So argument (1) is valid according to the theorem. Here is an intuitive reasoning to show you that this is the correct result. Either the conclusion of argument (1) is true or it is false. If it is true, then it is necessary, assuming that whatever is possible is necessarily possible, and if it is necessary then it follows from anything. So argument (1) is valid if its conclusion is true. On the other hand, if the conclusion is false, then it’s impossible that there be any reason for you to marry me, and in that case the premise is an imperative contradiction. Why? Because, just as a declarative contradiction is a proposition that is necessarily not true, an imperative contradiction is a prescription that is necessarily not supported by any reason. But if the premise is an imperative contradiction, then it entails anything, so argument (1) is again valid. That’s just an intuitive reasoning to suggest that this is the correct result, the result given by the theorem that argument (1) is valid.

14. FURTHER CONSEQUENCES OF EQUIVALENCE THEOREM 2
The following arguments are not valid:
(1) Marry me.
So: There is a reason for you to marry me.
(2) If he comes, leave the files open.
Do not leave the files open.
So: He will not come.
(3) Let it be the case that: he does not come, and you do not leave the files open.
(4) Marry Dan’s only daughter.
So: Dan has only one daughter.

15. MIXED-PREMISE ARGUMENTS
PART 3
Part 1:
THE GENERAL DEFINITION
Part 2:
CROSS-SPECIES ARGUMENTS
Part 3:
MIXED-PREMISE ARGUMENTS
I turn now to the third and final part of my talk, in which I examine some implications of the General Definition for mixed-premise arguments.

16. MIXED-PREMISE DECLARATIVE ARGUMENTS
Definition 6 (follows from GD): {I, P} entails P iff, necessarily, every fact that both supports I and guarantees P guarantees P.
Equivalence Theorem 3: {I, P} entails P iff P follows from the proposition that some fact which guarantees P possibly supports I.
E.g., the following argument is valid:
Either repent or undo the past.
It is impossible for you to undo the past.
So: It is possible for you to repent.
Take any mixed-premise declarative argument, namely an argument whose premises are both a prescription I and a proposition P, and whose conclusion is a proposition P. According to the General Definition, this argument is valid if and only if, necessarily, every fact that sustains its premises, namely every fact that both supports the prescription I and guarantees the proposition P, also sustains, namely guarantees, the conclusion. In the paper, I prove this theorem, Equivalence Theorem 3: the mixed-premise declarative argument from I and P to P is valid if and only if, necessarily, P follows from the proposition that some fact which guarantees P possibly supports I. Again, this tells us that a mixed-premise declarative argument is equivalent to a pure declarative argument: the first is valid if and only if the second is valid. Again, by using the tools of standard declarative logic, we can now find whether a mixed-premise declarative argument is or not valid. Here is an example. First premise: either repent or undo the past. This is a prescription. Second premise: it is impossible for you to undo the past. This is a proposition. Conclusion: it is possible for you to repent. This is a proposition. I will leave it as an exercise for you to show, by using Equivalence Theorem 3, that this argument is valid.

17. INCONSISTENCY BETWEEN PROPOSITIONS & PRESCRIPTIONS
Definition 7: P and I are inconsistent iff, neces-sarily, no fact both guarantees P and supports I.
Special cases: (1) P is impossible. (2) I is neces- sarily violated. (3) I entails the negation of P. (4) P entails the negation of I. (5) P entails that no fact supports I.
Examples:
(3) Repent. It is impossible for you to repent.
(4) Repent. The fact that you have sworn not to repent is a conclusive reason for you not to repent.
(5) Repent. There is no reason for you to repent.
As an aside, the General Definition also suggests a definition of inconsistency between propositions and prescriptions: a proposition and a prescription are inconsistent if and only if, necessarily, no fact sustains both of themin other words, no fact both guarantees the proposition and supports the prescription. Here are five special cases in which a proposition and a prescription are inconsistent. First case: the proposition is impossible. Second case: The prescription is necessarily violated; for example, “Run and don’t run”. Third case: the prescription entails the negation of the proposition; for example, “Repent, but it is impossible for you to repent”. Fourth case: the proposition entails the negation of the prescription; for example, “Repent, but the fact that you have sworn not to repent is a conclusive reason for you not to repent”. Fifth case: the proposition entails that no fact supports the prescription; for example, “Repent, but there is no reason for you to repent”.

18. MIXED-PREMISE IMPERATIVE ARGUMENTS
Definition 8 (follows from GD): {I, P} entails I iff, necessarily, every fact that both supports I and guarantees P supports I.
E.g., the following argument is valid:
Disarm the bomb. Every reason to disarm
the bomb is a reason to cut the wire.
So: Cut the wire.
Theorem 6: If I is unconditional and P is consistent with the proposition that some fact undefeatedly supports I&~I, then {I, P} does not entail I.
Take finally any mixed-premise imperative argument, namely an argument whose premises are both a prescription I and a proposition P, and whose conclusion is a prescription I. According to the General Definition, this argument is valid if and only if, necessarily, every fact that sustains its premises, namely every fact that both supports the prescription I and guarantees the proposition P, also sustains, namely supports, the conclusion. I don’t have an equivalence theorem to the effect that mixed-premise imperative arguments are equivalent to pure declarative arguments, and I suspect that they are not equivalent. But in the paper I provide some sufficient conditions, and one necessary condition, for validity. For example, in the paper I show that this argument is valid: “Disarm the bomb; every reason for you to disarm the bomb is a reason for you to cut the wire; so, cut the wire”. Theorem 6 here gives a necessary condition for validity, or equivalently a sufficient condition for invalidity: if I is unconditional and P is consistent with the proposition that some fact undefeatedly supports the conjunction of I with the negation of I, then the argument from I and P to I is not valid. This theorem has certain surprising consequences. So I come now to what is probably the highlight of my talk: I will argue that certain mixed-premise imperative arguments which seem to be obviously valid are in fact invalid.

19. INVALID MIXED-PREMISE IMPERATIVE ARGUMENTS1
(1) Either marry him or dump him.
You are not going to marry him.
So: Dump him.
(2) Either marry him or dump him.
You are not going to marry him,
and you are not going to dump him.
(2a) So: Dump him. (2b) So: Marry him.
If (1) is valid, then (2a) and (2b) are also valid. But (2a) and (2b) are not valid: from “Do X or Y” and “You are not going to do X or Y” neither “Do X” nor “Do Y” follows. So (1) is not valid.
Consider this argument: “Either marry him or dump him; you are not going to marry him; so, dump him”. I trust that this argument looks valid; in fact, one can imagine using it to give advice to a friend. And yet it is a consequence of Theorem 6 that this argument is invalid, as I show in the paper. How can this result be correct? To see how, note that if an argument is valid, and one adds extra declarative premises, then one gets a valid argument: if a conclusion follows from fewer premises, then it also follows from more premises. So let’s add to this argument the extra premise that you are not going to dump him. We get argument (2a): “Either marry him or dump him; you are not going to marry him, and you are not going to dump him; so, dump him”. But if this argument is valid, then by symmetry argument (2b) is also valid, in which the conclusion “Dump him” is replaced with the conclusion “Marry him”. It should be clear, however, that neither argument (2a) not argument (2b) is valid: from “Do X or Y” and “You are not going to do X or Y” neither “Do X” nor “Do Y” follows. Since these arguments are valid if argument (1) is valid, it follows that argument (1) is not valid after all.

20. INVALID MIXED-PREMISE IMPERATIVE ARGUMENTS2
(1) If the light is red, stop.
The light is red.
So: Stop.
(2) Stop.
So: If the light is green, stop.
(3) If the light is red, stop.
If (1) is valid, then (because (2) is valid) (3) is valid. But (3) is not valid. So (1) is not valid.
Many years ago, when I was in graduate school, a fellow graduate student asked me if I thought this argument is valid: “If the light is red, stop; the light is red; so, stop”. I said yes, I thought the argument is valid. Several years later, when I started thinking systematically about these issues, I thought that rendering this argument valid was a prerequisite for any satisfactory definition of validity. And yet I could not show that the argument is valid by using my definition. One day, the following reasoning occurred to me. Consider argument (2): “Stop; so, if the light is green, stop”. This is a pure imperative argument, and it is valid according to the definition of pure imperative validity I have defended in New Foundations II. Moreover, there is an intuitive explanation of why argument (2) is valid: the premise, namely “Stop”, is the conjunction of “If the light is green, stop” and “If the light is not green, stop”, and so entails the first conjunct. But if arguments (1) and (2) are both valid, then, given the transitivity of entailment, argument (3) is also valid: “If the light is red, stop; the light is red; so, if the light is green, stop”. But this is absurd. I still remember how stunned I was the day I came up with this reasoning. This is when I started entertaining seriously the possibility that argument (1) is invalid after all. Indeed, later on I was able to prove, using my definition, that argument (1) is invalid. But I still lacked an intuitive explanation of why the argument is invalid. I am happy to report that I did finally find such an explanation, which I would like to share with you now.

21. INVALID MIXED-PREMISE IMPERATIVE ARGUMENTS3
Many years ago, when I was in graduate school, a fellow graduate student asked me if I thought this argument is valid: “If the light is red, stop; the light is red; so, stop”. I said yes, I thought the argument is valid. Several years later, when I started thinking systematically about these issues, I thought that rendering this argument valid was a prerequisite for any satisfactory definition of validity. And yet I could not show that the argument is valid by using my definition. One day, the following reasoning occurred to me. Consider argument (2): “Stop; so, if the light is green, stop”. This is a pure imperative argument, and it is valid according to the definition of pure imperative validity I have defended in New Foundations II. Moreover, there is an intuitive explanation of why argument (2) is valid: the premise, namely “Stop”, is the conjunction of “If the light is green, stop” and “If the light is not green, stop”, and so entails the first conjunct. But if arguments (1) and (2) are both valid, then, given the transitivity of entailment, argument (3) is also valid: “If the light is red, stop; the light is red; so, if the light is green, stop”. But this is absurd. I still remember how stunned I was the day I came up with this reasoning. This is when I started entertaining seriously the possibility that argument (1) is invalid after all. Indeed, later on I was able to prove, using my definition, that argument (1) is invalid. But I still lacked an intuitive explanation of why the argument is invalid. I am happy to report that I did finally find such an explanation, which I would like to share with you now.
INVALID MIXED-PREMISE IMPERATIVE ARGUMENTS3
(1) If you drink, don’t drive.
You are going to drink.
So: Don’t drive.
(2) If you drink, don’t drive.
You are going to drink, but the fact that your
daughter will die if you don’t drive her to
the hospital is an undefeated reason for you
to drive and not drink.
If (1) is valid, then (2) is also valid. But (2) is not valid. So (1) is not valid.
The explanation uses again the method of adding declarative premises. Consider first a different argument: “If you drink, don’t drive; you are going to drink; so, don’t drive”. If this argument is valid, then the following argument is also valid: “If you drink, don’t drive; you are going to drink, but the fact that your daughter will die if you don’t drive her to the hospital is an undefeated reason for you to drive and not drink; so, don’t drive”. But this argument is clearly invalid, so the original argument is also invalid. Similarly, consider the following argument: If the light is red, stop; the light is red, but the fact that you are being chased by a tornado is an undefeated reason for you not to stop; so, stop”. This argument is clearly invalid, so the original argument on my previous slide is also invalid. Much more can be said about these arguments, and I do say it in the paper; but I will stop here and I will make further comments in the discussion period if you are interested.

22. CONCLUSION: VIRTUES OF THE GENERAL DEFINITION
The General Definition is:
(1) general (it provides unification);
(2) usable (it yields concrete results);
(3) useful (it transmits meriting endorsement);
(4) formally acceptable (reflexivity/transitivity);
(5) intuitively acceptable;
(6) principled (it goes beyond intuition); and
(7) in harmony with previous work.
Next step: Introduce a formal language & a proof procedure, and prove soundness & completeness.
To conclude, let me summarize what I take to be the main virtues of the general definition of argument validity that I proposed in this talk. (1) The definition is general: it provides a unified account of validity for pure, mixed-premise, and cross-species declarative and imperative arguments. (2) The definition is usable: it enables one to decide, via the theorems that I proved, whether specific arguments are valid. (3) The definition is useful: it satisfies transmission both of meriting pro tanto and of meriting all-things-considered endorsement. (4) The definition is formally acceptable: it satisfies reflexivity and transitivity. (5) The definition is intuitively acceptable: it yields intuitively acceptable results concerning the validity of a wide variety of arguments. (6) The definition is principled: it is motivated by considerations that go beyond a mere appeal to intuitions. (7) The definition is in harmony with previous work: it yields as special cases both the standard definition of validity for pure declarative arguments and my previously defended definition of validity for pure imperative arguments.
The natural next steps are to introduce a formal language and a proof procedure, and to prove the soundness and the completeness of the proof procedure with respect to a semantics based on the General Definition. These are the main tasks of the next and probably the last paper in the series.