Category: Logica Docens

The Notion of a Cumulative Case

 In a comment thread Tony Hanson asked me if I had written a post on cumulative-case arguments.  After some digging, I located one that I had written 24 August 2004.  Here it is for what its worth. 

……………
 
Suppose you have a good reason R1 to do X. Then along comes a second good reason R2 to do X. Does R2 remove the justificatory force of R1? Obviously not. Does R2 leave the justificatory force of R1 unchanged? No again. Clearly, R2 augments the force of R1. Any additional good reasons R3, R4, . . . Rn, would of course only add to the justification for doing X. What we have here is a cumulative case for doing X, a case in which the justificatory force of the good reasons is additive.
 
A thorough discussion would have to distinguish between cumulative case arguments in which each reason is sufficient to justify the action envisaged, and cumulative case arguments in which one or more or all of the reasons are individually insufficient to justify the action envisaged.
 
Suppose each reason in a cumulative case argument is individually sufficient to justify the action envisaged. Then in what sense are the reasons additive? They are additive in that each additional sufficient reason provides an additional fail-safe mechanism. If an agent has many reasons each of which is both good and sufficient for doing X, then, if one of the reasons should turn out to be either bad or insufficient, then the other reasons are available to shoulder the justificatory burden.
 
Apply this to the Iraq war. One reason for going to war was the widely shared belief that Saddam had WMDs. Another was that he was a known sponsor of Palestinian Arab terrorists and a reasonably surmised sponsor of other terrorists. (On the second point, see Stephen F. Hayes, The Connection: How al-Qaeda's Collaboration with Saddam Hussein Has Endangered America, Harper Collins, 2004) A third was humanitarian: the liberation of the Iraqi people from a brutal dictator and his sons. A fourth was to enforce unanimous U.N. resolutions that this august body did not have the cojones to enforce itself. A fifth was to end the ongoing hostilities, e.g., Iraqi attacks on coalition warplanes. Even if no one of these reasons is sufficient to justify the invasion, the five taken together arguably provide good and sufficient reason for the action.
 
The strategy of ‘Divide and Conquer’ cannot be used against a cumulative case argument. Suppose Jack has several reasons for marrying Jill: she’s nubile and pretty, moneyed and witty; they are physically and psychologically compatible; they share the same values; she has beautiful eyes, and there is beauty at the opposite pole of her being as well. So Jack has nine good reasons. It simply won’t do to point out that each of them, taken singly, is insufficient to justify the marriage. A good reason is not the same as a sufficient reason. A good reason can be either sufficient or insufficient. What then are examples of bad reasons? A bad reason would be her having a police record, or her having a doctorate in biology when her doctorate is in mathematics.
 
The point is that several good, but individually insufficient, reasons can add up to a good and sufficient reason. If so, then ‘Divide and Conquer’ is a fallacious form of refutation. But that is what many leftists do when they oppose the Iraq war. Suppose that the cumulative case consists of R1, R2, and R3, each of which is insufficient by itself to justify doing X. The ‘Divide and Conquer’ objector wrongly infers ‘no reason’ from ‘insufficient reason.’ Thus he thinks that if R1 is insufficient, then R1 is no reason, and similarly for R2 and R3. He then concludes: no reason + no reason + no reason = no reason. He fails to appreciate the additivity of individually insufficient but good reasons, just as the typical poor person fails to appreciate the additivity of the small amounts of money he throws away on cigarettes, lottery tickets, and overpriced convenience store items.
For example, if a conservative gives liberation of the Iraqi people as a reason for the invasion, the leftie is likely to object: "But then why don’t we liberate the North Koreans?" This is an asinine response since it it is based on a failure to appreciate that the liberation reason is only one part of a cumulative case, not to mention the fact that an attempted liberation of the North Koreans could easily lead to nuclear war. Granting that liberating the Iraqi people is an insufficient reason for the war, it does not follow that it is no reason at all. It is a good reason which, though insufficient taken by itself, is part of a cumulative case which amounts to a good and sufficient reason for the war.
Another mistake that leftists make is to confuse a reason with a motive. They do this when they say that a proffered reason is not the real reason. A reason is a motive when it plays a motivating role within the psychic economy of an agent. Suppose Jack has available to him an objectively good reason R for marrying Jill. But Jack is not consciously or subconsciously aware of R. Obviously, R can play no role in the etiology of his envisaged action. Yet R remains an objectively good reason for performing the act in question. A good reason need not be a motivating reason, and a motivating reason need not be a good reason. The expression ‘real reason’ should be avoided because it is ambiguous as between good reason and motivating reason.
 
Suppose Bush II’s sole motive for invading Iraq was to avenge Saddam’s assasination attempt on his father, Bush I. Even on this wildly counterfactual assumption, there were good reasons for the invasion. For an action to be justified, all that is required is that there be objectively good reasons for the action; it is not necessary that the agent’s motives be objectively good reasons. Even if an agent is not justified in doing X – because he is either not aware of or motivated by the good reasons for doing X – the act itself (the act-type itself) can have justification. Our man Jack, for example, may be driven to marry Jill by his lust and nothing besides; but this does not entail that his marrying her lacks justification. Jack’s father might say to him: "Son, you made the right decision, but for the wrong reason." The rightness of the decision is due to the availability of good reasons even if horny Jack did not avail himself of them.

At this point an objector might maintain that what I am calling good reasons are simply ex post facto rationalizations.But a rationalization after the fact is not the same as a good reason that plays no motivating role in bringing about the fact. For a rationalization is a bad reason. Suppose Ali physically assaults Benjamin because Benjamin is a Jew and Ali believes that Jews are the "sons of pigs and monkeys." After the fact, A explains his behavior by saying that B insulted him. Suppose B did insult A. A is rationalizing after the fact as opposed to giving a good reason after the fact. B’s insulting of A did not give A a good reason for initiating physical violence against B.

Now let us suppose that Bush II’s sole motive for ordering the Iraq invasion was his desire to deprive Saddam of the WMDs that he, Bush, believed Saddam to possess. Suppose, plausibly, that the belief is false. In that case, Bush II’s motivating reason was not an objectively good reason – based as it was on a false belief – but it could still count as a subjectively good reason in this sense: he had a reason that was a good reason based on the information he had available to him at the time of the decision. I would then argue that the other reasons, which are objectively good, bear the justificatory burden.

An astonishing number of people, some of them intelligent, believe that the motivating reason for the Iraq invasion was the desire to secure access to Iraqi oil. But if that was the motivating reason, it is was a very bad reason since (i) the oil was flowing; (ii) starting a war with an opponent believed to have WMDs and known to have ignited oil wells in the past is clearly a stupid way to secure access to Iraqi oil; (iii) the projected cost of the war would be scarcely offset by the value of the oil secured; and (iv) deposing Saddam and his sons was not at all necessary to insure the flow of oil. I would argue that since this oil reason is so obviously bad, it is not reasonable to impute it to Bush and his advisers as the motivating reason for the invasion.

To sum up. The case for invading Iraq was a cumulative case. A cumulative case cannot be refuted by ‘Divide and Conquer.’ A good reason need not be a sufficient reason. A reason is not the same as a motive: there can be objectively good reasons for an action even if the agent of the action is not motivated by any of these reasons. To find good reasons after the fact is not to engage in ex post facto rationalization. This is because a rationalization is the providing of a bad reason.  But of course, liberals and leftists are so blinded by their passionate hatred of Bush II, that patient analysis of the foregoing sort will be lost on them.

 

De Dicto/De Re

In the course of thinking about the de dicto/de re distinction, I pulled the Oxford Companion to Philosophy from the shelf and read the eponymous entry. After being told that the distinction "seems to have first surfaced explicitly in Abelard," I was then informed that the distinction occurs:

     . . . in two main forms: picking out the difference between a
     sentential operator and a predicate operator, between 'necessarily
     (Fa)' and 'a is (necessarily-F)' on the one hand, and on the other
     as a way of highlighting the scope fallacy in treating necessarily
     (if p then q) as if it were (if p then necessarily-q).

It seems to me that this explanation leaves something to be desired. I have no beef with the notion that the first distinction is an example of a de dicto/de re distinction. To say of a dictum that it is   necessarily true if true is different from saying of a thing (res) that it has a property necessarily. Suppose a exists in some, but not all, possible worlds, and that a is F in every possible world in which it exists. Then a is necessarily F, F in every possible world in which it exists. But since there are possible worlds in which a does not exist, then it will be false that 'a is F' is necessarily true, true
in all possible worlds.  So the de dicto 'Necessarily, a is F' is distinct from the de re 'a is necessarily F.'

So far, so good. But the distinction between

1. Nec (if p then q)

   and

2. If p, then Nec q

is situated entirely on the de dicto plane, the plane of dicta or propositions. The distinction between (1) and (2) is the well-known  distinction between necessitas consequentiae and necessitas consequentiis. To confuse (1) and (2) is to confuse the necessity of the consequence with the necessity of the consequent. Or you could think of the mistake as a scope fallacy: the necessity operator in (1) has wide scope whereas the operator in (2) has narrow scope. But what makes (2) de re? What is the res in question? Consider an example:

3. Necessarily, if a person takes Enalapril, then he takes an ACE inhibitor

does not entail

4. If a person takes Enalapril, then necessarily he takes an ACE  inhibitor.

A second example:

5. Necessarily, if something happens, then something happens

does not entail

6. If something happens, then necessarily something happens.

It can't be that easy to prove fatalism. The point, however, is that the distinction between (5) and (6) does not trade on the distinction between dicta and rei, between propositions and non-propositions: the  distinction is one of the scope of a propositional operator.  Our author thus seems wrongly to assimilate the above scope fallacy to a de dicto/de re confusion.

I conclude that the de dicto/de re distinction is a bit of a terminological mess. And note that it is a mess even when confined to the modal context as demonstrated above. If we try to apply the  distinction univocally across modal, doxastic, temporal, and other  contexts we can expect an even bigger mess. A fit topic for a future  post.

Terminological fluidity is a problem in philosophy.  It always has been and always will be.  For attempts at regimentation and standardization harbor philosophical assumptions and biases — which are themselves fit fodder for philosophical scrutiny.

Cf. Notes on Philosophical Terminology and its Fluidity

On Falsely Locating the Difference Between Deduction and Induction

One commonly hears it said that the difference between deductive and inductive inference is that the former moves from the universal to the singular, while the latter proceeds from the singular to the universal. (For a recent and somewhat surprising example, see David Bloor, "Wittgenstein as a Conservative Thinker" in The Sociology of Philosophical Knowledge, ed. Kusch (Kluwer, 2000), p. 4.) No doubt, some deductive inferences fit the universal-to-singular pattern, while some inductive inferences fit the singular-to-universal pattern.

But it does not require a lot of thought to see that this cannot be what the difference between deduction and induction consists in. An argument of the form, All As are Bs; All Bs are Cs; ergo, All As are Cs is clearly deductive, but is composed of three universal propositions. The argument does not move from the universal to the singular. So the first half of the widely-bruited claim is false.

Indeed, some deductive arguments proceed from singular premises to a universal conclusion. Consider this (admittedly artificial) example: John is a fat chess player; John is not a fat chess player; ergo, All chess players are fat. This is a deductive argument, indeed it is a valid deductive argument: it is impossible to find an argument of this form that has true premises and a false conclusion. Paradoxically, any proposition follows deductively from a contradiction. So here we have a deductive argument that takes us from singular premises to a universal conclusion.

There are also deductive arguments that move from a singular premise to an existentially general, or particular, conclusion. ‘Someone is sitting’ is a particular proposition: it is neither universal nor singular. ‘I am sitting’ is singular. The first follows deductively from the second.

As for the second half of the claim, suppose that every F I have encountered thus far is a G, and that I conclude that the next F I will encounter will also be a G. That is clearly an inductive inference, but it is one that moves from a universal statement to a statement about an individual. So it is simply not the case that every inductive inference proceeds from singular cases to a universal conclusion.

What then is the difference between deduction and induction if it does not depend on the logical quantity (whether universal, particular, or singular) of premises and conclusions? The difference consists in the nature of the inferential connection asserted to obtain between premises and conclusion. Roughly speaking, a deductive argument is one in which the premises are supposed to ‘necessitate’ the conclusion, making it rationally inescapable for anyone who accepts the premises, while an inductive argument is one in which the premises are supposed merely to ‘probabilify’ the conclusion.

To be a bit more precise, a deductive argument is one that embodies the following claim: Necessarily, if all the premises are true, then the conclusion is true. The claim is that the premises ‘necessitate’ the conclusion, as opposed to rendering the conclusion probable, where the necessity attaches to the inferential link between premises and conclusion, and not to the conclusion itself. (A valid deductive argument can, but need not, have a necessary conclusion: ‘I am sitting’ necessitates ‘Someone is sitting,’ even though the latter proposition is only contingently true.)

Equivalently, a deductive argument embodies the claim that it is impossible for all the premises to be true and the conclusion false. I say ‘embodies the claim’ because the claim might not be correct. If the claim is correct, then the argument is valid, and invalid otherwise. Since validity pertains to the form of deductive arguments as opposed to their content, we can define a valid (invalid) deductive argument as one whose form is such that it is impossible (possible) for any (some) argument of that form to have true premises and a false conclusion. Since the purport of inductive arguments is merely to probabilify, not necessitate, their conclusions, they are not rightly described as valid or invalid, but as more or less strong or weak, depending on the degree to which they render their conclusions probable.

Validity, Invalidity, and Contravalidity

If a deductive argument is valid, that does not say much about it: it might still be probatively worthless. Nevertheless, validity is a necessary condition of a deductive argument's being probative. So it is important to have a clear understanding of the notion of validity.  An argument is valid if and only if one of its logical forms is such that no argument of that form has true premises and a false conclusion.

Continue reading “Validity, Invalidity, and Contravalidity”

My Lately Posted Logic Problem Pondered . . .

. . . and pondered well by David Parker over at Pondering the Preponderance.  I challenged the reader to spot what is wrong in the following argument, an argument I thought was interesting because it is fairly seductive, as compared to the stock examples in logic texts:

The Argument
1. A necessary truth is true.
2. Whatever is true is possibly true.
3. Whatever is possibly true could be false.
Therefore
4. A necessary truth could be false.

(I hope it is clear that 'possibly' and 'could' are not being used epistemically in this argument.)  Since the conclusion is plainly false, the argument is unsound either in virtue of invalidity, or in virtue of one or more false premises, or both.  There is nothing wrong with the formal logic of the argument, so I pointed out, correctly, that while (1) and (2) are each true, (3) is false. 

But there is an alternative analysis which Parker notes (and I didn't just to keep the post short), namely that one can see the argument as trading on an equivocal use of 'possibly true.'  And this alternative analysis helps explain why the argument is seductive.  After all (3) would be true if 'possibly true' meant 'contingently true.'  That is not what it means, but one could be forgiven for thinking so.  One could then say that the argument goes wrong because it commits the informal fallacy of equivovation: 'possibly true' is used with different senses in (2) and (3).  On this alternative analysis one could say that all the premises are true, but the argument commits the informal fallacy of equivocation.

But there is another wrinkle, and one which Parker notes.   Equivocation is standardly classified as an informal fallacy, buy doesn't every case of equivocation in a deductive argeument induce a formal fallacy?  Yes it does.  The form of the above argument could be depicted as follows:

Every F is a G
Every G is an H
Every H is an I
Ergo
Every F is an I

The form just depicted is clearly valid, whence it follow that every argument instantiating this form is valid.  It is of course assumed that the terms are being used univocally.  But if there is an equivocation on 'possibly true,' then the form of the original argument is not the above, but this:

Every F is a G
Every G is an H
Every I is a J
Ergo
Every F is a J

which is plainly invalid.

One moral is that the distinction between formal and informal fallacies is not hard-and-fast. (Composition and Division would also be interesting to discuss in this connection).  One can analyze our original argument as involving an equivocation on  'possibly true' in which case the argument is invalid, or one can take the argument to be valid but reject it because of the falsity of premise (3).

Ah, the pleasures of analysis!

A Logic Problem

Consider this argument:

   1. A necessary truth is true.
   2. Whatever is true is possibly true.
   3. Whatever is possibly true could be false.
   Therefore
   4. A necessary truth could be false.
   
A sound argument is one that satisfies two conditions: its premises are all true, and the reasoning it embodies is correct. Is the above argument sound?

If not, what has gone wrong in the argument? Answer below the fold.

Continue reading “A Logic Problem”

Logic’s Limit

Logic is not to be denigrated, nor is it to be overestimated. It is an excellent vehicle for safe travel among concepts and propositions. It will save us from many an error and perhaps even lead us to a few truths. But it cannot move us beyond the plane of concepts and propositions and arguments. It aids safe passage from thought to thought, but cannot  transport us beyond thought to the source of thoughts, to their thinker, the transcendental condition without which there would not be any thoughts.  It cannot transport us to the Transdiscursive.  For that a different vehicle is needed, meditation.

Collective Inconsistency and Plural Predication

We often say things like

1. The propositions p, q, r are inconsistent.

Suppose, to keep things simple, that each of the three propositions is self-consistent.  It will then be false that each proposition is self-inconsistent. (1), then, is a plural predication that cannot be given a distributive paraphrase.  What (1) says is that the three propositions are collectively inconsistent.  This suggests to many of us  that there must be some one single entity that is the bearer of the inconsistency.  For if the inconsistency does not attach distributively to each of p, q, and r, then it attaches to something distinct from them of which they are members.  But what could that be?

If you say that it is the set {p, q, r} that is inconsistent, then the response will be that a set is not the sort of entity that can be either consistent or inconsistent.  Note that it is not helpful to say

A set is consistent (inconsistent) iff its members are consistent (inconsistent).

For that leaves us with the problem of the proper parsing of the right-hand side, which is the problem with which we started.

And the same goes for the mereological sum (p + q + r).  A sum or fusion is not the sort of entity that can be either consistent or inconsistent.

What about the conjunction p & q & r?  A conjunction of propositions is itself a proposition.  (A set of propositions is not itself a proposition.) This seems to do the trick. We can parse (1) as

2. The conjunctive proposition p & q & r is (self)-inconsistent.

In this way we avoid construing (1) as an irreducibly plural predication.  For we now have a single entity that can serve as the logical subject of the predicate ' . . . is/are inconsistent.'  We can avoid saying, at least in this case, something that strikes me as only marginally intelligible, namely, that there are irreducible monadic non-distributive predicates.  My problem with irreducibly plural predication is that I don't know what it means to say of some things that they are F if that doesn't mean one of the following: (i) each of the things is F; (ii) there is a single 'collective entity' that is F; or (iii) the predicate 'is F'  is really relational. 

One could conceivably object that in the move from (1) to (2) I have 'changed the subject.'  (1) predicates inconsistency of some propositions, while (2) predicates (self)-inconsistency of a single conjunctive proposition.  Does this amount to a changing of thr subject?  Does (2) say something different about something different?

A Problem With the Multiple Relations Approach to Plural Predication

Consider

1. Sam and Dave are meeting together.

2. Al, Bill, and Carl are meeting together.

3. Some people are meeting together.

Obviously, neither (1) nor (2) can be decomposed into a conjunction of singular predications.  Thus (2) cannot be analyzed as 'Al is meeting together & Bill is meeting together & Carl is meeting together.'  So it is natural to try to analyze (1) and (2) using relational predicates.  (1) becomes

1R. Meeting(Sam, Dave)   In symbols: Msd

But if 'meeting' is a dyadic (two-place) predicate, then we should expect (2) to give way to

2R. Mab & Mbc & Mac.

Unfortunately, (2R) is true in circumstances in which (2) is false.  Suppose there are three separate meetings.  Then (2R) is true and (2) false.  To get around this difficulty, we can introduce a triadic relation M* which yields as analysans of (2):

2R*. M*abc.

But then we need a tetradic relation should Diana come to the meeting.  And so on, with the result that 'meeting together' picks out a family of relations of different polyadicities.  But what's wrong with that?  Well, note that (1) and (2) each entail (3) by Existential Generalization in the presence of the auxiliary premise 'Al, Bill, Carl, Dave, and Sam are people.' 

But then we are going to have difficulty explaining the validity of the two instances of Existential Generalization.  For the one instance features a dyadic meeting relation and  the other a triadic.  If two different relations are involved, then what is the logical form of (3) — Some people are meeting together — which is the common conclusion of both instances of Existential Generalization?  If 'meeting together picks out a family of relations of different 'adicities, then (3) has no one definite logical form.

Does this convince you that the multiple relations approach is unworkable?

REFERENCE:  Thomas McKay, Plural Predication (Oxford 2006), pp. 19-21.

 

Irreducibly Plural Predication: ‘They are Surrounding the Building’

Let's think about the perfectly ordinary and obviously intelligible sentence,

1. They are surrounding the building.

I borrow the example from Thomas McKay, Plural Predication (Oxford 2006), p. 29.  They could be demonstrators.  And unless some of them have very long arms, there is no way that any one of them could satisfy the predicate, 'is surrounding the building.'  So it is obvious that (1) cannot be analyzed in terms of 'Al is surrounding the building & Bill is surrounding the building & Carl is surrounding the building & . . . .'  It cannot be analyzed in the way one could analyze 'They are demonstrators.'  The latter is susceptible of a distributive reading; (1) is not.  For example, 'Al is a demonstrator & Bill is a demonstrator & Carl is a demonstrator & . . . .'  So although 'They are demonstrators' is a plural predication, it is not an irreducibly plural predication.  It reduces to a conjunction of singular predications.

Continue reading “Irreducibly Plural Predication: ‘They are Surrounding the Building’”

Necessitas Consequentiae versus Necessitas Consequentiis

Take the sentence, 'If I will die tomorrow, then I will die tomorrow.' This has the form If p, then p, where 'p' is a placeholder for a proposition. Any sentence of this form is not just true, but logically true, i.e., true in virtue of its logical form. Now every sentence true in virtue of its logical form is necessarily true. (The converse, however, does not hold: there are necessary truths that are not logically true.) Thus we can write, 'Necessarily(if p, then p)'  or

1. Nec (p –>p).

The parentheses show that the necessity attaches to the consequence, represented by the arrow, and not to the consequent, represented by the terminal 'p.' (When speaking of conditionals, logicians distinguish the antecedent from the consequent, or, trading Latin for Greek, the protasis from the apodosis.) Thus the above is an example of the necessitas consequentiae. This, however, must not be confused with the necessitas consequentiis, which is exemplified by

2. p–>Nec p.

In (2) the necessity attaches to the consequent. It should be obvious that (1) does not entail (2), equivalently, that (2) does not follow from (1). For example, although it is necessarily true that if I will die tomorrow, then I will die tomorrow, it does not follow, nor is it true, that if I will die tomorrow, then necessarily I will die tomorrow. Proving fatalism cannot be that easy. For even if I do die tomorrow, that will be at best a contingent occurrence, not something logically necessitated. (Think about it.)

To confuse (1) and (2) is to confuse the necessity of the consequence with the necessity of the consequent. This is an example of what logicians call a fallacy, i.e., a typical error in reasoning, and in particular a modal fallacy in that it deals with the (alethically) modal concepts of necessity and possibility and their cognates.

Class dismissed.

Amphiboly

Amphiboly is syntactic ambiguity.  "The foolish fear that God is dead."  This sentence is amphibolous because its ambiguity does not have a semantic origin in the multiplicity of meaning of any constituent word, but derives from the ambiguous way the words are put together.  On one reading, the construction is a sentence: 'The foolish/ fear that God is dead.'  On the other reading, it is not a sentence, does not express a compete thought, but is a sentence-fragment: ' The foolish fear/that God is dead.'

A good writer avoids ambiguity except when he intends it.

Is There a Paradox of Conjunction?

There are supposed to be paradoxes of material and strict implication. If there are, why is there no paradox of conjunction? And if there is no paradox of conjunction, why are there paradoxes of material and strict implication? With apologies to the friends and family of Dennis Wilson, the ill-starred original drummer of the Beach Boys, let's take this as our example:

1. Wilson got drunk, fell overboard, and drowned.

Translating (1) into the Propositonal Calculus (PC), we get

2. Wilson got drunk & Wilson fell overboard & Wilson drowned. 

Now the meaning of the ampersand (or the dot or the inverted wedge in alternative notations) is exhausted by its truth table. This meaning can be summed up in two rules. A conjunction is true if and only if all of its conjuncts are true. A conjunction is false if and only if one or more of its conjuncts is false. That is all there is to it. The ampersand, after all, is a truth-functional connective which means that the truth-value of any compound proposition formed with its aid is a function (in the mathematical sense) of the TVs of its components and of nothing besides. You will recall from your college calculus classes that if f is a function and y = f(x), then for each x value there is a unique y value.

Now are the conjuncts of (2) related? Well, they are related in that they all have the same truth-value, namely True. But beyond this they are not related qua components of a truth-functional compound proposition. The 'conjuncts' — note the inverted commas! — of (1), however, are related beyond their having the same truth-value. For it is because Wilson got drunk that he fell overboard, and it is because he fell overboard that he drowned. So causal and temporal relations come into play in (1), relations that are not captured by (2).

Note also that the ampersand has the commutative property. But this is not so for the comma and the 'and' in (1). Tampering with the order of the clauses in (1) turns sense into nonsense:

3. Wilson drowned, fell overboard, and got drunk.

We should conclude that the ampersand abstracts from some of the properties of occurrences of the natural language 'and' and cognates. Despite this abstraction, (1) entails (2), which means that (2) does capture part of the meaning of (1), that part of the meaning relevant to the purposes of logic. But surely there is no 'paradox' here. Any two propositions can be conjoined, and the truth-value of the compound can be computed from the TVs of the components. It is the same with material implication: any two propositions can be connected with a horseshoe or an arrow and the TV of the result is uniquely determined by the TVs of the component propositions. Thus we get a curiosity such as

4. Snow is red –> Grass is green

which has the value True. This is paradoxical only if you insist on reading the arrow as if it captured all the meaning of the natural language 'if' or 'if…then___.' But there is no call for this insistence any more than there is call for reading the ampersand as if it captures the full meaning of 'and' and cognates in ordinary English.

What I am suggesting is that, just as there is no paradox of conjunction, there is no paradox of material implication either.

Geach on Assertion

The main point of Peter Geach's paper, "Assertion" (Logic Matters, Basil Blackwell, 1972, pp. 254-269) is what he calls the Frege point: A thought may have just the same content whether you assent to its truth or not; a proposition may occur in discourse now asserted, now unasserted; and yet be recognizably the same proposition. This seems unassailably correct. One will fail to get the Frege point, however, if one confuses statements and propositions. An unstated statement is a contradiction in terms, but an unasserted proposition is not. The need for unasserted propositions can be seen from the fact that many of our compound assertions (a compound assertion being one whose content is propositionally compound) have components that are unasserted.

To assert a conditional, for example, is not to assert its antecedent or its consequent. If I assert that if Tom is drunk, then he is unfit to drive, I do not thereby assert that he is drunk, nor do I assert that he is unfit to drive.  I assert a compound proposition the components of which I do not assert.  The same goes for disjunctive propositions. To assert a disjunction is not to assert its disjuncts. Neither propositional component of Either Tom is sober or he is unfit to drive is asserted by one who merely asserts the compound disjunctive proposition.

What bearing does this have on recent discussions?  I am not sure I understand William of Woking's position, but he seems to be denying something that Geach plausibly maintains, namely, that "there is no expression in ordinary language that regularly conveys assertoric force." (261)  Suppose I want to assert that Tom is drunk.  Then I would use the indicative sentence 'Tom is drunk.'  But there is nothing intrinsically assertoric about that sentence.  If there were, then prefixing 'if' to it would not remove its assertoric force as it does.    As I have already explained, an assertive utterance of 'If Tom is drunk, then he is unfit to drive'  does not amount to an assertive utterance of 'Tom is drunk.'  'If' cancels the assertoric force.  And yet the same proposition occurs in both assertions, the assertion that Tom is drunk and the assertion that if Tom is drunk, then he is unfit to drive.  I conclude that there is nothing intrinsically assertoric about indicative sentences.  If so, there is no semantic component of an indicative sentence that can be called the assertoric component.

'If' prefixed to an indicative sentence does not alter its content: it neither augments it nor diminishes it.  But it does subtract assertoric force.  Given that the meaning of an indicative sentence is its content, and the semantics has to do with meaning, then there is no semantic assertoric component of an indicative sentence or of the proposition it expresses.  Assertion and assertoric force do not belong in semantics; they belong in pragmatics.  Or so it seems to me.

Five Grades of Self-Referential Inconsistency: Towards a Taxonomy

Some sentences, whether or not they are about other things, are about themselves. They refer to themselves. Hence we say they are 'self-referential.' The phenomenon of sentential self-referentiality is sometimes benign. One example is 'This sentence is true.' Another  is 'Every proposition is either true or false.' Of interest here are the more or less malignant forms of self-reference. One example is the so-called Liar sentence:

1. This sentence is false.

If (1) is true, then it is false, and if false, then true. This is an example of an antinomy. In pursuit of a taxonomy, we might call this Grade I of self-referential inconsistency. Grade I, then, is the class
of self-referentially inconsistent sentences that issue in antinomies.

There are other self-referential sentences that are not antinomies, but imply their own necessary falsehood. These are such that, if true, then false, and if false, then false, and are therefore necessarily false. For example,

2. All generalizations are false.

If (2) is true, then, since (2) is itself a generalization, (2) is false. But its falsity does not imply its truth. So, if false, then false. Assuming Bivalence, it follows that (2) is necessarily false, whence it follows that its negation — Some generalizations are true  – is necessarily true, and moreover an instance of itself. A second example might be

3. There are no truths.

If (3) is true, then it is false. And if false, then false. So, (3) is necessarily false, whence it follows that its negation — There are truths — is necessarily true.

Examples (2) and (3) belong to Grade II in my tentative taxonomy. These are self-referential sentences that entail their own necessary falsehood. Grade III comprises those self-referential sentences that are such that if true, then neither true nor false, and if false, then false. For example,

4. There are no truth-bearers.

If (4) is true, then, since (4) is a truth-bearer, (4) is neither true nor false. But if false, then false. If we define the cognitively meaningful as that which is either true or false, then (4) is either cognitively meaningless or false. A more interesting example that seems to belong in Grade III is the Verifiability Principle of the Logical Positivists:

5. Every cognitively meaningful sentence is either analytic or empirically verifiable in principle.

If (5) is supposed to be cognitively (as opposed to emotionally) meaningful, and thus not a mere linguistic recommendation or pure stipulation, then it applies to itself. So if (5) is true, then (5) —
which is clearly neither analytic nor verifiable — is meaningless. So, if true, then meaningless, and if false, then false. Therefore, either meaningless or false. Not good!

Grade IV comprises those self-referential sentences that can be described as self-vitiating (self-weakening) though they are not strictly self-refuting. For example,

6. All truths are relative.

If (6) is true, then (6) is relative, i.e., relatively true. It is not the case that if (6) is true, then (6) is false. So (6) is not self-refuting. Nevertheless, (6) is self-vitiating in that it relativizes and thus weakens itself: if true, it cannot be absolutely true; it can only be relatively true. It is therefore a mistake, one often made, to say that he who affirms (6) contradicts himself.  He does not.  He would contradict himself only if he maintained that it is nonrelatively true that all truths are relative.  But no sophisticated relativist would say such a thing.  Other examples which seem to fall into the category of the self-vitiating:

7. Every statement is subject to revision. (Quine)
8. Every theory reflects class interests. (Marxism)
9. All theory is ideology. (Marxism)
10. Nothing can be known.
11. Nothing is known.
12. Nothing is certain.
13. All truth is historical.
14. All is opinion.

What is wrong with self-vitiating propositions? What does their weakness consist in? Consider (8). If (8) is true, then the theory that every theory reflects class interests itself reflects class interests. Suppose (8) reflects the class interests of the proletariat. Then what is that to me, who am not a proletarian? What is it to anyone who is not a proletarian? If (8) is true only for you and those with your interests, and your interests are not my interests, then I have been given no reason to modify my views.  The trouble with (7)-(14) and their ilk is that they make a claim on our rational attention, on our common rational interest, while undercutting that very claim.

It seems we need a fifth category. The sentences of Grade V are such that, if they are true then they are, not false, and not self-vitiating, but non-assertible. Consider

15. No statement is negative.

(15) applies to itself and so at first appears to refute itself: if (15) is true, then it is false. And if false, then false; hence necessarily false. But consider a possible world W in which God destroys all negative statements and makes it impossible for anyone to make a negative statement. In W, (15) is true, but non-assertible. (15) does not prove itself to be false; it proves itself to be non-assertible.

Can the same said of

16. All is empty (Buddhism)?

I think not, for reasons supplied here.

Finally, we consider

17. All memory reports are deceptive.

This is subject to the retort that one who asserts (17) must rely on memory, and so must presuppose the reliability of the faculty whose reliability he questions by asserting (17). For if anyone is to be in a position responsibly to affirm (17), to affirm it with a chance of its being true, he must remember that on some occasions he has misremembered. He must remember and remember correctly that some of his memories were merely apparent. It seems obvious, then, that the truth of (17) is inconsistent with its correctly being affirmed as true. If true, it is unaffirmable as true. But this is different from saying that (17), if true, is false. Although (17) is unaffirmable or non-assertible if true, it seems that (17) could be true nonetheless.