Deviant Logic is based on the work Susan Haack completed for a PhD dissertation at Cambridge University. The book was published by Cambridge University Press in 1974. It is a brave book that provides penetrating critiques of some of the most cherished doctrines of the most prominent philosophers of logic of the previous generation; including Quine, Putnam, Dummett, Popper, and Lukasiewicz. While recommending what she calls a ‘radical’ approach to non-classical logics, in the end she opts for classical logic as the logic which best satisfies her pragmatist criteria of simplicity and economy.
While her book, as suggested by the title, considers alternatives to what she calls ‘classical logic,’ she nowhere in the body of the text says exactly what she means by this term. In an appendix she provides a (very handy) semantic summary of a series of logics. This appendix provides matrices which define the truth tables for the various logical connectives in each system; negation, conjunction, disjunction, implication, and identity. Here she presents “2-valued (‘classical’) logic” as the first system considered. But for all of her attempts at clarity, she leaves the system that she is defending otherwise undefined. It is safe to assume that what she has in mind is primarily bivalence (which she abbreviates to ‘PB’ for the principle of bivalence), but that she also includes here the laws of (non-) contradiction (LNC), excluded middle (LEM), and identity, in addition to the truth values for the connectives given in the appendix.
The introductory chapter sets out her goals: to establish the senses in which there could be alternative logics to bivalent logic and to systematically review the plausibility of these alternatives. The alternatives considered include: intuitionist logic, Post’s multi-valued logic, minimal logic, Lukasiewicz’s many-valued logic, van Frassen’s propositional languages, and quantum logics. She also considers systems which supplement classical logic: Lewis’s modal logic, epistemic logic, deontic logic, and tense logic. Since the book reviews only logics proposed up to the decade of the 1960s, it does not consider some logics which have become seriously considered in the last 40 years: this includes the general category of paraconsistent logics including the dialetheism of Graham Priest, relevant logic, or fuzzy logic. The latter are considered in papers appended to a subsequent edition of Deviant Logic, but the current review considers only the original text. Given what has come afterwards, her work becomes even more important in being one of the first systematic attempts to address the general question of rivals to standard bivalent logic in a clear and rigorous manner.
The first of the giants whom Haack takes on is Quine. In the first chapter she considers Quine’s argument that alternative logics are not really rivals to classical logic at all because their apparent inconsistency can be explained as involving a change of meaning of the logical constants. This argument of Quine’s derives from his famous indeterminacy theory of translation. Haack characterizes this as:
QIT: Alternative, and mutually incompatible, translations may conform to all data concerning dispositions to speaker’s behavior.
She evaluates three theses that she finds in Quine:
1) There is inductive uncertainty in the translation of even observation sentences
2) There is radical uncertainty in the translation of words and phrases
3) There is radical uncertainty in the translation of theoretical sentences
Her argument, found in section 3 of Chapter 1, accuses Quine of circularity. When Quine says that he is just trying to “save the obvious” by retaining bivalence she responds that this is obvious only if one assumes bivalent logic to begin with. Much the same response could be given to several of her own arguments against the plausibility of the alternative logics considered in her book. But at least she establishes from the start that the very idea of an alternative logic is within the realm of possibility.
In her second chapter she develops her pragmatist conception of logic, which in spite of her criticism of Quine in Chapter 1, derives from Quine’s Two Dogmas article of 1952 (included in Quine, from a logical point of view, 1952.) She finds the Two Dogmas argument foreshadowed in Duhem’s 1904 book, The Aim and Structure of Physical Theory. Her position is summarized in four principles:
1) No statement is conclusively verifiable by experience
2) No statement is conclusively falsifiable by experience
3) No statement is immune from revision in the light of experience
4) The criteria for deciding which statements to retain, and which to abandon, in the face of recalcitrance, are pragmatic ones, notably simplicity and economy.
The first of the above principles she finds established by the many critics of the justificationist program of Descartes, Frege, and Carnap; for example, Quine and Popper. Her acceptance of thesis 2 follows her interpretation of Duhem’s argument that “no hypothesis in physics is conclusively falsifiable, because there are always auxiliary assumptions involved in the derivation of observational consequences from a physical hypothesis, so that if these consequences fail to obtain the most one is entitled to conclude is that either the hypothesis or the auxiliary assumption is mistaken.” She considers and rejects two objections to this approach: 1) that the view is incoherent (the critique by Quine) and 2) that it is methodologically vicious (argued by Popper and Feyerabend.)
In the midst of her defense against 1) she repeats the standard argument against inconsistency that from a contradiction one could deduce anything. Popper, also, recapitulates Aristotle’s original argument along these lines in an article in Conjectures and Refutations. These arguments were prior, of course, to the development of paraconsistent logics like Priest’s LP (Logic of Paradox) and Anderson and Belnap’s 4-valued relevant logic, in which this consequence is avoided. Anderson and Belnap show that the argument rests on the disjunctive syllogism which they argue commits a fallacy of relevance or ambiguity (Entailment, Volume 1, p 165 ff.)
She finds Popper’s argument against 2) to be circular: Logic must not be revised because to do so would impede the progress of science, because logic is outside of science, and hence provides a place from which to falsify hypotheses, but logic is excluded because “it is un-falsifiable . . . The argument has come full circle.”
In her third chapter, she presents the four ways in which non-classical logics offer solutions to problem sentences (or statements):
1) The ‘no item’ hypothesis: the items in question are not of the kind with which logic is, or should, be concerned.
2) The ‘misleading form’ thesis: the items in question, though within the scope of logic, do not really have the form that they appear to have.
3) The ‘truth-value gap’ thesis: The items in question, though within the scope of logic, are neither true nor false, but truth-valueless.
4) The ‘new truth value’ thesis: the items in question, though within the scope of logic, are neither true nor false, but have some other truth-values.
For the problem of future contingents, for example, she cites the following approaches to explaining the issue: Kneale and Kneale (The Development of Logic, 1964) argue the ‘no-item’ thesis, Prior (Time and Modality, 1957) argues the ‘misleading form’ thesis, Aristotle (de Interpretatione) and van Frassen (‘Presupposition, implication, and self-reference’, Journal of Philosophy, 1968 ) argue the ‘truth-value gap’ thesis, and Lukasiewicz (‘Many valued systems of logic’, 1930 in McCall, Polish Logic, 1929-1939, 1967) the ‘new truth value’ thesis. She cites the following approaches to the problem of non-denoting terms (‘The King of France is bald’); Strawson (‘On referring’, Mind, 1950) the first and Strawson (‘Identifying reference and truth-values’, Theoria, 1964) the third, Frege (‘On Sense and Reference’, Philosophical Writings, 1960) and van Frassen (‘Presuppositions, supervaluations and free logic’, in The Logical Way of Doing Things, 1969) the third, Russell (‘On Denoting’, Mind,1905) the second, and Keenan (‘Two kinds of presupposition . . .’, Studies in Linguistic Semantics, 1971) the third.
She goes to some trouble to distinguish between approaches 3) and 4). Approach 3) is the ‘truth-value gap’ thesis: a sentence has no truth value. Frege takes this route in explaining sentences with non-denoting terms like “The King of France is bald.” According to his theory of sense and reference, if one of the components of a sentence lacks reference, then so does the sentence. Since non-denoting terms have no truth value, the sentence has none either. By thesis 4) the problematic element is given a truth value other than true or false, such as intermediate, or partially true.
Here she considers two questions which she considers important:
1) What kind of system is appropriate for the truth value gap thesis (3)?
2) Does a 3-valued system necessarily commit one to the new truth value thesis (4)?
To the first question she considers the systems of van Frassen, which she likes, and Kleene. She finds van Frassen’s supervaluation system especially appropriate to Aristotle’s future contingents’ problem. In general she is more accepting of thesis 3) than 4). In consideration of 4) she finds some affinities with it in Popper’s concept of ‘verisimilitude’ which she interprets as ‘closer to the truth’, where he considers two systems both of which are ‘really’ false. She finds an error in Popper’s argument here, which finding she credits to ‘D. Miller’. The error is just that if both systems are false and seeing that Popper is an adherent of bivalence, ‘closer to the truth’ is a non-sequitor.
She then goes on to conclude the core part of her book: Consequences for the theory of truth. Here she explores how three principles of classical logic are related:
1) Principle of bivalence (PB): Tp v Fp
2) The law of the excluded middle (LEM): p v ~p
3) Tarski’s hierarchical theory of truth (T): Tp = p.
She finds that
1) LEM & T => PB
2) T => PB, unless a non-classical meta-language is adopted (!)
Here ‘&‘ represents ‘and’ or conjunction, ‘v’ represents ‘or’ or disjunction, ‘~’ represents negation, ‘=’ represents identity, ‘p’ represents an arbitrary proposition or sentence, ‘Tp’ and ‘Fp’ are meant to represent ‘p is true’ and ‘p is false’, and ‘=>’ is understood as material implication, which is equivalent to ~p v q.
This last point highlights a question that seems to follow from her acceptance of Quine’s Two Dogmas thesis; if there is no difference in the end between ‘analytical’ and ‘empirical’ truths, then there are ultimately no grounds upon which to choose which deductive system, which logic, you prefer. She praises a pragmatist approach – take the system that is most economical and simple. But how to choose that? She finds the alternatives to classical logic unsatisfactory, but can give no better reason for adopting a given logic, in the end, than this. This leaves the door open to those who appreciate other values (like conformance with natural speech) in choosing alternative logics.
Haack proceeds in Part 2 of her book to a consideration of a series of candidates for replacement of classical logic. These include chapters devoted to Aristotle and Lukasiewicz’s problem of future contingents, intuitionism, vagueness, non-denoting singular terms, and quantum mechanics.
Her systematic critique of alternative systems of logic takes her first to Aristotle’s questions about the applicability of bivalence to future contingent events and Lukasiewicz’s many-valued logic motivated by the same question. She represents Aristotle’s argument in de Interpretatione, Chapter IX as follows:
1) If every future tense sentence is either true or false, then, of each pair consisting of a future tense sentence and its denial, one must be true, the other false.
2) If, of each pair consisting of a future tense sentence and its denial, one must be true, the other false, then everything that happens, happens ‘of necessity’.
3) But not everything that happens, happens of necessity; some events are contingent.
4) Therefore, not every future tense sentence is true or false.
This is a reductio ad absurdum argument. Premise 1) is the question at issue. If premise 2) and premise 3) are true then 4) represents the conclusion. Haack says that this is a valid argument, but challenges the supporting premises, namely 2). She also says that she has doubts about 3). She maintains that the argument from 2) to 3) constitutes a modal fallacy. What is a modal fallacy? Well, Aristotle explains what this is in the very same Chapter IX of de Interpretatione, in the middle of what Haack takes to be his argument! Aristotle says ‘Now that which is must needs be when it is, and that which is not must needs not be when it is not. Yet it cannot be said without qualification that all existence and non-existence is the outcome of necessity.’ That a fact is true or false does not imply that it is necessarily true or false. Haack explains this by saying that the fallacy committed here is to assume that L(p=>q) |- p=>Lq, where here ‘=>’ is understood to be material implication, which is identical to –p or q, L means ‘it is necessary that’, and ‘|-‘ represents semantic validity. The modal fallacy is commonly held to be the flaw in any argument in favor of fatalism and this is the argument that Aristotle employs in his reduction of universal bivalence.
It is good to get clear what she means by the important terms in her discussion. By the principle of bivalence (PB) she means ‘every wff is either true or false’. By the law of the excluded middle (LEM) she means “the wff ‘p or not p’”. Here, ‘wff’ means ‘well-formed formula’.
One possibility regarding truth values that Haack absolutely will not countenance in Deviant Logic is the possibility that a statement or sentence could be both true and false. In her discussion of Lukasiewicz’s consideration of what he proposes as the intermediate truth value ‘possible’ she concludes that this would lead to contradiction: ‘since |p&~p| = i, when |p|=|~p|=i, it looks as if |p&~p| must be regarded as possible if its conjuncts are, individually, possible’ which she says is ‘a quite unacceptable consequence’ (p. 87).
Her next candidate for alternative logic is intuitionism, developed in Brouwer’s PhD dissertation of 1907. She begins by showing how radical was Brouwer’s critique of the logicism of Frege and Russell. For while they tried to show that mathematics could be grounded in logic, Brouwer thinks otherwise; the intuitionist view is that mathematics is primary and logic secondary. Logic is just a compilation of the rules that are found, a posteriori, to be true of mathematical reasoning. But Brouwer held unusual views about mathematics too, which had elements of both psychologism and constructivism. First, numbers are mental entities, constructed out of ‘the sensation of time.’ Only constructible entities are admitted to the world of mathematical reality. This has the consequence that some mathematical postulates of classical (formalist) mathematics are not intuitionistically valid. For example, the LEM has counter examples.
She presents the structure of intuitionist mathematics as follows:
1) A subjectivist, constructivist, view of mathematics supports the thesis that
2) Some parts of classical mathematics are unacceptable
3) Logic is a description of the valid forms of mathematical reasoning which concludes that
4) Some parts of classical mathematics are mistaken
Haack recognizes that the set-theoretical paradoxes and Gödel’s theorems provide some incentive for adoption of a constructivist view, but in the end she finds intuitionism lacking. Her argument here is contra Dummett. Dummett’s “aim was to show that classical logic is in some respects mistaken, and to do so in a way which made no special appeal to the subject matter of mathematics. But this means that if his arguments were sufficient to establish the Intuitionist’s view of mathematics, they would be sufficient to establish anti-realism with respect to any subject matter. Anti-realism is less plausible when applied to other subject matters – e.g. geography – than when applied to mathematics. Dummett seems to recognize this . . . . So, on confession of its advocate, the ‘strongest argument’ for Intuitionism is less than conclusive.” The ellipses in the quotation are a quote from Dummett (‘Wittgenstein’s philosophy of mathematics,’ Philosophical Review, 1959) to the effect that his considerations to establish Intuitionism apply to all discourse, but that “they surely do not imply outside mathematics the extreme of subjective idealism – that we create the world.” Haack’s argument seems to be that Intuitionism entails idealism, even though Dummett explicitly denies it. What seems to lead to the difference in these conclusions seems to be the assumption of bivalence by Haack, but not by Dummett.
Haack’s third candidate as a rival to classical logic is vagueness. She considers several types of vagueness. One is the sentence where it may be indeterminate that a vague predicate applies to certain subjects. Another type case would be the Sorites paradox, and example of which is the question:”Bertrand Russell was young in 1890 and old in 1960. At what age did he become old?” Here it would seem that an adherent to bivalence would insist that there must be a true or false answer to the question, “Was Professor Russell old in, say, 1952?” (She doesn’t use this example, but the argument Russell used concerning the sentence “Socrates is bald.”) But isn’t this a question to which there is no answer which is wholly true or wholly false, but one in which there are grades of truth? In other words, this must fall in Haack’s category 4), the new truth value category. She thinks not. She considers two ways in which predicates could be vague:
1) The qualifications for being F are imprecise
2) The qualifications for being F are precise, but there is difficulty in determining whether certain subjects satisfy them
She considers Dummett’s argument (‘Wang’s paradox’, unpublished paper, 1970) that LEM may not apply. He considers a sentence “O is orange.” where it may be uncertain whether the object is orange or red. She agrees that if Dummett’s argument were sound, it would suggest a non-classical logic like van Frassen’s ‘conventional so far as theoremhood is concerned but non-bivalent’. She considers arguments that vagueness requires a non-classical logic to be unsound, but that logic should include vague predicates, against Russell and Whitehead’s exclusion. She considers Duhem’s argument that science, because it can offer more precise measurements, actually makes statements that are more vague (less certain) than more conventional statements. For example ‘Jones is tall’ can be more certain than ‘Jones is 6 ft 4.0625 in. high’ because the specificity of the more precise measurement means the allowable error is less. She finds Carnap’s argument (Logical Foundations of Probability, 1950) for ‘precisifying’ common language, replacing vague sentences by more precise ones, feasible because, even though it may lead to more uncertainty, this uncertainty does not threaten bivalence, according to Carnap (and her).
Her next topic is the problem of singular terms and existence. Haack suggests two “troublesome assumptions” of classical logic:
1) That all singular terms denote [|- Fa => (Ex)Fx]
2) That the universe of discourse is non-empty [|- (x) (Fx=>Fa)]
Where ‘|-‘ can be read ‘it is a theorem’, ‘Fa’ and ‘Fx’ are meant to represent ‘The function F of a or x’, ‘Ex’ is meant to mean ‘there exists an x’, and ‘=>’ is understood as material implication.
She finds four possible strategies to deal with these problems, similarly to her treatment of how to manage the question of a third truth value in Chapter 3:
1) the ‘no item’ thesis
2) The ‘misleading form’ thesis
3) Modify logic at the predicate calculus level
4) Modify logic at the propositional calculus level
She suggests that Frege falls in the ‘no item’ camp, Russell in the ‘misleading form’ camp, several philosophers including Schock (Logics without existence assumptions, 1968), Leonard (‘The logic of existence,’ Philosophical Studies, 1956), and Hintikka (‘Existential presuppositions . . . , Journal of Philosophy, 1959) in the camp that want to modify the predicate calculus, and lastly (surprisingly) she finds Frege exploring, but ultimately rejecting, the fourth approach. In the course of this discussion she presents an interesting discussion of Russell’s quarrel with Meinong and of Frege’s sense / reference distinction and his famous identity:
The Morning Star = The Evening Star.
Russell finds that “The King of France is bald.” suffers from a failure to provide a bona fide subject. The sentence fails to provide a proper logical form. He argues against Frege who proposed to provide a real object, for example the number zero, for otherwise non-denoting predicates to reference. Russell (‘Review of Meinong,’ Mind, 1904) also attacked Meinong (Untersuchungen zur Gegendstandtheorie und Psychologie, translation reprinted in Logic and Philosophy, 1968) who allowed non-denoting terms to stand for unreal objects. Russell objects that Meinong’s theory leads to contradiction. But Meinong denied that existence is a property of an object. Haack finds that Meinong would not be worried by Russell’s claim of inconsistency. Deviant Logic was completed at about the same time as the defense of Meinong’s approach by Richard Routley (See Graham Priest’s recent book Towards Non-being, 2005.) As a result, she is not able to comment upon it here.
She ends this chapter with a consideration of a ‘solution’ to the problem of non-denoting terms and empty worlds: replacing the ‘objectual’ conception of quantification by a ‘substitutional’ interpretation. She considers an objection to this by Quine which she ultimately rejects. It is interesting that the possibility that these ‘problems’ arise as a result of bivalence and that a more tolerant attitude to non-bivalent logic would also serve as a ‘solution’ to the problems of emptiness and singular terms is not a possibility that she gives much concern.
Her last chapter is devoted to an analysis of the ‘quantum logics’ of Birkhoff and von Neumann (B v N), Destouches-Février (D-F), and Reichenbach (Philosophical Foundations of Quantum Mechanics, 1944). These all propose a three-valued logic with the third value generally ‘indeterminate.’ This alternative logic is proposed to accommodate the ‘causal anomalies’ of quantum mechanics, for example the ambiguity of a particle or wave description of a Young’s slit interference experiment. Haack presents three objections to these proposals:
1) It is methodologically vicious (Popper and Feyerabend.)
2) It is simpler to use bivalence to deal with these anomalies (Quine.)
3) The ‘causal anomalies’ aren’t really derivable in quantum mechanics (Feyerabend.)
4) Reichenbach’s logic doesn’t avoid the anomalies (Feyerabend and others.)
She rejects Popper’s criticism (‘A realist view of physics, logic and history’, 1970) on the same grounds of circularity that she develops in Chapter 2. She doubts that Quine’s judgment (in Philosophy of Logic, 1970) on the supposed simplicity and familiarity of classical logic is defensible. She presents an argument to rebut Feyerabend’s contention (‘Reichenbach’s interpretation of quantum mechanics’, 1958) that Reichenbach smuggles in a classical assumption to demonstrate the ‘causal anomalies.’ An example of this classical assumption would be the assumption that an electron possesses a well-defined position and a well-defined momentum. Haack shows that Reichenbach’s argument needs no such classical assumption. She is sympathetic to the fourth criticism and finds that indeed Reichenbach’s three-valued system ends up with the same problem of ‘causal anomalies’ as bivalent logic. But one might say that the problem is no worse.
Lastly Haack considers the quantum logics of B v N and D-F. She defends B v N against Popper’s claim that their logic collapses into classical logic. But she is less sympathetic to Putnam’s defense of B v N (‘Is Logic Empirical?’, 1969) to the effect that the only laws of classical logic given up by B v N are the distributive laws and that by doing so ‘every single anomaly vanishes’. She finds this argument inconclusive. But she finds D-F likewise absent of any ‘complete formalization.’ Finally, she considers a fifth objection; that quantum logics are not ‘really’ logics. She generally rejects this argument since she has accepted Quine’s argument that logic and empirical theory amount to the same thing. So logic is as revisable as any other theory.
So here Deviant Logic rather abruptly ends. In spite of her openness to the idea of the revisability of logic, she really finds no acceptable candidate of which she whole-heartily approves. The book is in the end conservative with respect to classical bivalent logic. Perhaps the best summary for this review is her own post script included in the preface to her subsequent book Philosophy of Logics:
“It is, I find, irritating to be unsure whether, or how, an author has modified views he previously put forward; but, on the other hand, it is tedious to be subjected to frequent discussions of an author’s earlier mistakes. By way of compromise, therefore, I indicate here, briefly, where, and how, I have modified the ideas put forward in Deviant Logic. First: I have, I hope, made the distinction between metaphysical and epistemological questions about the status of logic rather clearer; and this has led me to distinguish more carefully between the question of monism and pluralism, and the question of revisability, and to support a qualified pluralism rather than the monism somewhat confusedly assumed in Deviant Logic. Second: I have come to appreciate that the consequences for ontology of the substitutional interpretation of the quantifiers are somewhat less straightforward than I used to suppose; and this has led me to a more subtle, or at any rate more complex, account of the respective roles of quantifiers and singular terms. I dare say, though, that I shall have missed some old mistakes, besides making some new ones.”
This seems to be a relatively fair assessment. The current reviewer finds that most of the arguments offered in defense of classical logic as uniquely valid fall short of establishing this goal. The choice of logic may be one based on pragmatic grounds, as Haack thinks in Deviant Logic. A skeptical attitude may serve better: the choice of logic seems to often mirror the chooser’s metaphysics. Realists typically choose bivalence, anti-realists intuitionist logic. The logic for skeptics seems to be a multi-valued logic which allows values of true and false, but also neither, intermediate, or both.