The Normative status of Logic

A Logic is a set of rules yielding valid inferences provided the conditional tautologies on which they are based are themselves true. Such rules are not value judgments. We don't say the 'excluded middle' rule is very naughty coz it is excluding the middle from its snobby playgroup. We might say, as the Intuitionists do, that it permits illegitimate inferences and thus should be discarded. But that is not because the rule is naughty but because the inferences it permits are defective.

Can a Logic have a 'normative status'? If so, why not a particular number system- Arabic rather than Hindu- or an alphabet- Latin against Devnagari?

Currently, we think 'Logical Pluralism' is 'normative'- in the sense of being useful and conforming to real world doxastic reasoning- and, moreover, that Dialethia & paraconsistent logics can permit superior pragmatics.

Given these considerations, it seems strange that there are some academics who still think it worthwhile to go on gassing on about the soi disant 'normative status of logic'.

Consider the following introductory paragraph from the Stanford Encyclopedia of Philosophy on this subject-

We consider it to be a bad thing to be inconsistent. Similarly, we criticize others for failing to appreciate (at least the more obvious) logical consequences of their beliefs.

Is this really true? No doubt, as small children we accuse our parents of being inconsistent because the want us 'to do as they say, not do as they do'. However, we soon outgrow such childishness. By the time we are in High School, we have learned to feel contempt for the kid who is constantly putting his hand up to show he is smarter than the teacher coz of some supposed inconsistency in what she is saying.
But, the thing is puerile. We understand that the 'Structural Causal Model' being invoked is merely heuristic; it is deliberately under-specified for reasons of communicative economy, and thus applying Logic to it won't get us to valid inferences.  Instead, they will make us stupider and more obstreperous than our ignorance would otherwise militate for.

It is not the case that we act inconsistently when our SCM is underspecified and our 'cheap talk' reasoning does not conform to any 'logical structure' upon that model simply because it would be too cognitively costly to bother.

The Stanford article, takes the opposite view-

In both cases there is a failure to conform one’s attitudes to logical strictures.

 Logical structures only exist upon a fully specified SCM which may itself not be effectively 'computable'- i.e. have an accessible mathematical representation. Still, they can give us an idea of the direction in which a Schelling focal solution to a coordination problem exists. Attitudes orient us towards those focal points and have mimetic value. There is no great metaphysical 'scandal' or epistemological 'failure' here. The thing is economic simply.

We generally take agents who fall short of the demands of logic to be rationally defective.

Nonsense! I can't put my wife in a lunatic asylum just because I think her to be 'rationally defective' in preferring to sleep with much  uglier and poorer men than myself, protesting all the while that she cares about me deeply and wants us to work on our marriage.

This suggests that logic has a normative role to play in our rational economy; it instructs us how we ought or ought not to think or reason.

 This is the suggestion of our bad angel who will cause us to fuck up every relationship we get into. A sociopath or paranoiac may be able to 'reason' much more logically than we can. A Computer may be wholly logical in its operations and yet come to crazy conclusions.

The test of how we ought or ought not to think or reason is empirical. If we are successful in pursuing the sort of aims a Court would find it 'reasonable' for us to have, then we are not 'rationally defective' even if we claim that our actions are based upon Astrology or Numerology or some other such 'rationally defective' belief system.

The notion that logic has such a normative role to play is deeply anchored in the way we traditionally think about and teach logic. To consider just two examples, Kant characterizes what he calls “pure general” logic as embodying the “absolutely necessary rules of thought without which there can be no employment whatsoever of the understanding” (A52/B76), which instruct us not “how the understanding is and thinks” but “how it ought to proceed” (Kant 1974 [1800]: 16).

But Kant came to crazy conclusions- e.g. slavery was cool but wanking was a terrible violation of Human Freedom.

How thought ought to proceed is something we can only decide by looking at the likely outcome of actions based on that thought.

Of course, Logic could be normative, if we live in an Occasionalist Universe (i.e. one where everything is pre-ordained) governed by a crazy God who will punish us horribly in the after-life if we don't think the way He has decided we ought to.

Similarly, Frege, in his vehement opposition to the psychologistic tendencies of his time, classifies logic, “like ethics” as a “normative science” (Frege 1897/1979: 228), one whose laws “prescribe universally how one should think if one is to think at all” (Frege 1893/1903/2009: xv).

But Frege's project collapsed because of Godel & Tarski & Turing and so forth. Logic is not a 'normative science' because it can't include its own metalanguage and thus attach a normative label to its own operations.

This entry is concerned with the question as to whether the tradition and the intuitions that appear to underwrite it are correct. In other words, it is concerned with the question as to whether logic has normative authority over us? And if so, in what sense exactly it can be said to do so?

The  question is a foolish one. The answer is no; logic has no 'normative authority' over us. No logician has ever claimed it does or can do. Logic itself can't contain its own meta-language- i.e. it isn't normative even with respect to itself.

Let us see how a worthless branch of the Academy can continue to pretend otherwise.

Before we can hope to make any headway with these questions a number of clarifications are in order. First and foremost, in asking after the normative status of logic, we had better get clearer on what we mean by “logic”. For present purposes, I will take a logic to be a specification of a relation of logical consequence on a set of truth-bearers.

Since 'a truth-bearer is an entity that is said to be either true or false and nothing else' and if propositions are 'modally neutral'- i.e. don't differentiate between possible worlds and thus have no description as a probability distribution- they are themselves the output, or logical consequence, of the operation of a system of logic.

What is the point of defining something in terms of itself?  If I say- 'I will take a boojandi to be a specification of a relationship of boojandiness on a set of boojandis' - in what manner have I clarified the nature of a boojandi? 

By contrast, if we say that a Logic specifies the set of valid inferences that can be made from a given Structural Causal Model which has an axiomatic representation, then though my statement is defeasible, it is not self-referential or, practically, meaningless.

 Furthermore, because, an Intuitionist or Fuzzy Logic, or Bayesian approach, can be 'observationally equivalent' to one which features 'truth bearers', it is foolish to 'take Logic to be' something it doesn't necessarily have to be.

Moreover, I will assume consequence relations to necessarily preserve truth in virtue of logical form.

Why assume that paradoxes, fallacies, or cases of undecidability never arise? The history of mathematics shows the opposite to be the case.

For simplicity, I will use “⊨$\models$” to denote such a consequence relation. My default assumption will be to take the double turnstile to denote the semantic consequence relation of the classical first-order predicate calculus. But not much hangs on this. Partisans of other types of non-classical consequence relations may read “⊨$\models$” as referring to their preferred consequence relation.

So, the 'default assumption' here is that we are dealing with a set of truths- some of which are given, others of which are logically inferred- valid in all possible worlds.

But, if we lived in a world where we had access to such truths, why would we need a concept of 'normativity'? Consider the story of Moses and Al Khidr. The former sees the latter doing 'bad' things. But the latter know the truth about everything. What appears to 'bad' is actually 'good' because Al Khidr knows the future. Indeed, he is univocal with the Occasionalist God of Ghazali.

Presumably, if logic is normative for thinking or reasoning, its normative force will stem, at least in part, from the fact that truth bearers which act as the relata of our consequence relation and the bearers of other logical properties are identical to (or at least are very closely related in some other way) to the objects of thinking or reasoning: the contents of one’s mental states or acts such as the content of one’s beliefs or inferences, for example.

How can 'the normative force' of anything stem from something impossible- viz. truth-bearers existing as 'objects of thinking or reasoning'? Why think or reason about something true in all possible worlds? It is a waste of cognitive resources.

Norms exist because of Uncertainty, Information asymmetry & the Stochastic nature of 'regret minimization'.

What is the point of assuming none of these exist? Can anything more foolish be asserted? Yes- look at this next sentence-

For present purposes I will assume the identity between truth-bearers and the contents of our attitudes, and I will assume them to be propositions.

Wow! My attitude of contempt towards this article arises because I somehow have access to 'truth-bearers' which assert that Philosophy is shite in all possible worlds!

However, no such 'truth-bearers' can exist. Still, there is a sense in which Logic is normative. There are 'focal point' solutions to coordination games- things like Courts of Law where disputes can be adjudicated- and protocol bound 'juristic' reasoning is normative in such places. People have to show that they acted as a 'reasonable person' would do and this necessarily entails casting their assertions in a coherent logical form.

Academic Philosophy, it seems, takes a different view-

.. let us ask what it is that logic is normative for, if indeed it is normative. The paradigmatic objects of normative appraisal are actions, behaviors or practices. What, then, is the activity or practice that logical norms apply to?
2.1 Logic as normative for reasoning
One response—perhaps the most common one—is that logic sets forth norms for (theoretical) reasoning.

 'Norms are concepts (sentences) of practical import, oriented to effecting an action, rather than conceptual abstractions that describe, explain, and express.'
A logic- i.e. a set of rules for valid inference- can't cause us to do theoretical reasoning instead of watching Kung Fu videos. By contrast, Mom hitting you with a rolling pin and threatening to break your legs if you fail your exams and don't go off to College, is setting forth a norm- viz. watching Kung Fu videos is naughty. You ought to be studying instead.

Unlike thinking, which might consist merely of disconnected sequences of conceptual activity, reasoning is presumably a connected, usually goal-directed, process by which we form, reinstate or revise doxastic attitudes (and perhaps other types of states) through inference.

Not necessarily. Reasoning can be wholly mimetic. Indeed, in its origins, it always is. Furthermore, it always has a strategic aspect.

Consider the following two examples of how logic might give rise to norms. First, suppose I am trying to find Ann and that I can be sure that Ann is either in the museum or at the concert.

If you are trying to find Ann, you must think you ought to find Ann. That is a norm. But it has nothing to do with any system of logic.

I am now reliably informed that she is not in the museum. Using logic, I conclude that Ann is at the concert.

This is not a safe conclusion. Ann may be planning to leave the concert and teleport to the museum the moment she is reliably informed that you have been reliably informed that she is not in the museum.

Thus, by inferring in conformity with the valid (by the standards of classical logic) logical principle of disjunctive syllogism, I have arrived at a true belief about Ann’s whereabouts.

No you haven't.

Second, if I believe that Ann is either at the concert or the museum, while at the same time disbelieving both of the disjuncts, it would seem that there is a tension in my belief set, which I have reason to rectify by revising my beliefs appropriately.

Tension in one's belief set is not a good reason to do something foolish. 'Rectifying and revising my beliefs' w.r.t something which does not concern me at all is foolish. It involves an opportunity cost. Suppose I skip work coz I'm rectifying and revising my beliefs about the ancient Aztecs, I'll get fired and end up a homeless bum.

There is a norm for not thinking about stupid shite. There is none which forbids me having incompatible beliefs- e.g. that we evolved by Natural Selection and also possess immortal souls endowed by a Divine Creator.  Why? Normativity can only exist if Dialethia prevails.

Logic may thus be thought to normatively constrain the ways we form and revise doxastic attitudes.

No. If Logic can normatively constrain its own operations then Godel & Tarski & Turning an so forth must have been wrong- i.e. Logic must be wrong.

And it does so, presumably, in our everyday cognitive lives (as in our example), as well as in the context of more self-conscious forms of theoretical inquiry, as in mathematics, the sciences, law, philosophy and so on, where its normative grip on us would seem to be even tighter.

Sheer nonsense. Bayesian Decision, or Learning, theory, not Logic, explicates such 'normative constraints' which can lead to cognitive biases. What matters is getting to a better specified Structural Causal Model rather than making inferences on the basis of a foolish caricature of one.

Thought may have an algorithmic expression. But, most often, to achieve robustness, it is likely to be non-linear, use 'oracles', and have a stochastic 'regret minimizing' representation.

It simply can't be logical for purely mathematical reasons well known since the early Seventies. Yet, academic philosophers, pretending otherwise, can write of-

 Logic as constitutively normative for thought

 In other words, if a thought has a logical form then it quite properly becomes action guiding. Thus, if you have the thought- cake exists to be eaten, there is a cake in the fridge- then you should eat the cake even if Mom will beat you for ruining her dinner party.

Other philosophers have taken the normativity of logic to kick in at an even more fundamental level. According to them, the normative force of logic does not merely constrain reasoning, it applies to all thinking. The thesis deserves our attention both because of its historical interest—it has been attributed in various ways to Kant, Frege and Carnap[6]—and because of its connections to contemporary views in epistemology and the philosophy of mind. 

To get a better handle on the thesis in question, let us agree to understand “thought” broadly as conceptual activity. Judging, believing, inferring, for example, are all instances of thinking in this sense. It may seem puzzling at first how logic is to get a normative grip on thinking: Why merely by engaging in conceptual activity should one automatically be answerable to the strictures of logic? After all, at least on the picture of thought we are currently considering, any disconnected stream-of-consciousness of imaginings qualifies as thinking. One answer is that logic is thought to put forth norms that are constitutive for thinking. That is, in order for a mental episode to count as an episode of thinking at all, it must, in a sense to be made precise, be “assessable in light of the laws of logic” (MacFarlane 2002: 37). Underlying this thesis is a distinction between two types of rules or norms: constitutive ones and regulative ones.
The distinction between regulative and constitutive norms is Kantian at root. Here, however, I refer primarily to a related distinction due to John Searle. According to Searle, regulative norms “regulate antecedently or independently existing forms of behavior”, such as rules of etiquette or traffic laws. Constitutive norms, by contrast create or define new forms of behavior. The rules of football or chess, for example, do not merely regulate playing football or chess but as it were they create the very possibility of playing such games. 

Logic can be neither constitutive nor regulative of itself- at least, that is its own verdict on itself. Thus either it isn't 'thinking' or else it can't regulate or constitute even thinking about itself- let alone anything else.

Take the case of traffic rules. While I ought to abide by traffic rules in normal circumstances, I can choose to ignore them. Of course, rowdy driving in violation of the traffic code might well get me in trouble. Yet no matter how cavalier my attitude towards traffic laws is, my activity still counts as driving. Contrast this with the rules governing the game of chess. I cannot in the same way opt out of conforming to the rules of chess while continuing to count as playing chess; in systematically violating the rules of chess and persisting in doing so even in the face of criticism, I forfeit my right to count as partaking in the activity of playing chess.

Yet we know that a rigid designation of 'chess playing' must have featured chess players- probably the greatest ones of their time- who did precisely that without 'forfeiting' the right to 'count as partaking in the activity of playing chess'.

If I am asked to play chess with a 5 year old child, I insist the little fellow handicap himself by removing those horsy things which frighten me. Since little kids enjoy humiliating elderly Uncles, they comply and win anyway, though I cheat incessantly.

Unless one’s moves are appropriately assessable in light of the rules of chess, one’s activity does not qualify as playing chess.

They may do, if you are a 5 year old kid determined to show that the boring old Uncle is completely stupid.

According to the constitutive conception of logic’s normativity the principles of logic are to thought what the rules of chess are to the game of chess: I cannot persistently fail to acknowledge that the laws of logic set standards of correctness for my thinking without thereby jeopardizing my status as a thinker (i.e., someone presently engaged in the act of thinking).
Two important clarifications are in order. For one, on its most plausible reading, the thesis of the constitutive normativity of logic for thought must be understood so as to leave room for the possibility of logical error: an agent’s mental activity may continue to count as thinking, despite his committing logical blunders.
That is, although one may at times (perhaps even frequently and systematically) stray from the path prescribed by logic in one’s thinking, one nevertheless counts as a thinker provided one appropriately acknowledges logic’s normative authority over one’s thinking. Consider again the game of chess. In violating the rules of chess, deliberately or out of ignorance, I can plausibly still be said to count as playing chess, so long, at least, as I acknowledge that my activity is answerable to the rules; for example, by being disposed to correct myself when an illegal move is brought to my attention. Similarly, all that is necessary to count as a thinker is to be sensitive to the fact that my practice of judging, inferring, believing, etc., is normatively constrained by the laws of logic. It is not easy to specify, in any detail, what the requisite acknowledgment or sensitivity consists in. A reasonable starting point, however, is provided by William Taschek who, in his interpretation of Frege, proposes that acknowledging the categorical authority of logic will involve one’s possessing a capacity to recognize—when being sincere and reflective, and possibly with appropriate prompting—logical mistakes both in one’s own judgmental and inferential practice and that of others. 

So what? With hindsight, one could do the same for the 'categorical authority' of Astrology or Numerology or the I Ching or the Grand Wizard or the Great Leader or anyone or anything real or imaginary that can be named or alluded to.

A second point of clarification is that the agent need not be able to explicitly represent to herself the logical norms by which she is bound.

So, if captured by ISIS, I can show that I've always acted as true adherent of their abhorrent creed though I can't say what it is.

For instance, it may be that my reasoning ought to conform to disjunctive syllogism in appropriate ways. I may be able to display the right kind of sensitivity to the principle by which I am bound (with the right prompting if need be), without my having to possess the conceptual resources to entertain the metalogical proposition that ¬A,A∨B⊨B¬A,A∨B⊨B. Nor must I otherwise explicitly represent that proposition and the normative constraint to which it gives rise.
With these clarifications in place, let us turn to a central presupposition of the approach I have been sketching. What is being presupposed, of course, is a conception of thinking that does not reduce to brute psychological or neurophysiological processes or events. If this naturalistic level of description were the only one available, the constitutive account of the normativity of logic would be a non-starter.

It is a non-starter iff Godel, Tarski, Turing et al were bag logicians. Watering down what 'constitutive' means turns it into a 'use as you like' word.  Anything at all can be said to be constitutive of anything else.

What is being presupposed, therefore, is the permissibility of irreducibly normative levels of descriptions of our mental lives.

Why not 'irreducibly Catholic', or 'irreducibly Freudian', or 'irreducibly Marxian' levels of descriptions of our mental lives?

These things are 'impermissible' because they are the stuff that cults and totalitarian dictators use to turn people into amoral robots.

In particular, it is assumed that the boundary between the kinds of mental activity that constitute thinking and other kinds of mental activity (non-conceptual activity like being in pain, for instance) is a boundary best characterizable in normative terms.

Why make such a foolish assumption? What is sauce for the goose is sauce for the gander. If there is a boundary 'best characterizable in normative terms', then there can be another boundary some one with superior autocritas (derivable, perhaps, in a coercive manner) between who can use those normative terms to direct others and those who must hear and obey.

This is not to deny that much can be learned about mental phenomena through descriptions that operate at different, non-normative levels—the “symbolic” or the neurological level of description, say—the claim is merely that if we are interested in demarcating conceptual activity from other types of mental phenomena, we should look to the constitutive norms governing it.

If the information set can be usefully enlarged by looking on the other side of the boundary, then the boundary can't be 'normative'. A Judge can't say 'this evidence is not admissible but we should listen to it anyway'.

If we are interested in demarcating phenomena, we take an extensional approach- i.e. accumulate data and develop Structural Causal Models. We don't look for intensional 'constitutive norms'- unless we are monks living in the Dark Ages.

Davidson (1980, 1984), Dennett (1987), and Millar (2004) all hold views according to which having concepts and hence thinking requires that the agent be interpretable as at least minimally sensitive to logical norms.

'Minimally sensitive' is not a term useful to Logic. A fallacy may be minimally or even moderately, or indeed hyper, sensitive to logical norms but it is still a fallacy.

Also, certain contemporary “normativist approaches” according to which accounts of certain intentional states involve ineliminable appeals to normative concepts may advocate the constitutive conception of logic’s normativity (e.g., Wedgwood 2007, 2009; Zangwill 2005).

Yes. If you are talking bollocks, it is helpful to believe something false and foolish. You can then exploit -Ex falso quodlibet- to give the appearance of logical consistency to your stupid shite.

2.3 Logic as normative for public practices
So far the answers to the question “What is logic normative for?” we considered had in common that the “activities” in question—reasoning and thinking—are internal, mental processes of individual agents. But logic also seems to exert normative force on the external manifestations of these processes—for instance, it codifies the standards to which we hold ourselves in our practices of assertion, rational dialogue and the like. While much of the literature on the normativity of logic focuses on internal processes of individuals, some authors have instead emphasized logic’s role as a purveyor of public standards for normatively regulated practices.

'Public practices'- e.g. the practice of Law or Medicine or the conduct of a Business Enterprise- do involve a duty to act, and to show you have acted, in a reasonable manner and in keeping with established epistemic protocols in the relevant field.

However, hiring a logician, rather than an expert in the relevant field, to tell you what you should or shouldn't do would be very foolish. This is because evidence of what is normative is wholly empirical or extensional.

Take the practice of asserting. Assertion is often said to “aim at truth” (or knowledge, Williamson 2000: Ch. 11) as well as being a “matter of putting forward propositions for others to use as evidence in the furtherance of their epistemic projects” (Milne 2009: 282). Since I take the asserted propositions to be true and since truths entail further truths, I am “committed to standing by” the logical consequences of my assertions or else to retract them if I am unable to meet challenges to my assertion or its consequences.

Nonsense! No assertion has any 'logical consequences' whatsoever unless it is a 'truth-bearer'. But if it is, and if Logic has been perfected, then all 'truth-bearing' propositions can be algorithmically derived. We would have a Mathesis Universalis. History would end because we'd have a Theory of Everything and could change the Multiverse in any manner we pleased.

Similarly, if the set of propositions I assert is inconsistent at least one of my assertions must fall short of being true and the set as a whole cannot be regarded as part of my evidence.

Is this true? Suppose there is an inconsistency in a witnesses testimony. Would it necessarily cause the whole of it to be dismissed? No. What matters is if the inconsistency was 'material' or it threw into question the good faith or cognitive or other capacity of the witness.

Plausibly, therefore, logic does have a normative role to play in governing the practice of assertion.

If it 'governs' the practice of assertion- i.e. assertions are valid inferences from 'truth makers'- nobody in history has ever asserted anything. What sort of 'normative role' is this?

Peter Milne takes an interest in assertion mainly in order to “work back” from there to how logic constrains belief. He concludes that logic exerts normative force at least on the stock of beliefs that constitute the agent’s evidence (Milne 2009: 286).

Milne and his colleagues are counter-evidence. Their 'stock of beliefs' are wholly incompatible with modern Logic- or, indeed, common sense.
What constrains them is the desire to hang on to their salary checks and professorial prestige.
In any case, it is cognitive dissonance, not logic, which causes people to prune their stock of beliefs.

Other authors explicitly prioritize the external dimension of reasoning, conceived of as a social, inter-personal phenomenon.

This is either 'process monitoring'- which has to do with information asymmetry- or 'error checking'. It has nothing to do with logic whose inputs are 'truth-makers'- i.e. error free and unequivocal.

According to them, it is reasoning in this external sense (as opposed to intra-personal processes of belief revision, etc.) that is the primary locus of logical normativity (MacKenzie 1989). The norms govern our rational interactions with our peers. For instance, they might be thought to codify the permissions and obligations governing certain kinds of dialogues.

But, these are essentially defeasible dialogues- not logical at all.

Viewed from this perspective, logic’s normative impact on the intra-personal activity of reasoning is merely derivative, arrived at through a process of interiorization. A view along these lines has been advanced by Catarina Dutilh Novaes (2015).

This view can be given an economic description. However, it doesn't have any great utility. Vide http://socioproctology.blogspot.com/2013/01/this-is-great-mathematician-terry-tao.html

In a similar vein Sinan Dogramaci (2012, 2015) has proposed a view he calls “epistemic communism”. According to epistemic communism our use of “rational” applied to certain deductive rules has a specific functional role. Its role is to coordinate our epistemic rules with a view to maximizing the efficiency of our communal epistemic practices. On the basis of this view, he then elaborates an argument for the pessimistic conclusion that no general theory of rationality is to be had.

This is perfectly reasonable. That is why, for this academic availability cascade to trundle on, the 'bulk of the literature' must ignore it and carry on regardless.

We will here follow the bulk of the literature in asking after the normative role logic might play in reasoning understood as an intra-personal activity. Yet, much of the discussion to follow applies mutatis mutandis to the other approaches.

The Encyclopedia article next takes up Gilbert Harman's sensible assertion that reasoning is wholly distinct from deductive logic. It is stupid to pretend otherwise.

The author takes a typically perverse view-

saying that deductive logic and theories of reasoning are distinct is one thing, affirming that there could not be an interesting normative connection between them is quite another.

'Interesting normative connections' can be made between anything and everything. By refusing to make these connections, we save ourselves from talking or listening to shite.

As a first stab at articulating such a connection, we might try the following simple line of thought: theoretical reasoning aims to provide an accurate representation of the world.

Nonsense! Theoretical reason may be wholly ontologically dysphoric. Practical reason aims at a more useful representation of the world.

We accurately represent the world by having true (or perhaps knowledgeable) beliefs and by avoiding false ones.

Rubbish! A Cartographer who produces a better map might also have absurd beliefs. There is no necessary connection between 'representation' and doxastic states. I may not be able to accurately paint this beautiful sun-set whereas you might be able to do so, yet consider the result to be trashy kitsch.

But our doxastic states have contents—propositions—and these contents stand in certain logical relations to one another.

Quite false! I believe my Mom to be the best cook in the world though I acknowledge that she used less salt than is proper because my father's side of the family had a history of high blood pressure.

Having an awareness of these logical relations would appear to be conducive to the end of having true beliefs and so is relevant to theoretical reasoning.

It does not so appear to anyone with even the slightest knowledge of the history of philosophy.

In particular, the logical notions of consequence and consistency seem to be relevant. If I believe truly, the truth of my belief will carry over to its logical consequences.

And you will be truly damned.

Conversely, if my belief entails a falsehood it cannot be true.

So what? Beliefs only have strategic, that is survival, value if they aren't true. All our loving relationships are based on it being common knowledge that we think the other to be the bestest Mommy or Baby or Cutie-pie ever. We may not know the exact weight and height of the other- indeed, it is possible to love a person for many years without being able to say what color their eyes are- but that is irrelevant. Beliefs are a first step to, not Truth, but Faith.

Similarly, if the set of propositions I believe (in general or in a particular domain) is inconsistent, they cannot possibly afford an accurate representation of the world;

If you believe this then you can't believe a set of propositions accurately represents what you believe because this belief would be a member of that set and it says that some other member of the set isn't a member of the set.
The author doesn't get this. He says-

at least one of my beliefs must be false.

Not necessarily- the world may be inconsistent. However, you are still in trouble because at least one of your beliefs is that you don't believe something you do believe.

Harman may be able to agree with all of this. His skepticism pertains also (and perhaps primarily) to the question whether logic has a privileged role to play in reasoning; that the principles of logic are relevant to reasoning in a way that principles of other sciences are not (Harman 1986: 20). However, I want to set this further issue to one side for now.
Notice that this simple reflection on the connection between logic and norms of reasoning leads us right back to the basic intuitions at the beginning of this entry: that there is something wrong with us when we hold inconsistent beliefs or when we fail to endorse the logical consequences of our beliefs (at least when we can be expected to be aware of them). Let us spell these intuitions out by way of the following two principles. Let S
S be an agent and P
P a proposition.[13]
Logical implication principle (IMP): If S
S’s beliefs logically imply AA, then S
S ought to believe that AA.
Logical consistency principle (CON): S
S ought to avoid having logically inconsistent beliefs.
Notice that on the face of it IMP and CON are distinct. IMP, in and of itself, does not prohibit inconsistent or even contradictory beliefs, all it requires is that my beliefs be closed under logical consequence.

This can't happen if logically inconsistent beliefs are possible- i.e. if CON is not empty- because then you have a belief about your beliefs- a meta-belief- so 'closure under logical consequence' is off the table.

CON, on the other hand, does not require that I believe the consequences of the propositions I believe, it merely demands that the set of propositions I believe be consistent.

In which case, if you believe your belief set can be closed under logical consequence, then you can't have a logically consistent belief system for reasons Godel, Tarski etc clarified long before I was born.

What happens if you assume something necessarily, logically, false- viz. that Set theory is immune to Russell's paradox- is that you get ex falso quodlibet an explosion of nonsense. As witness-

However, given certain assumptions, IMP does entail CON. Against the background of classical logic, the entailment obtains provided we allow the following two assumptions: (i) one cannot (and, via the principle that “ought” implies “can”, ought not) both believe and disbelieve one and the same proposition simultaneously; and (ii) that disbelieving a proposition is tantamount to believing its negation.

 For let 
{A1,…,An}$\{A_1,\dots ,A_n\}$ be S$S$’s inconsistent belief set. By classical logic, we have A1,…,An−1⊨¬An$A_1,\dots ,A_{n-1}\models \mathrm{\neg }{A}_n$. Since S$S$’s beliefs are closed under logical consequence, S$S$ believes ¬An$\mathrm{\neg }{A}_n$ and hence, by (ii), disbelieves An$A_n$. So, S$S$ both believes and disbelieves An$A_n$.

There is no great scandal in both believing and not believing a proposition. We do so all the time. The thing has survival value.  What is utterly pointless is appealing to Set theory and forgetting Russell's paradox or invoking Logic and forgetting Godel & Tarski.

Still, if that's what you are paid to do, then, I guess, it is normative.