Analytic Self-Management & Attention Portfolio Analysis

Introduction

Attention is scarce while possible demands on attention are innumerable. A fire alarm setting off in a building, allowing most occupants to exit safely, places about the same amount of demand on their attention as would a false alarm triggered due to a problem with the wiring. The two cues differ, however, in the value of return on attention they generate subsequent to elicitation of cued responsive dispositions.

In a burning building, people responding quickly to a well-functioning fire alarm escape with relatively less harm compared to those who don’t, and, who thereby suffer loss in life, limb, or physical quality of life. In a building with a malfunctioning fire alarm producing a false positive, however, people responding promptly to the misleading cue take on non-negligible unrewarded risk of loss of life, limb, or wellbeing [for instance, by getting crushed in a stampede, or, stumbling precipitously down a flight of stairs while making a mad dash to the exit, etc.]. If there really is a fire, then, fortuitously, the risk of injury or death from other hazards is less salient than the risk from fire-which is the better foreseeable, and estimable scenario of the two.

Demands on attention arising from such exogenous sources, and, the risk, return, and feasibility characteristics of the responsive dispositions they subsequently elicit are largely beyond individuals’ control. Accordingly, the practical objective of analytic self-management is to discover, or fabricate, methods for maximizing return on attention available for voluntary activities, and determining acceptable risk thresholds for various classes of voluntary activities when possible. The theoretical objective of analytic self-management is the elaboration of effective, efficient, and easy rationing, valuation, and attention allocation strategies and methods for identifying the best [highest return on attention generating] uses of endogenously available attention.

Agenda Setting
Voluntary activities range from answering nature’s call to commenting on a facebook thread. The difference between these consists in their locus of causality. You piss voluntarily but you couldn’t feasibly do otherwise because of the causal architecture of the world and the plumbing of the human digestive system. You comment on a facebook thread voluntarily but you could do otherwise. Voluntary activities akin to excretion have an external locus of causality while those akin to commenting on a thread have an internal locus of causality. Informed analytic self-management is concerned with attention allocation and budgeting for voluntary activities with an internal locus of causality.

Measuring and Valuating Endogenous Attention

Attention available for voluntary activities with an internal locus of causality is measured as time available per day divided by the activities’ duration. The duration is valued at the hourly wage rate discounted at the risk free rate.

Case Study

Abtidoon is a 21 year old marketing professional, working 40 hrs per week, and making Rs. 60,000 per month. The risk free rate is 4%. Outside of necessary rest, and domestic and leisure activities, he has only 74 hrs {with 90 hrs expensed on work and sleep, and 4 hrs on miscellaneous tasks} left over to attend to voluntary activities with an internal locus of control.

Abtidoon’s hourly wage is Rs. 375. So, his leisure time is worth at least 375 / (1.04) =   > Rs. 360. He has 74 hrs of voluntarily expendable attention, which is worth 74*360 = Rs. 26,640.  If he’s equally interested in learning how to program and in writing a novel, and realises that both tasks are time and effort intensive, though prima facie the second option is cheaper, he’ll face the decision of how to pick between them.

The method recommended here is as follows:

Compare expected completion times, and money value of completion times

Compare present value of returns on task post completion

Assume it’s expected to take 8 hrs every week over a year to become an entry level programmer, who can command Rs. 500 an hour. Also, that it’s expected to take anywhere between 3 to 6 years of writing 10 hrs every week to complete writing and publish a novel. The expected return on publishing the novel is a single lump sum of Rs. 50,000.

Using the recommended method, observe:

Expected completion times for learning to program and writing a novel respectively are 416 hrs and 1560-3120 hrs. The money values of completion time, accordingly, are 416*360 = Rs. 1,49,760 for learning to program, and Rs5,61,600-11,23,200 for becoming a novelist.

The expected return on learning to program is Rs.500*8hrs*52 weeks = Rs2,08,000. That’s 17333.33 per month. The net present value of this monthly return given it’ll be received only after a year is Rs. 10,826.34 = 17333.33 / (1.04)^12.

The expected return on writing the novel is Rs.50,000. The net present value of this return given it’ll be received only after 3-6 year is between Rs. 12183.43 to Rs. 2968.72. Let’s assume it’ll be received in 3, and the discount factor is (1.04)^36.

Comparison of the Net Present Value of Return on Activities

  1. Return / Cost of Learning Programming = 10,826.34 / 1,49,760 = 0.07229
  2. Return / Cost of Writing a Novel = 12183.43 / 5,61,600 = 0.021694

Since 1 > 2 above, Abtidoon must go with learning to program in his free time as this is the preferred activity with highest return on attention expended holding all else equal.

If Abtidoon chooses activity 2 despite being apprised of the above facts then his demand for programming expertise is relatively elastic compared to his relatively inelastic demand for becoming a novelist.

Own-Attention Elasticity of Demand for Voluntary Activities

Measures responsiveness of quantity of attention allocated to a voluntary activity with respect to changes in cost of completing it. It is calculated as the ratio of percentage change in money value of attention allocated for a given task to percentage changes in its expected completion cost-it is a cognate of the arc elasticity of demand measure familiar from microeconomics.

Own-attention demand for an activity is perfectly elastic iff any increase in its expected completion cost reduces quantity of attention allocated to zero. Own-attention demand for an activity is perfectly inelastic iff any increase/decrease leaves the quantity of attention allocated to the activity unchanged.

More on cross attention elasticity of demand in the next installment!

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What’s The Price of a Conversation?

Introduction

‘Talk is cheap’ and ‘Time is money’ are but two underappreciated platitudes repeaters of proverbs are apt to remind us of. Yet, most stop shy of the observation that repeating a proverb takes time too. Talk may be cheap, but it isn’t free. As with anything that has value, talk can be worth more or less depending on its characteristics, and circumstances. In general, talk’s value is the talkers’ opportunity costs of time spent talking. We pay for talk with an expense whose value is equal to the expected value of activities foregone to spend time talking.


Analysis

Talk has value. But, how much? And, how should it be measured?

If a one hour chat with a mathematics tutor-in the setting of a contractual agreement- is worth about $75, then a minute of the tutor’s time on facebook has an intrinsic value of $75/60minutes = $1.25.

The vignette above has the moral down right, though it could stand to be spelled out in its most general form. In what follows note that the value added by initiators and responders is computed in terms of their respective investments and returns on the talk.

The value of a talk, Vi(T), to its initiator, i, is:

Vi(t)Hourly Wage ÷ Minutes Spent Talking +/- Value Added by Responder

The value of talk, Vr(T), to its responder, r, is:

Vr(t)Hourly Wage ÷ Minutes Spent Talking +/- Value Added by Initiator

Clearly, for the initiator, if Vi(t)Hourly Wage  ÷  Minutes Spent Talking +/- Value Added by Responder is greater than the expected return on an alternative use of time spent talking, the talk is worth initiating. If it is less than that, the talk isn’t worth initiating. Paralleley, for the responder, if Vr(t)Hourly Wage  ÷ Minutes Spent Talking +/- Value Added by Initiator is greater than the expected return on an alternative use of time spent responding to the talk, the talk is worth responding to. If it is less than that, the talk isn’t worth responding to.


Further Refinements

The value added by initiators and responders differs from each parties’ perspective. The lawyer responding to a shit-post in his leisure time is paying money to the tune of the sum of his marginal cost of leisure time spent responding to the shit-post plus the value added [in this case negative] by the shit-poster’s responses. The shit-poster is paying with his opportunity cost of shit-posting, [positive or negative], equal to value lost by foregoing feasible alternative activities [positive or negative].

The lesson is perfectly general: time has value, so whatever takes time has value. Any activity that takes time costs money. The cost of any activity, then, is sum of its execution price [if applicable] and the cost of foregone alternative activities that could be handled in the time the activity takes up. If you value your time, you must strive to maximize the return on every activity so that the value of the return on time expenditure on it is positive, and greater than that expected from feasible alternative activities. As with all general lessons, particular applications call for attention to the relevant constraints, and affordances, embedded in the target situation. With suitable modifications, the principle can be extended to the valuation of any temporally distributed activity.


Synthesis

If initiating [or responding to] a talk is more valuable [in the sense discussed above] than doing something else, then initiate [respond].  If it isn’t, don’t. Shit-posting can be expensive if the time spent shit-posting could be used to reap returns greater than the return on shit-posting. Responding to shit-posts may be more or less valuable depending on the identity, characteristics, and circumstances of the shit-poster. If your love interest habitually shit-posts, and you love them more than you love receiving the time adjusted compensation of returns foregone by responding to them, then it’s valuable for you to respond to their shit-posts though this might destroy economic value.  Certainly, if the shit-post is by an unknown party it might be most valuable to leave its bid for your attention unattended.

Takeaway

Not all talk is cheap. Some ruinously expensive talks may still be worth having. Some cheap talk may yet be best avoided. Respond only to those bids for interaction which match or exceed your asking price, and initiate only those bids which match or exceed the responder’s asking price (i.e. opportunity cost of foregone alternatives to interaction).

Materiality & Relevance: Conditional Reasoning in Practice

0.

Some simple propositions have a surface resemblance to compound propositions, but not all is as it appears. Do all propositions that appear to be compound need to be made simpler?
I show in sections 2-3 they do not by showing that doing so results in logically valid but semantically unsound, and practically misleading results. The takeaway is that 28,754 / 24,950  is not about 1.076 simply because you profit by 1.076% when you sell what cost you 24,950 for 26,852 rupees, and by 11.52% when you sell it for 28,754 rupees.  Restricting or extending your logic can’t, and shouldn’t, change the way the world is and how much profit you’re entitled to at the market price at the time of sale on account of your commitment to one rule of inference or another.

1.

Classical logic permits inferences from one argument to other new arguments. It also proffers such inference rules for producing new arguments from old ones as guarantee the new ones are sound if the old ones are.

Consider a. below.

a. If you’re to my left, then I’m to your right.

Say you were standing to my left you’d see that I really was, and also see that you were standing to my left. So, intuitively, a.’s premises are true, and a. is a valid assertion. Your intuitions coincide nicely with the classical inference rule for the conditional. These state:

P Q ├ Q v ~P

Think of P as the fact that ‘you’re to my left’ and Q as the fact that ‘I’m to your right’. You’d think Q couldn’t be true if P weren’t. The classical inference rule for the conditional agrees. It formalizes that idea in its most general sense as follows:

It is true that “If you’re to my left, then I’m to your right”├ It is true that “Either I’m to your right or you’re not to my left.”

2.

Why care about conditionals?

It is fortuitous that the classical conditional discussed above is also called the material conditional. In finance, and economics, we say some fact or the other is material information when its possession can influence the behaviour and decisions of market participants, and consequently influence market outcomes.

Consider the case vignette, suggestive of many similar situations in financial practice, below.

Say you’d acquired 10 grams of gold at 24,950 rupees last year and this year the market price had risen to 28754 rupees.  You could either sell your gold for the market price today to make a whopping 3804 rupees, which is a profit of over 11.52%. Alternatively, you could hold on to the gold hoping the prices will rise further next year. May be the bigger profit you could make from higher gold prices would come in handy to pay for the wedding of your daughter’s wedding next year.

Say you thought you ought to sit tight with your gold, till the prices were higher than 11.52% in addition to your outlay of 24,950 rupees. But come next year, the price of gold has dropped to 26852 rupees. Now you could sell for 1902 rupees of underwhelming profit—a little over 1.076% profit on your initial outlay. Alternatively, you could wait for prices to rise.

Assessing which course of action is feasible for you given your goals and resources calls for conditional reasoning.

3.

When are Conditionals Material?

Classical logic tells you:

p. = If it’s true that you sell what cost you 24,950 rupees for 28,754 rupees, then, it’s true that your profit is over 11.52%

q. = If it’s true that you sell what cost you 24,950 rupees for 26,852 rupees, then, it’s true that your profit is over 1.076%.

Some logicians, like Graham Priest, add that classical inference rules allow, and necessitate, the inference from p. and q. above to r. below.

r. = If it’s true that you sell what cost you 24,950 rupees for 28,754 rupees, then, it’s true that your profit is over 1.076%, or If it’s true that you sell what cost you 24,950 rupees for 26,852 rupees, then, it’s true that your profit is over 11.52%.

r. is straightforwardly incorrect though it bears a surface resemblance to the classical inference rule of the form (A B) & (C D) ├ (A D) v (C B). The key thing to note here is that the resemblance is superficial; although p. and q. seem to contain connective occurrences [‘→’, ‘&’] they are actually simple propositions.

Note that the conditional occurring in each of p. and q. is not material in the economic sense. For it to be material in the economic sense the world would have to be otherwise than it is. Specifically, it would have to be a world where 28,754 /  24,950 is about 1.076, and where 26,852 / 24,950 is about 11.52. Our world is nothing like that. If it were you could sell in the second year and end up with a tidy profit of 3804. But, actually, you can only make that much if you sell in the first year; and, if you sell in the second year you will end up with only 1902 rupees in profit.

Why do some logicians bite this bullet, or shoot classical logicians who don’t with it? In logic p. and q. are called compound propositions. These contain simpler, or what are called atomic, propositions which don’t have connective occurrences. In p. and q. the connective ‘→’ standing for ‘If…then’ occurs, so they are compound rather than atomic. Logicians who think all compound propositions need to be made simpler, by removing connective occurrences, for analytical purposes would obviously think they ought to do so because p. and q. are compound.

But what analytical purpose is served by splitting p. and q. into atomic propositions? I dare say none. For all intents and purposes for p. to be true just is for both the implicit antecedent and the implicit consequent to be true. If it were not then you would among other things profit over 11.52% even if you did not sell what cost you 24,950 rupees for 28,754 rupees because P Q ├ Q v ~P. Similarly, if q. were divided up artificially into its antecedent and consequent, then, it would appear you’d profit by over 1.076% even if you didn’t sell what cost you 24,950 rupees for 26,852 rupees. Would that it were so!

4.

CONCLUSION

Some simple propositions have a surface resemblance to compound propositions, but not all is as it appears. Logic only promises the deduction of truth from truth; to ensure you end up with true conditional inferences simpliciter, you need to begin reasoning with true antecedents and true consequents. As we saw 28,754 / 24,950  is not about 1.076 simply because you profit by 1.076% when you sell what cost you 24,950 for 26,852 rupees, and by 11.52% when you sell it for 28,754 rupees. To sell what cost you 24,950 rupees for 28,754 just is what it is for your profit to be over 11.52%. Likewise, to sell what cost you 24,950 rupees for 26,852 rupees just is what it is for your profit to be over 1.076%.  Changing your logic can’t, and shouldn’t change the way the world is and how much profit you’re entitled to on account of the market price on the day of your sale.

REFERENCES

Goodman, V., & Stampfli, J. (2009). The Mathematics of Finance: Modeling and Hedging. Providence, RI: American Mathematical Society.

Priest, G. (2013). An Introduction to Non-Classical Logic: From If to Is. New York, NY: Cambridge University Press.

Logics of Self-Hood

0.  INTRODUCTION

Why is it that you are disappointed when you order fish and chips but receive a coke-float instead? The cause of your disappointment is the receipt of the wrong item but the reason you’re disappointed is that you thought: “If I order fish and chips then I must receive fish and chips.” Why might you have such a queer idea? Do you also think if you mustn’t receive fish and chips then you do not order fish and chips? You might. If you didn’t, then you wouldn’t think you must receive fish and chips if you order fish and chips; and, so, you would have neither cause nor reason to be disappointed. Thus, prima facie, it appears you have very elaborate ideas about what else must happen if anything does at all. You have an implicit belief architecture comprised of implicit axiom and rule schemes, whose instances, jointly or singly, determine what you think you say and do when you think you say and do anything. You are a logic. But, which logic? It depends on who you are, and what it is to be yourself.

1. WHAT IS THE SELF

The self is a house of many mansions. In biology, the self is the locus of individuating differentia: anatomic borders, harmonious communication between organs, hierarchical systems of dominance and control, and division of labour between parts that distinguish it from other organisms. In immunological terms, it is the privileged recipient of protection and beneficiary of concinnity. In an economic idiom, the self is an agent willing [and able] to supply [demand] a specified amount of some good in exchange for a specified amount of another. Self-determination theory identifies the self as an endowment of autonomy, competence, and relatedness in service of a cluster of goals, which may be externally or internally motivated. In daily life, the self is a token of account for commitments and entitlements attributed to it by others.

2. LOGICS OF SELF-HOOD

Although these disparate accounts of what it is to be a self are replete with uniquely salient details, their compossible elements cohere to furnish a rich picture of self-hood. The biological notion of self-hood is minimally relevant to self-definition, and so to the present inquiry. However, the immunological, economic, psychological, and practical notions of self-hood are of obvious interest. Specifically, the expanded vocabulary and theoretical constraints and affordances afforded by the combination of the compatible aspects of immunological, economic, psychological, and practical notions of self-hood furnish a more nuanced way to conceive of selves than provided by blind intuition, or any single disciplinary orientation.

Loosely speaking, an expanded vocabulary of the sort alluded to can be fleshed out by bringing the keystone concepts of each disciplinary account into interaction with the others. Social selves attribute to each other deontic statuses which settle or change matters of fact about who is entitled or committed to say what about themselves, others, or anything. Economic selves transact with each other to maximize the receipt of goods they value most in exchange for goods they values less. Psychological selves manage their endogenous budget of constraints and affordances to detect, appraise, pursue, or abandon autonomy, competence, and relatedness maximising goals-by selecting and executing tasks associated with goal attainment or abandonment. Thus, we may use the notion of immune selves as an organising principle around which assessments of the performances of other self-conceptions are normed.

Immune selves in such an expanded sense might be thought to protect economic selves from hazardous transactions and mismanagement, while promoting informed market participation. Analogously, they might be thought to protect social selves from the immediate and distal consequences of others’ negative deontic status attributions. The performances of immune selves, then, would be subject to assessment with respect to their influence on selves’ economic and social well-being. Obviously, not just immune selves but any other coherent self-conceptions are apt for duty as organising principles, or fixed points, for further theorising, and promise differentially individuated patterns of insight into the hydra-headed phenomenon that is self-hood.

3. PLAN OF ACTION

In subsequent posts this blog will bring psychology, economics, management science, and classical and non-classical logic, into conversation on matters salient to informed self-management. The purpose of these inquiries is to understand functional aspects of self-hood as manifested in agenda setting, goal identification, and, task selection and execution behaviours with a view to managing oneself better.

REFERENCES

Brandom, R. (1994). Making it Explicit: Reasoning, Representing, and Discursive Commitment. New York, NY: Harvard University Press.

Gagné, M. (2014). The Oxford Handbook of Work Engagement, Motivation, and Self-Determination Theory. New York, NY: Oxford University Press.

Gabbay, Dov, M., Gerbrandy, J., & Mineur, A-M. (1994). I am a Logic. < http://icr.uni.lu/Gabbay_interview.html >.

Mankiw, G. (2011). Principles of Economics. Thousand Oaks, CA: Cengage Learning.

Tauber, A. (2015). The Biological Notion of Self and Non-self. Updated: May 21, 2015. Retrieved: November 27, 2016. <http://plato.stanford.edu/entries/biology-self/ >.

From Is to If: Reasoning with Material Conditionals

0.

Some Worries about the Material Conditional Are Immaterial

In his monumental expository wrkAn Introduction to Non-Classical Logic: From If to Is, Graham Priest (2013) prefaces the discussion of the classical material, indicative, conditional with the worry that it doesn’t belong in natural language reasoning. To drive home the urgency of the problem, and motivate a revisionist solution, he offers classical inferences in natural language as heuristic falsifiers for the rules of classical connectives; particularly, the conditional. These are intended to demonstrate that classical inferences in natural language produce results that are either logically invalid, factually incorrect, or at variance with common sense. But, as will become clear shortly, these falsifiers work only if classical inferences in natural languages are restricted so that only elementary propositions can do duty as premises in any arguments. In this essay I show one could do otherwise without producing invalidities-whether they be logical, factual, or intuitive. Compound variants of Priest’s atomic premises {SEE Heuristic Falsifiers: 1.1; 2.1; 3.1.} are used to deduce classically valid, and intuitively correct results {SEE Heuristics Falsified: 1.2; 2.2; 3.2}. I conclude with the suggestion that worries about the material conditional mentioned are immaterial.

1.

(A → B) & (C → D) ├ (A → D) v (C → B)

Priest offers the following argument to undermine the applicability of the above inference to natural language reasoning.

Heuristic Falsifier 1.1: “If John is in Paris he in France, and if John is in London then he is in England. Hence, it is the case that if John is in Paris he is in England, or that if he is in London he is in France” (2013, 1.7, p. 12).

This argument is set up so that:

John is in Paris = A;
John is in France = B;
John is in London = C;
John is in England = D.

But, of course [1], we can set it up another way:
A’ = If John is in Paris he in France;
B’ = If John isn’t in Paris he isn’t France
C’ = If John is in London then he is in England.
D’ = If John isn’t in London then he isn’t in England.

Voila! The material conditional gives the correct result. Namely, (A’ → B’) & (C’ → D’) ├ (A’ → D’) v (C’ → B’) produces the valid and intuitive result:

Heuristic Falsified 1.2: John is in Paris when in France but not otherwise, and he is in London when in England but not otherwise. Hence, either he is in Paris when in France  and only then, or he is in London when in England and then only.

2.

(A & B) → C ├ (A → C) v (B → C)

Heuristic Falsifier 2.1: “If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on. [Imagine an electrical circuit where switches x and y are in series, so that both are required for the light to go on, and both switches are open]” (2013, 1.9: p. 14).

Again, it appears, a simple tweak in the propositions being reasoned about defangs the bad inference in 2.1.  Consider:

A” = You close both switches x and y.
B” = You close one of switches x and y.
C” = The light will go on if both switches are closed.

Heuristic Falsified 2.2: If you close switches x and y, and also close one of switches x and y, then, the light will go on if both switches are closed. Hence, either the light will go on if both switches are closed when you close switches x and y, or one of x and y, or the light will go on if both switches are closed.

Now, clearly, (A’’ & B’’) → C’’ ├ (A’’ → C’’) v (B’’ → C’’) gives the correct answer! So, we needn’t be in any hurry to be rid of inferences of the form (A & B) → C ├ (A → C) v (B → C).

3.

¬(A → B) ├ A.

Heuristic Falsifier 3.1: “It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god” (2013, 1.9.1p. 15).

This argument is set up so that:

A = There is a good God
B= The prayers of evil people are answered.

But nothing prevents us from formulating the premises like this:

A”’ = If there is a good God the prayers of evil people are not answered.
B”’ = The prayers of evil people are answered.

Heuristic Falsified 3.2: It isn’t that the prayers of evil people are answered when the prayers of evil people are not answered if there is a good God. Hence, if there is a good God the prayers of evil people are not answered.

Clearly, ¬(A’’’ → B’’’) ├ A’’’ is the expected answer.

4.

‘If A then B’ is true iff ‘A ⊃ B’ is true.’

Heuristic Falsifier 4.1: “First, suppose that ‘If A then B’ is true. Either ¬A is true or A is. In this first case, ¬A ∨ B is true. In the second case, B is true by modus ponens. Hence, again, ¬A ∨ B is true. Thus, in either case, ¬A ∨ B is true.”

Now, consider the flowing reformulation of the premises.

A* = ‘If A then B’ is true iff ‘A ⊃ B’ is true.

B* = ‘If A then B’ is false iff ‘A ⊃ B’ is false.

Heuristic Falsified 4.2: Suppose it’s true that if “If A then B’ is true iff ‘A ⊃ B’ is true” then If A then B’ is false iff ‘A ⊃ B’ is false“. Then, either it’s not true “If A then B’ is true iff ‘A ⊃ B’ is true” or it’s true  that “‘If A then B’ is false iff ‘A ⊃ B’ is false” or that “‘If A then B’ is false iff ‘A ⊃ B’ is false“. In the second case, it follows by modus ponens that “If A then B is false iff ‘A ⊃ B’ is false.” So, in all cases, (¬A* ∨ B*) is true.

There’s no danger in retaining (¬A* ∨ B*) ├  (A*  → B*) as a rule of reasoning, clearly.

5.

CONCLUSION

Examples discussed, and falsified, show inferences using the mentioned classical connectives, and classical inference forms, falsify our heuristics. Such heuristic falsifiers really don’t go any way towards showing the problem to lie with the underlying logic, rather than with the complexities of premise formulation and appraisal. Abandoning well-behaved logical connectives, classical or otherwise, isn’t a good fix for filling the gap between logic and intuition. The deduction of true conclusions from non-exhaustively descriptive premises, or premises that need and admit of sensible alternative reformulations, are certainly interesting topics in their own right. But, to tweak logical connectives specifically for failing to deliver deep insight on those issues is to treat the wrong disease.

REFERENCE

Priest, G. (2013). An Introduction to Non-Classical Logic: From If to Is. New York, NY: Cambridge University Press.

Notes:

[1] Provided we are not committed to the idea that only atomic and not compound sentences may be used as premises in an argument.