Quantification without Quantifiers (or Epsilons): A New Scholardarity Article

I believe I've  found a way to say everything that one would normally say in predicate logic using quantifiers, but without actually using any, and also without using Hilbert’s epsilon operator. (See "Hilbert's Epsilon Calculus", here.) To begin with, instead of saying that something holds for all x by binding x with a universal quantifier, one can say it by capitalizing all of x’s occurrences in a sentence that would, in “standard notation”, be in the scope of the quantifier. And instead of saying that something holds for some y by binding y with an existential quantifier, one can say it by leaving all of y’s occurrences un-capitalized. Thus, in “variable notation”—so called because it expresses quantification by modifying variables—instead of ∀x (Fx) one would write FX, and instead of ∃y (Gy) one would write Gy.

Immediately a problem presents itself. In standard notation the order of the quantifiers in a sentence can make a big difference to its meaning. How can one make up for this loss if we discard quantifiers? My solution is to use scope brackets, ‘[’ and ‘]’. For every distinct variable in a sentence there is a pair of scope brackets that determine the scope of the claim involving that variable. Where, in standard notation, a quantifier would be within the scope of another quantifier, in variable notation the scope brackets of the former quantified claim are inside the scope brackets of the latter quantified claim. The brackets should enclose no more than they need to.

Some examples will make this clear. Consider (1) and (2), and their counterparts (1’) and (2’):

1. ∀x ∃y (Rxy)                        1’. [RX[y]]

2. ∃y ∀x (Rxy)                        2’. [R[X]y]

In (1’) the scope brackets for y enclose only y, and the scope brackets for X enclose the entire sentence, indicating that the existential claim is governed by the universal claim. In (2’) the reverse is true, indicating that the universal claim is governed by the existential claim.

If you want to see this idea worked out in more detail, please see the full article at Scholardarity, Quantification without Quantifiers (or Epsilons): A New Notation for Quantification in Predicate Logic.

A new article on Singer’s “Famine, Affluence and Morality" at Scholardarity.com

I've published a new article at Scholardarity.com, Singer’s “Famine, Affluence and Morality”: Exposition and Appraisal.

Quote of the Day: The Real Source of Moral Motivation

Nobody sane generally or standardly acts so as to realise utility, or “on the motive of Duty” (Kant’s own phrase), or “for the sake of virtue itself” (Aristotle’s own phrase). Even people who only act occasionally in these ways can be in danger of looking priggish, and out of touch with what really matters in life, to the rest of us.

            As I hinted in the last section, what really motivates most of us, most of the time, at least if we are moderately good people or better and are not being distracted by false motives like concern about “what others will think”, is love: love for spouses, love for children and parents, love for friends, love for God, love for ideals, love for valued places or artworks or possessions, love for pet projects, love for pets. The centrality of love is a striking feature of any typical credible ethical outlook. The marginality of love in typical moral theories, and more broadly in contemporary philosophical research, is equally striking. Love is at the heart of our ethical outlooks; it is love, and not concern with what is right and wrong, that mostly drives us into action. In that sense love puts us “beyond good and evil”, and beyond morality; while morality is a constraint on what motivates us (and a constraint: there are others), love is the very engine of motivation. 

[footnote omitted]--Timothy Chappell “Ethics beyond moral theory” Philosophical Investigations 32 (3):206-243.

Natural Patterns: The Nonad — A New Book For Sale through Scholardarity.com

In a new book for sale through Scholardarity.com, John Krey, a chemistry instructor with many years of experience, presents a discovery of patterns in nature as well as patterns in the organization of knowledge, which are helpful for the educational purposes. You can check out his book here: Natural Patterns: The Nonad.

A Primer on Logic: Interlude -- A New Scholardarity.com Article

I've published a new article on Scholardarity, A Primer on Logic: Interlude, the latest installment of my introduction to formal logic. In it I explain some of the inadequacies of Aristotelian logic.

Also, in case you missed Parts 1, 2 and 3, which respectively cover logical preliminaries, propositional logic, and Aristotelian logic, you can check them out here:

Part 1


Part 2
 

Part 3

On Searle's Chinese Room Argument -- A New Scholardarity.com Article

I've posted a new article in Scholardarity.com, On Searle's Chinese Room Argument: A Defense of the Systems Reply.

Scholardarity.com is now publishing papers for free!

Scholardarity.com, a new e-publishing website for scholars, is now publishing papers for free!


Peter Krey and I, as co-founders of Scholardarity.com, are happy to announce that we are now accepting submissions. Subscribe now and publish your work free of charge. You can choose whether to sell your work or make it available for free. This promotional offer lasts only until June 2nd. We invite you to subscribe to Scholardarity, and put your writing to work for you!

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