Some Strong Conditionals for Sentential Logics (Circulation Draft)

Here is my latest draft of a paper attempting to give an account of a stronger-than-material conditional which can be adapted to various sentential logics. An abstract is provided below. To see the draft, go here:

Some Strong Conditionals for Sentential Logics (Circulation Draft)

Please keep in mind that the paper is a work in progress, and is still in a fairly rough state. That being said, I appreciate any comments and / or criticism that those who are interested in the subject have to offer.

Abstract:

In this article I define a strong conditional for classical sentential logic, and then extend it to three non-classical sentential logics. It is stronger than the material conditional and is not subject to the standard paradoxes of material implication, nor is it subject to some of the standard paradoxes of C. I. Lewis’s strict implication. My conditional has some counterintuitive consequences of its own, but I think its pros outweigh its cons. In any case, one can always augment one’s language with more than one conditional, and it may be that no single conditional will satisfy all of our intuitions about how a conditional should behave. Finally, I will make no claim that the strong conditional is a good model for any particular use of the indicative conditional in English or other natural languages, though it would certainly be a nice bonus if some modified version of it could serve as one.

The basic idea is this: In general, one starts out with with the logic that one  wants to define the conditional for, and describes a meta-language for it. The metalanguage contains ┌(q|p)┐, a “conditional designator” which designates the truth value that q takes given that p is true, i.e., given that p has the value 1. It is to be read as ┌ the value of q given p┐ or ┌the value of q conditional on p┐. The stroke, |, is not a connective; it merely serves to separate the letter q from the letter p. The designator works like this: If p never takes the value 1, then ┌(q|p)┐ designates nothing—for q cannot take a value given that p is true if p can never be true—and is said to be empty. It is also empty if the value of q varies when the value of p is 1, for in that case q doesn’t take a unique value given that p is true. If q always takes the value 1 when p takes the value 1, then ┌(q|p)┐ designates 1, and in our meta-language we can say that ┌(q|p)┐ = 1, which is another way of saying that ┌(q|p)┐ designates 1. Similarly, if q always takes the value 0 when p takes the value 1, then ┌(q|p)┐ designates 0, and in our meta-language we can say that ┌(q|p)┐ = 0. 

With our meta-linguistic conditional designator ready to hand, one can now define what I call the strong conditional, or strong implication, for which I will use the symbol ‘→’. Its definition is (where ‘v( )’ is the valuation function, which gives the semantic value of an expression):

If ┌(q|p)┐= 1, then v(p → q) = 1

If ┌(q|p)┐= 0, then v(p → q) = 0

If ┌(q|p)┐ is empty, then v(p → q) = 0

I shall begin by exploring some of the disadvantages of the material conditional, the strict conditional, and some relevant conditionals. I proceed to define a strong conditional for classical sentential logic. I go on to adapt this account to Graham Priest’s Logic of Paradox, to S. C. Kleene’s logic K3, and then to J. Łukasiewicz’s logic Ł, a standard version of fuzzy logic.

Notes on Timothy Williamson’s Lecture “Logics as Scientific Theories”

Over at Scholardarity I've posted my notes on Timothy Williamson’s lecture “Logics as Scientific Theories” which was given at U.C. Berkeley on November 1st.

A Primer on Logic: Part 4.1

I've published a new article on Scholardarity, A Primer on Logic: Part 4.1, the latest installment of my introduction to formal logic. In it I address some preliminaries to Predicate Logic.

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

Part 1

Part 2
 
Part 3

Interlude

If you have any comments / criticism, by all means share it!

Quantification without Quantifiers (or Epsilons)

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 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

A Primer on Logic Part 3 (A New Scholardarity.com Article)

I've posted a new article, A Primer on Logic: Part 3, my new Scholardarity piece in which I give a brief introduction to Aristotelian logic. It's the latest entry in my introduction to formal logic.

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

Part 1


Part 2


If you have any comments / criticism, by all means share it!

A Primer on Logic: Part 2 (A New Scholardarity.com Article)

Check out "A Primer on Logic: Part 2", my new Scholardarity.com article which is essentially a crash-course in propositional logic. It's the latest entry in my introduction to formal logic. (Also available as a PDF.)

Also, in case you missed Part 1, which covers logical preliminaries and vocabulary, you can check it out here. (Also as a PDF.)

If anyone has any comments / criticism, by all means share it!

A Dilemma for Dialetheism

I've just published a revised version of my article "A Dilemma for Dialetheism" on Scholardarity.com, which was originally published in the Spring 2010 edition of the Stanford undergraduate philosophy journal The Dualist (vol. 15). In the article I argue that dialetheists, who believe that some sentences are both true and false, either cannot express the notion that some sentences are not both true and false, or else that their accounts suffer from "revenge" liar paradoxes that not even they can regard as being both true and false. If you like logic and paradoxes as much as I do, please check it out and let me know what you think.

Another new variant of the liar paradox?

I've thought of another variant of the liar to add to my collection--I'll leave it to my readers to tell me if someone has already thought of it.

Consider the following statement:

(*): Nothing entails that (*) is true.

Suppose (*) is false. In that case, it is false that nothing entails that (*) is true. So something entails that (*) is true. But if something entails that (*) is true, then (*) is true. But then what (*) says must be the case, and hence it follows that nothing entails that (*) is true. So if (*) is false, it is true both that something entails that (*) is true and that nothing entails that (*) is true, which is a contradiction. (*) must, in consequence, be true. So it is true that noting entails that (*) is true. (*), however, is not only true, it is necessarily true, for its falsity would entail a contradiction. However, if (*) is necessarily true, its truth is entailed by every statement whatever. So if (*) is true, it is true both that nothing entails that (*) is true and that everything entails that (*) is true. This too is a contradiction. So no matter whether (*) is true or false, it must be both true and false.