- Instead of showing just one counterexample, clicking the "Counterexample" button more than once causes the application to cycle through different counterexamples for the current statement.
- There is a new button: the "Example" button, which is the opposite of the "Counterexample" button.
- Propositions (nullary predicates) can now be lower-case letters as well as upper-case letters.
- There is now a "therefore" operator -- ',' -- apostrophe comma apostrophe, for representing logical arguments.

## Saturday, March 31, 2018

### Improvements to somerby.net/mack/logic

After a spate of good input from commenters, I've improved somerby.net/mack/logic in several ways:

## Sunday, October 15, 2017

### The Video Shooter Fuzz: a Fuzz Pedal Based on the S9014 Transistor

I built a fuzz pedal for my guitar using parts from an old, off-brand Nintendo accessory and a broken Fender Frontman amplifier. It's based on the Bazz Fuss, which is a simple fuzz circuit that is easy to modify. I wanted to use parts from the Nintendo accessory as much as possible, so I designed the circuit around the accessory's enclosure and the S9014 transistors I found inside. One S9014 did not put out enough gain to work as a guitar effect, so, after reading this forum thread, I joined two transistors into what is called a "Darlington transistor". The happy result was sufficient gain and a very well-behaved fuzz pedal. Unlike some other Bazz Fuss derivatives I have encountered, it doesn't fizzle or sputter when connected to a guitar with high-impedance pickups and or when I roll off the volume on the guitar. Its gain is moderate. I call it the "Video Shooter Fuzz" in honor of the NES accessory.

Video Shooter Fuzz Schematic |

This is the fuzz circuit by itself, minus the 50k Ohm pot:

Hand-Soldered, Point-to-Point Wiring! |

And here are some sound clips: (1), (2). Interestingly, when I was first wiring a Bazz Fuss circuit, I connected the collector and the pot to the negative end of a battery but not to ground and the resulting circuit had no distortion, instead behaving like a buffer.

Labels:
Bazz Fuss,
effects pedal,
S9014,
schematics

## Monday, September 04, 2017

### Alternative Hexadecimal Digits: Published

Valdis and I wrote up our final version of the hexadecimal digits in the form of a proposal and got it published in the IJCSET. See here. The proposal includes descriptions and assessments of various other proposed sets of hexadecimal digits. Unfortunately, we missed a good set called Birkana that are rune-like symbols.

In case anyone wants to see our digits used for practical purposes, I made a JavaScript digital clock that uses our digits to display hours, minutes and seconds as hexadecimal numbers.

In case anyone wants to see our digits used for practical purposes, I made a JavaScript digital clock that uses our digits to display hours, minutes and seconds as hexadecimal numbers.

Labels:
hexadecimal numbers

## Sunday, April 17, 2016

### A Cancel Button for somerby.net/mack/logic

Good news: somerby.net/mack/logic now has a cancel button. If a decision is taking too long, you can cancel it and continue working. You don't have to close the browser anymore.

Bad news: somerby.net/mack/logic is not working in Internet Explorer, or at least it isn't working in Internet Explorer 11 on my computer. I'll try to fix it soon.

It works for me in Chrome and Firefox. I'd appreciate it if anyone tells me whether or not its working for them in other browsers. Just leave a comment on this post.

Bad news: somerby.net/mack/logic is not working in Internet Explorer, or at least it isn't working in Internet Explorer 11 on my computer. I'll try to fix it soon.

It works for me in Chrome and Firefox. I'd appreciate it if anyone tells me whether or not its working for them in other browsers. Just leave a comment on this post.

## Monday, February 15, 2016

### A New Web App: "Expanding Quine's Definitions"

I've been messing around with code that creates symbolic definitions of the number one for quite a while: these blog posts show some of the results. All this messing around has culminated in a web app:

http://somerby.net/mack/definitions/. With it, you can expand Quine's definition of the number one interactively in various ways, and also many other of his definitions. What fun!

http://somerby.net/mack/definitions/. With it, you can expand Quine's definition of the number one interactively in various ways, and also many other of his definitions. What fun!

Labels:
foundations of mathematics,
software

## Tuesday, December 29, 2015

### The Number One, Part Three

In an earlier post, "What it Means to be Number One", I presented Quine's definition of the number one, which is defined in terms of four basic mathematical constructs: class membership (⌜α ϵ β⌝), universal generalization (⌜(α)ϕ⌝), joint denial (⌜(ϕ ↓ ψ)⌝), and class abstraction (⌜α^ϕ⌝). It is six pages of dense text. In another post, "The Number One, Part Two", I claimed that the length of the definition is "the result of giving the number one a precise, complete, logical
definition without resorting to using any numbers except zero". This is not quite true. I've since realized that the size comes not from precision, completeness nor logicality, but from Quine's definition of negation, which is

⌜∼ϕ⌝

This is definition is important because it enables his system to define all truth-functional connectives in terms of just one truth-functional connectives, thereby minimizing the number of basic constructs in his system. But it does have the effect of making expanded definitions quite large. You see, any time a negation is expanded, the expression which is negated (ϕ) is duplicated in the resulting expansion. If that expression also contains negated expressions, then those expressions are quadrupled, and if those negated expressions contain negated expressions, they are octupled, and so on. The result is an exponential relationship between the size of an expanded definition and the number of layers of negation in the definition. Those of you who know computer science know that exponential relationships mean huge outputs for all but the smallest inputs. Hence the 6-page definition of one. If we expand all constructs contained in Quine's definition of one

which is not shockingly complicated. Even if we expand statements of membership in and of class abstractions in this definition (which is something I did not do in "What it means to be..."), the definition of one is still just

which is more than anyone would want to try to write or memory, but still not embarrassingly long. If we don't expand any of the truth-functional connectives, nor existential quantification, the result is something that it almost readable by a person who is familiar with symbolic logic:

⌜∼ϕ⌝

*for*⌜(ϕ ↓ ϕ)⌝This is definition is important because it enables his system to define all truth-functional connectives in terms of just one truth-functional connectives, thereby minimizing the number of basic constructs in his system. But it does have the effect of making expanded definitions quite large. You see, any time a negation is expanded, the expression which is negated (ϕ) is duplicated in the resulting expansion. If that expression also contains negated expressions, then those expressions are quadrupled, and if those negated expressions contain negated expressions, they are octupled, and so on. The result is an exponential relationship between the size of an expanded definition and the number of layers of negation in the definition. Those of you who know computer science know that exponential relationships mean huge outputs for all but the smallest inputs. Hence the 6-page definition of one. If we expand all constructs contained in Quine's definition of one

*except*negation, the result is not so big. It is*x*^∼(*y*)∼(∼(*y*ϵ*x*) ↓ ∼(α^(∼(α ϵ*x*) ↓ ∼(α ϵ α′^(∼(α′ ϵ α′′^((α′′′)(∼∼(∼(α′′′ ϵ α′′) ↓_{▪}α′′′ ϵ*y*) ↓ ∼∼(∼(α′′′ ϵ*y*) ↓_{▪}α′′′ ϵ α′′))))))) ϵ α^((α′)(∼∼(∼(α′ ϵ α) ↓_{▪}α′ ϵ*x*′^(∼((α′′)(∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′) ↓ ∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′))))) ↓ ∼∼(∼(α′ ϵ*x*′^(∼((α′′)(∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′) ↓ ∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′))))) ↓_{▪}α′ ϵ α)))))which is not shockingly complicated. Even if we expand statements of membership in and of class abstractions in this definition (which is something I did not do in "What it means to be..."), the definition of one is still just

*x*^∼(*y*)∼(∼(*y*ϵ*x*) ↓ ∼(∼(γ)∼(∼(∼(β)∼(∼((α)(∼∼(∼(α ϵ β) ↓_{▪}∼(γ′)∼(∼(α ϵ γ′) ↓ ∼(α′)∼(∼(α′ ϵ γ′) ↓ (∼(α′ ϵ*x*) ↓ ∼(∼(γ′′)∼(∼(α′ ϵ γ′′) ↓ ∼(α′′)∼(∼(α′′ ϵ γ′′) ↓_{▪}∼(∼(γ′′′)∼(∼(α′′ ϵ γ′′′) ↓ ∼(α′′′)∼(∼(α′′′ ϵ γ′′′) ↓_{▪}(α′′′′)(∼∼(∼(α′′′′ ϵ α′′′) ↓_{▪}α′′′′ ϵ*y*) ↓ ∼∼(∼(α′′′′ ϵ*y*) ↓_{▪}α′′′′ ϵ α′′′)))))))))))) ↓ ∼∼(∼(∼(γ′)∼(∼(α ϵ γ′) ↓ ∼(α′)∼(∼(α′ ϵ γ′) ↓ (∼(α′ ϵ*x*) ↓ ∼(∼(γ′′)∼(∼(α′ ϵ γ′′) ↓ ∼(α′′)∼(∼(α′′ ϵ γ′′) ↓_{▪}∼(∼(γ′′′)∼(∼(α′′ ϵ γ′′′) ↓ ∼(α′′′)∼(∼(α′′′ ϵ γ′′′) ↓_{▪}(α′′′′)(∼∼(∼(α′′′′ ϵ α′′′) ↓_{▪}α′′′′ ϵ*y*) ↓ ∼∼(∼(α′′′′ ϵ*y*) ↓_{▪}α′′′′ ϵ α′′′)))))))))))) ↓_{▪}α ϵ β))) ↓ ∼(β ϵ γ))) ↓ ∼(α)∼(∼(α ϵ γ) ↓_{▪}(α′)(∼∼(∼(α′ ϵ α) ↓_{▪}∼(γ′)∼(∼(α′ ϵ γ′) ↓ ∼(*x*′)∼(∼(*x*′ ϵ γ′) ↓_{▪}∼((α′′)(∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′) ↓ ∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′)))))) ↓ ∼∼(∼(∼(γ′)∼(∼(α′ ϵ γ′) ↓ ∼(*x*′)∼(∼(*x*′ ϵ γ′) ↓_{▪}∼((α′′)(∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′) ↓ ∼∼(∼(α′′ ϵ*x*′) ↓_{▪}α′′ ϵ*x*′)))))) ↓_{▪}α′ ϵ α))))))which is more than anyone would want to try to write or memory, but still not embarrassingly long. If we don't expand any of the truth-functional connectives, nor existential quantification, the result is something that it almost readable by a person who is familiar with symbolic logic:

*x*^(∃*y*)(*y*ϵ*x*_{▪}(∃γ)((∃β)((α)(α ϵ β_{▪}≡_{▪}(∃γ′)(α ϵ γ′_{▪}(α′)(α′ ϵ γ′_{▪}⊃ (α′ ϵ*x*_{▪}(∃γ′′)(α′ ϵ γ′′_{▪}(α′′)(α′′ ϵ γ′′_{▪}⊃_{▪}∼((∃γ′′′)(α′′ ϵ γ′′′_{▪}(α′′′)(α′′′ ϵ γ′′′_{▪}⊃_{▪}(α′′′′)(α′′′′ ϵ α′′′_{▪}≡_{▪}α′′′′ ϵ*y*))))))))))_{▪}β ϵ γ)_{▪}(α)(α ϵ γ_{▪}⊃_{▪}(α′)(α′ ϵ α_{▪}≡_{▪}(∃γ′)(α′ ϵ γ′_{▪}(*x*′)(*x*′ ϵ γ′_{▪}⊃_{▪}∼((α′′)(α′′ ϵ*x*′_{▪}≡_{▪}α′′ ϵ*x*′))))))))
Labels:
foundations of mathematics,
the number one

## Saturday, July 11, 2015

### Binary Operators in somerby.net/mack/logic

Binary Operators in somerby.net/mack/logic now have different precedences. See here.

Labels:
UPL

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