Help a poor soul with his truth tables
December 22, 2008 7:40 AM Subscribe
I'm confused about Wittgenstein's Tractatus and the truth table for the Sheffer stroke.
It's been a minute since I read the Tractatus and I'm trying to get back in with Mounce's introduction. Mounce claims that Wittgenstein uses only one operator, the Sheffer stroke. For him the Sheffer stroke p|q means "neither p nor q". As I understand it, and Wikipedia agrees with me, the Sheffer stroke actually means "not both" and its truth table is:
T T F
T F T
F T T
F F T
Mounce seems to disagree with this. For him Wittgenstein's notation (----- T), which for two propositions p and q would mean (FFFT), "takes us...to the Sheffer stroke -- neither p nor q or -p&-q. Thus:
T T F
F T F
T F F
F F T
I understand how this is the truth table for -p&-q. However, according to Wikipedia, this is the truth table for the Peirce arrow, not the Sheffer stroke. (I understand that you could express the same thing using Sheffer strokes.) But in 5.101 Wittgenstein defines p|q as -p&-q or as (FFFT)(p,q). Does the sign | not mean the Sheffer stroke? Is Wittgenstein himself confusing the Sheffer stroke with the other thing? Or are Wikipedia and my own recollection wrong?
I am not a logician. I am interested in Wittgenstein's views on ethics. All I want to know is: what the fuck operator does he use in the Tractatus, and what is its truth table? I turned to Mounce for clarity and it's just making me winded and dull.
It's been a minute since I read the Tractatus and I'm trying to get back in with Mounce's introduction. Mounce claims that Wittgenstein uses only one operator, the Sheffer stroke. For him the Sheffer stroke p|q means "neither p nor q". As I understand it, and Wikipedia agrees with me, the Sheffer stroke actually means "not both" and its truth table is:
T T F
T F T
F T T
F F T
Mounce seems to disagree with this. For him Wittgenstein's notation (----- T), which for two propositions p and q would mean (FFFT), "takes us...to the Sheffer stroke -- neither p nor q or -p&-q. Thus:
T T F
F T F
T F F
F F T
I understand how this is the truth table for -p&-q. However, according to Wikipedia, this is the truth table for the Peirce arrow, not the Sheffer stroke. (I understand that you could express the same thing using Sheffer strokes.) But in 5.101 Wittgenstein defines p|q as -p&-q or as (FFFT)(p,q). Does the sign | not mean the Sheffer stroke? Is Wittgenstein himself confusing the Sheffer stroke with the other thing? Or are Wikipedia and my own recollection wrong?
I am not a logician. I am interested in Wittgenstein's views on ethics. All I want to know is: what the fuck operator does he use in the Tractatus, and what is its truth table? I turned to Mounce for clarity and it's just making me winded and dull.
Yeah, what's going on with that Wikipedia article? The Sheffer stroke does not represent NAND at all.
The reason it's called the Sheffer stroke is because Henry Sheffer showed that all other truth-functional operators from it. (Although this is also true of NAND, I think). Apparently, he showed this to Bertrand Russell, who was thrilled, thought that Sheffer was some sort of logical genius, and got him a tenured position at Harvard. In retrospect, this finding is utterly, utterly trivial, and Sheffer never did anything important again.
The truth table of the operator that Wittgenstein uses is obviously FFFT. He says this straight-out. He is clear that he uses the downward stroke to symbolize this. And Wittgenstein doesn't use the term 'Sheffer stroke' himself (it postdates the Tractatus). So there's definitely no confusion on Wittgenstein's part.
The symbol '|' might be used in some branches of computer science to represent the NAND operator, but it's more commonly used to represent NOR. It's only the Sheffer stroke if it's representing the NOR operator -- otherwise, it's just a downward stroke.
posted by painquale at 9:06 AM on December 22, 2008 [1 favorite]
The reason it's called the Sheffer stroke is because Henry Sheffer showed that all other truth-functional operators from it. (Although this is also true of NAND, I think). Apparently, he showed this to Bertrand Russell, who was thrilled, thought that Sheffer was some sort of logical genius, and got him a tenured position at Harvard. In retrospect, this finding is utterly, utterly trivial, and Sheffer never did anything important again.
The truth table of the operator that Wittgenstein uses is obviously FFFT. He says this straight-out. He is clear that he uses the downward stroke to symbolize this. And Wittgenstein doesn't use the term 'Sheffer stroke' himself (it postdates the Tractatus). So there's definitely no confusion on Wittgenstein's part.
The symbol '|' might be used in some branches of computer science to represent the NAND operator, but it's more commonly used to represent NOR. It's only the Sheffer stroke if it's representing the NOR operator -- otherwise, it's just a downward stroke.
posted by painquale at 9:06 AM on December 22, 2008 [1 favorite]
I believe Sheffers stroke in his 1913 paper was originally used by Sheffer as a NOR, but common usage (at least on the tubes and nearly every online directionary/pedia I could find - even Wolfram, who knows far more maths than I do regardless of his hubris) has replaced it with a NAND. You get twice as many google hits for Sheffer Stroke NAND as NOR if that's any indication. Both NOR and NAND are able to derive all other operators, possibly enabling their confusion.
Strictly speaking, if I recall that bit correctly, W is pointing out a similarlity between his N operator and NOR, but the N operator is extendable to more than two arguments so it's not quite the same anyway.
posted by Sparx at 10:50 AM on December 22, 2008
Strictly speaking, if I recall that bit correctly, W is pointing out a similarlity between his N operator and NOR, but the N operator is extendable to more than two arguments so it's not quite the same anyway.
posted by Sparx at 10:50 AM on December 22, 2008
Yes, how odd. My copy of A Wittgenstein Dictionary has it (the Scheffer stroke) as NOR, but the accepted definition nowadays seems to be NAND. I wonder how that happened.
posted by king walnut at 11:03 AM on December 22, 2008
posted by king walnut at 11:03 AM on December 22, 2008
I'm surprised, even Wolfram's mathworld is wrong about this. To expand on what painquale said, in the original Sheffer, p|q is clearly NOR, i.e. "neither-nor" -- defined as "~(p V q). The original article is here if you have jstor, and it is pretty clear (see the proof of theorem 1). (On preview, it isn't a matter of accepted definition, I think anyone calling nand the Sheffer stroke is not consistent with Sheffer's work. Wittgenstein is completely consistent with this -- my edition says "~p.~q or p|q", where "." means "and" (5.1241); the first formula is ~(p V q) via de Morgan's law.
posted by advil at 11:10 AM on December 22, 2008
posted by advil at 11:10 AM on December 22, 2008
but common usage -- has replaced it with a NAND
Huh, that's interesting. I never knew that. At least as normally taught in philosophy, the Sheffer stroke is still normally used to represent the NOR function (which I guess is why leibniz and I were so sure of ourselves).
Actually, it appears that in his original paper, Sheffer used the | symbol to represent NAND and the downward dagger to represent NOR, just as wikipedia has it (I haven't looked up the original paper to verify this, though). However, in the Principia, Russell and Whitehead used | as Wittegnstein does, and they credit Sheffer. I guess some philosophers have followed the Tractatus/Principia tradition, and mathematicians and computer scientists have followed Sheffer's original symbolization (renaming the dagger 'the Pierce arrow' to give Pierce his due).
Of course, all of this is pretty much arbitrary. You can use whatever symbol to represent whatever function you want, as long as you're clear.
This mailing list discussion gives an interesting bit of history.
posted by painquale at 11:15 AM on December 22, 2008
Huh, that's interesting. I never knew that. At least as normally taught in philosophy, the Sheffer stroke is still normally used to represent the NOR function (which I guess is why leibniz and I were so sure of ourselves).
Actually, it appears that in his original paper, Sheffer used the | symbol to represent NAND and the downward dagger to represent NOR, just as wikipedia has it (I haven't looked up the original paper to verify this, though). However, in the Principia, Russell and Whitehead used | as Wittegnstein does, and they credit Sheffer. I guess some philosophers have followed the Tractatus/Principia tradition, and mathematicians and computer scientists have followed Sheffer's original symbolization (renaming the dagger 'the Pierce arrow' to give Pierce his due).
Of course, all of this is pretty much arbitrary. You can use whatever symbol to represent whatever function you want, as long as you're clear.
This mailing list discussion gives an interesting bit of history.
posted by painquale at 11:15 AM on December 22, 2008
Oops, advil shows that I'm wrong about what Sheffer said in his original paper. Scratch that previous comment.
posted by painquale at 11:17 AM on December 22, 2008
posted by painquale at 11:17 AM on December 22, 2008
Yeah, downward dagger is just a footnote symbol in the paper. But on p. 488 there is a dagger footnote that points out that the results hold for nand.
posted by advil at 11:34 AM on December 22, 2008
posted by advil at 11:34 AM on December 22, 2008
Response by poster: All of your answers are the best. I started to mark them all as best, but decided that was stupid. Thank you everyone.
So in conclusion: Wittgenstein's N is a multi-variable version of NOR, its truth table for two variables is FFFT, you can call it the Sheffer stroke if you want (especially in philosophy circles), it's usually written as p|q, it's equivalent to -p&-q, and Mounce and Wittgenstein are on the same page. My entire confusion stems from the fact that computer scientists in Wikipedia have a convention of calling NAND the Sheffer stroke instead of NOR.
I suppose in retrospect I shouldn't have been so confused. There was a frustrating moment yesterday where I was flipping back and forth between Mounce, Tractatus and Wikipedia and trying to parse the meaning of "neither nor" and "not both" in my head and I got very discouraged.
posted by creasy boy at 1:57 AM on December 23, 2008
So in conclusion: Wittgenstein's N is a multi-variable version of NOR, its truth table for two variables is FFFT, you can call it the Sheffer stroke if you want (especially in philosophy circles), it's usually written as p|q, it's equivalent to -p&-q, and Mounce and Wittgenstein are on the same page. My entire confusion stems from the fact that computer scientists in Wikipedia have a convention of calling NAND the Sheffer stroke instead of NOR.
I suppose in retrospect I shouldn't have been so confused. There was a frustrating moment yesterday where I was flipping back and forth between Mounce, Tractatus and Wikipedia and trying to parse the meaning of "neither nor" and "not both" in my head and I got very discouraged.
posted by creasy boy at 1:57 AM on December 23, 2008
I'm taking Symbolic Logic at college right now, and I decided I was going to "find" the Sheffer stroke. I derived the other functions from NAND, and thought I had found it, but I guess NOR works just as well.
posted by Picklegnome at 4:38 PM on February 5, 2009
posted by Picklegnome at 4:38 PM on February 5, 2009
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posted by leibniz at 8:38 AM on December 22, 2008