Wittgenstein's Truth Opertor

Wittgenstein's Truth Opertor
Furrowed Brow
Newbie

Usergroup: Members
Joined: Feb 09, 2007

Total Topics: 5
Total Posts: 0
#1 - Quote - Permalink
Posted Feb 9, 2007 - 9:11 AM:
Subject: Wittgenstein's Truth Operator
At point 5.5 in the Tractatus Wittgenstein introduced a truth-operator for the production of truth function. The operator takes the following form: (- - - - -T)(x,...). In the left parentheses five dashes represent false and a T represents true. So instead of (- - - - - T) it is also possible to write (FFFFFT).The letter x is the sign for the variable, whilst the dots leave space to introduce further signs as values of the variable. The operation is meant to be one of joint negation. But notice what’s missing! If reading Wittgenstein literally, the following truth table results.

Xab -x-a-b “-xa & -Xb”
TTT...F.F.F...........F
FTT...T.F.F
TFT...F.T.F...........F
FFT...T.T.F...........F
TTF...F.F.T...........F
FTF...T.F.T...........F
TFF...F.T.T
FFF...T.T.T...........T

The signs a and b are names which refer to objects in the world. The missing truth values are then fundamental. If a and b are to be values of the variable they cannot both be false when the variable is true, or not true when the variable is false. If they could then “There is an x” would attempt to express an existential commitment, and be saying there is a value of x that is true, whilst in fact there was no existing object. Thus the truth places left empty must remain so, otherwise the relationship of value and variable is negated.

Here I have used names because the basic relationship between variable and name should be obvious. Wittgenstein intends this operation to be applied to elementary propositions. Things get can get a bit more complicated, but the lesson is that Wittgenstein is using the truth-operator to express the formal relationship between variable and its value. We could dig into this more, but for the moment I’d liked to ask whether anyone else thinks this is the correct way to understand point 5.5. Is Wittgenstein being deliberate here? Does he mean there to be missing truth-values?

[Since starting this thread I realise the way I have put things here is still not quite right - I put things better here - http://www.freewebs.com/furrowedbrow/index.htm ]

Edited by Furrowed Brow on Feb 24, 2007 - 2:04 PM. Reason: typos
Owen
Resident

Usergroup: Members
Joined: Oct 17, 2004

Total Topics: 25
Total Posts: 287
#2 - Quote - Permalink
Posted Feb 9, 2007 - 10:29 AM:

Furrowed Brow wrote:
At point 5.5 in the Tractatus Wittgenstein introduced a truth-operator for the production of truth function. The operator the following form: (- - - - -T)(x,...). In the left parentheses five dashes represent false and a T represents true. So instead of (- - - - - T) it is also possible to write (FFFFFT).The letter x is the sign for the variable, whilst the dots leave space to introduce further signs as values of the variable. The operation is meant to be one of joint negation. But notice what’s missing! If reading Wittgenstein literally, the following truth table results.

Xab -x-a-b “-xa & -Xb”
TTT...F.F.F...........F
FTT...T.F.F
TFT...T.F.F...........F
FFT...T.T.F...........F
TTF...F.F.T...........F
FTF...T.F.T...........F
TFF...F.T.T
FFF...T.T.T...........T

The signs a and b are names which refer to objects in the world. The missing truth values are then fundamental. If a and b are to be values of the variable they cannot both be false when the variable is true, or not true when the variable is false. If they could then “There is an x” would attempt to express an existential commitment, and be saying there is a value of x that is true, whilst in fact there was no existing object. Thus the truth places left empty must remain so, otherwise the relationship of value and variable is negated.

Here I have used names because the basic relationship between variable and name should be obvious. Wittgenstein intends this operation to be applied to elementary propositions. Things get can get a bit more complicated, but the lesson is that Wittgenstein is using the truth-operator to express the formal relationship between variable and its value. We could dig into this more, but for the moment I’d liked to ask whether anyone else thinks this is the correct way to understand point 5.5. Is Wittgenstein being deliberate here? Does he mean there to be missing truth-values?



Wittgenstein extends the 'Pierce Arrow', ie. the 'Nor' function, to include propositional logic and monadic predicate logic functions.

N(p) = (p nor p) = ~p.
N(p,q) = (p nor q) = ~p & ~q.
N(p,q,r) = ~p & ~q & ~r.
N(Fa,Fb,Fc,...) = ~Fa & ~Fb & ~Fc & ... = ~ExFx.
etc.

See: 5.502, 5.51, etc.

He claims that all propositions are truth functions of atomic propositions.

See: 6, 6.001, etc.
becomingagodo
Socratic method?
Avatar

Usergroup: Members
Joined: Feb 09, 2007

Total Topics: 14
Total Posts: 0
#3 - Quote - Permalink
Posted Feb 9, 2007 - 11:03 AM:

Does this escape Godels uncertaity laws and paradoxs.
Owen
Resident

Usergroup: Members
Joined: Oct 17, 2004

Total Topics: 25
Total Posts: 287
#4 - Quote - Permalink
Posted Feb 9, 2007 - 11:12 AM:

becomingagodo wrote:
Does this escape Godels uncertaity laws and paradoxs.


Yes, both propositional logic and monadic predicate logic are decidable.
becomingagodo
Socratic method?
Avatar

Usergroup: Members
Joined: Feb 09, 2007

Total Topics: 14
Total Posts: 0
#5 - Quote - Permalink
Posted Feb 9, 2007 - 11:20 AM:

Yes, both propositional logic and monadic predicate logic are decidable.

What thats impossibleconfused

I will look up Wittgenstein Truth opertor. Have any of you got any links to a good page on this.
Furrowed Brow
Newbie

Usergroup: Members
Joined: Feb 09, 2007

Total Topics: 5
Total Posts: 0
#6 - Quote - Permalink
Posted Feb 9, 2007 - 11:26 AM:

Owen wrote:



Wittgenstein extends the 'Pierce Arrow', ie. the 'Nor' function, to include propositional logic and monadic predicate logic functions.

N(p) = (p nor p) = ~p.
N(p,q) = (p nor q) = ~p & ~q.
N(p,q,r) = ~p & ~q & ~r.
N(Fa,Fb,Fc,...) = ~Fa & ~Fb & ~Fc & ... = ~ExFx.
etc.

See: 5.502, 5.51, etc.

He claims that all propositions are truth functions of atomic propositions.

See: 6, 6.001, etc.


You are quite right about 5.502, 5.503; but here he has already moved on to use the N notation. The underlying structure of what is going on with the N notation is given at 5.5. How would you write out that truth table?

When Wittgenstein writes ‘(- - - - - T)(x, …)’ should we take him literally?

‘5.5 [b] This operation negates all the propositions in the right hand pair of brackets…’.

So should we be asking what the variable sign is doing in the right hand pair of brackets, and why the five dashes? Are they just a mistake? Not significant? Does that sound like Wittgenstein to you?

Wittgenstein was laying the ground for the general form of the proposition much earlier in the book.

3.31 [c] Everything essential to their sense that propositions can have in common with one another is an expression.

3.311 An expression presupposes the forms of all the propositions in which it can occur. It is the common characteristic mark of a class of propositions.

3.313 Thus an expression is presented by means of a variable whose values are the propositions that contain the expression.

If we follow through on how Wittgenstein has already set things up in the 3s, and given he has placed the variable in the right hand pair of brackets, the resulting truth function should also present the common expression shared by the propositions that are the values of the variables. In other words I am saying Wittgenstein is being utterly consistent here.

Furrowed Brow
Newbie

Usergroup: Members
Joined: Feb 09, 2007

Total Topics: 5
Total Posts: 0
#7 - Quote - Permalink
Posted Feb 9, 2007 - 11:37 AM:

becomingagodo wrote:
Does this escape Godels uncertaity laws and paradoxs.


Yes. Look again at the table in the first post. The variable is truth functionally bound to its values. This is a bound variable not a free variable. Godel's results rely on the free variable, and an isomorphism between Godel's numbers and the formal system being translated.
Owen
Resident

Usergroup: Members
Joined: Oct 17, 2004

Total Topics: 25
Total Posts: 287
#8 - Quote - Permalink
Posted Feb 9, 2007 - 1:33 PM:

becomingagodo wrote:
Yes, both propositional logic and monadic predicate logic are decidable.

What thats impossibleconfused


Not so. Dyadic predicate logic is not decidable, see Church (1936)
But, monadic predicate logic is complete and decidable.
I have developed a truth table procedure for monadic logic, if you are interested.

I will look up Wittgenstein Truth opertor. Have any of you got any links to a good page on this.


Tractatus is available on line:

http://www.kfs.org/~jonathan/witt/ten.html
Owen
Resident

Usergroup: Members
Joined: Oct 17, 2004

Total Topics: 25
Total Posts: 287
#9 - Quote - Permalink
Posted Feb 10, 2007 - 4:58 AM:

Owen wrote:
Wittgenstein extends the 'Pierce Arrow', ie. the 'Nor' function, to include propositional logic and monadic predicate logic functions.

N(p) = (p nor p) = ~p.
N(p,q) = (p nor q) = ~p & ~q.
N(p,q,r) = ~p & ~q & ~r.
N(Fa,Fb,Fc,...) = ~Fa & ~Fb & ~Fc & ... = ~ExFx.
etc.

See: 5.502, 5.51, etc.

He claims that all propositions are truth functions of atomic propositions.

See: 6, 6.001, etc.


Furrowed Brow wrote:


You are quite right about 5.502, 5.503; but here he has already moved on to use the N notation. The underlying structure of what is going on with the N notation is given at 5.5. How would you write out that truth table?

When Wittgenstein writes ‘(- - - - - T)(x, …)’ should we take him literally?


Yes we should.

(FT)(p) = N(p) = ~p.
(FFFT)(p,q) = N(p,q) = ~p & ~q.
(FFFFFFFT)(p,q,r) = N(p,q,r) = ~p & ~q & ~r.
(FFFFFFFF..T)(Fa,Fb,Fc, ..) = N(Fa,Fb,Fc,...) = ~Fa & ~Fb & ~Fc & ... = ~ExFx.

In 5.5, Wittgenstein's (- - - - - T) expresses the truth table for the 'nor' function applied to (x, …).
Furrowed Brow
Newbie

Usergroup: Members
Joined: Feb 09, 2007

Total Topics: 5
Total Posts: 0
#10 - Quote - Permalink
Posted Feb 10, 2007 - 6:39 AM:

Owen wrote:
Yes we should.

(FT)(p) = N(p) = ~p.
(FFFT)(p,q) = N(p,q) = ~p & ~q.
(FFFFFFFT)(p,q,r) = N(p,q,r) = ~p & ~q & ~r.
(FFFFFFFF..T)(Fa,Fb,Fc, ..) = N(Fa,Fb,Fc,...) = ~Fa & ~Fb & ~Fc & ... = ~ExFx.

In 5.5, Wittgenstein's (- - - - - T) expresses the truth table for the 'nor' function applied to (x, …).



Lets call ‘~xa’ a proposition. Characterising it as a single proposition we are no longer recognising its internal structure. So we can substitute the sign ‘~xa’ with ~p and ‘~xb’ with ~q.

0Xab -x-a-b “-xa & -Xb”
1TTT...F.F.F...........F
2FTT...T.F.F
3TFT...F.T.F...........F
4FFT...T.T.F...........F
5TTF...F.F.T...........F
6FTF...T.F.T...........F
7TFF...F.T.T
8FFF...T.T.T...........T

If we are to write a table that does not make the internal structure of the proposition explicit then we lose the distinction seen at rows 3/4, and rows 5/6. So if we derive a truth function in terms of p and q we get (FFFT)N(p,q) = ~p & ~q. Just as Owen points out. But unlike the N operation which deals with propositions as a whole, the truth operator in its explicit form is saying something about the relationship that must hold between variable and value.

If we completed rows 2 and 7, the resulting truth function would say it is possible for all values of x to be true when x is false, and it is possible for all values of x to be false when x is true. So we cannot acknowledge these possibilities with F for false, because that admits the possibility they could be true. One way to indicate the threatened contradiction is just not to write in these possibilities. In effect this is a three valued system. True, False and contradiction.

Looking at this the other way round, if you wanted to write a truth table that expressed a formal relationship between variable and value, you would be forced to write a table like the one above. Pertinently Wittgenstein spends some resource detailing the relationship between variable and value, and when one takes a closer look at the truth operator we find missing truth values.

So we come back to the same set of questions. Is Wittgenstein being systematic here? Are the five dashes significant? Was Wittgenstein being sloppy, or adept?





Edited by Furrowed Brow on Feb 10, 2007 - 12:56 PM
locked
Download thread as
  • 0/5
  • 1
  • 2
  • 3
  • 4
  • 5



This thread is closed, so you cannot post a reply.