Monadic tautologies and contradictions?

[Arising_uk]

What do they represent or symbolise?

Some background for those who haven't studied truth-functional logic(TFL).

Think of one object, thing or state of affairs call it 'p'

It has two conditions and they are shown in TFL by T and F or 1 and 0. Normally like this,

p
T
F

or

p
1
0

That is, it is the case or it is not the case.

From this we can produce four truth-functions,


p 1 2 3 4
T T T F F
F T F T F

2. is p itself, i.e. it is true when p is true and false when p is false.
3. is called Negation or NOT, i.e. it is false when p is true and true when p is false.

What are 1. and 4? It's obvious that 1 is a tautology and 4 is a contradiction but what can they be symbolised by? The best I guess is 1. must be (P OR NOT P) and 4. must be (P AND NOT P) so 1. is the OR operator and 4. the AND operator but can one do or think like this with a monadic function?

[Wyman]

p is an undefined term.

One interpretation of the undefined term is "p is a proposition and a proposition is either true or false, exclusive.' Another interpretation is 'p is a switch and can be set to either 1 or 0.'

'p v q' is not an interpretation of 'p' but a wff (well formed formula, or statement of proper syntax in PL), even where q = p.

Interpretations of formal systems involve meta-systematic considerations - i.e. they are not created from within the system, but from without, for aesthetic or pragmatic reasons. wffs are formed mechanically, from within the system pursuant to algorithmic rules of syntax.

That the interpretation (aka 'model' if true for the formal system) of 'p' in terms of a truth table as T or F (and not both) resembles the wffs (and also axioms of PL) which are called the rule of non-contradiction (not(p AND notp)) and the law of the excluded middle (p or notp) is confusing and creates confusions which I attempted to address with your interpretation of 'a thing cannot exist and not exist at the same time' as an axiom of logic. Not that I think I am right in that argument by any stretch of the imagination, but I think I am right in placing that dispute in terms of the interplay between interpretations or models of PL, and its axioms.

I realize the above paragraph is not clearly written and maybe I can return to it some other time. But take an example. In basic geometry, points and lines are usually interpreted as dots and dashes on paper. An axiom might be 'two separate points may have one and only one straight line which passes through both.' That could be a wff of this geometry. The fact that, if I try, with a straight edge ruler, to pass more than one line through two points that I draw on my paper and find that I cannot, is not a repetition of the axiom, but shows that my interpretation of the system is true, since it is true for that (and the other) axiom(s). Similarly, the interpretation of p as T or F (and not both) is not a repetition of the law of the excluded middle, but shows that the truth table model of PL is a true interpretation of the system.

[Owen]

What do they represent or symbolise?

Some background for those who haven't studied truth-functional logic(TFL).

Think of one object, thing or state of affairs call it 'p'

It has two conditions and they are shown in TFL by T and F or 1 and 0. Normally like this,

p
T
F

or

p
1
0

That is, it is the case or it is not the case.

From this we can produce four truth-functions,


p 1 2 3 4
T T T F F
F T F T F

2. is p itself, i.e. it is true when p is true and false when p is false.
3. is called Negation or NOT, i.e. it is false when p is true and true when p is false.

What are 1. and 4? It's obvious that 1 is a tautology and 4 is a contradiction but what can they be symbolised by? The best I guess is 1. must be (P OR NOT P) and 4. must be (P AND NOT P) so 1. is the OR operator and 4. the AND operator but can one do or think like this with a monadic function?

There is no "must be" here. 1=any tautology and 4=any contradiction.

1 is true when p is true and true when p is false.
4 is false when p is true and false when p is false.

1=(p v ~p)=(p -> p)=(p <-> p)=(p xor ~p)=(p nand ~p).
4=(p & ~p)=(p ~> p)=(p <-> ~p)=(p xor p)=(p nor ~p).

Note: (p ~> q) def ~(p -> q).

For example, Wittgenstein chose 1=(p -> p) and 4=(p &~p) in his Tractatus. See TPL page 77, (5.101).
http://semweb.ch/_/wordpress/wp-content ... en_D-E.pdf

I agree that 1=(p v ~p) and 4= (p & ~p) are practical choices.

[Arising_uk]

Ah! Thank you for that Owen, wasn't thinking straight.

Still, interesting(to me) that the monadic truth functions produce dyadic tautologies off the bat? Is it because there are two possible states of affairs with one object?

[thedoc]

Does anyone here remember WFF 'N PROOF, the game, and what is your opinion of it's relationship to logic?

[Wyman]

Ah! Thank you for that Owen, wasn't thinking straight.

Still, interesting(to me) that the monadic truth functions produce dyadic tautologies off the bat? Is it because there are two possible states of affairs with one object?

No, I think you were right in asking the question in the first case. 1,2,3,4 represent single value(monadic) 'functions' where a truth value is plugged in for 'p' and a unique truth value is spit out. I.e. 1(p), 2(p), 3(p), 4(p)

Although it is easy to interpret 'not' as a monadic function, it is not so for sentence connectives - they connect two sentences. Two value functions will have truth tables with four combinations of truth values, as you know. Each combination represents two-value functions - 5(p,q), 6(p,q), etc.. The 5(p,q) could be, for instance, the conjunction connective so for each combination of p and q, it spits out one unique value;i.e. 5(f,t) = f, 5(t,t) = true and so on. We represent that function with the conjunction symbol.

So 1 does not equal (p v notp) as Owen said, as disjunction is a sentence connector representing (in a two value truth table model) a certain combination(two value function) of truth values. 1 is a single value function without a name, technically. 'Tautology' is just a description of certain statements or argument forms which always give the value 't' no matter what value are assigned to the variables.

So (p or notp) is a tautology, but does not represent the one-value truth function 1(p). It represents the two-value function f(p,q) where q = notp.

[Wyman]

The answer to your last question is: there are only two possible values for the variables of sentence calculus - t or f; we have an interest in exploring all combinations of those two values as they occur in any combination of an infinite number of variables (p,q,r, etc.); when we begin with the first variable, p, the possible combinations are tt, tf, ff,ft. We give names to some of these combinations and in general, call these combinations truth functions. Allowable relations of variables include
AND, OR, NOT, etc..

An analogy is with algebra: there are an infinite number of possible values for any variable (the domain is the class of real numbers). We have an interest in exploring (at least recursively) all combinations of those values as they occur in any possible combination of variables. Variables may be combined by a set of either undefined or defined relations, +, -, <,>, etc..

[Arising_uk]

Does anyone here remember WFF 'N PROOF, the game, and what is your opinion of it's relationship to logic?

Had never heard of it but thanks for this as it looks a fun and interesting way of teaching the principles of logic and logical thinking.

[Owen]

Ah! Thank you for that Owen, wasn't thinking straight.

Still, interesting(to me) that the monadic truth functions produce dyadic tautologies off the bat? Is it because there are two possible states of affairs with one object?

You are very welcome.

Yes, the two states of affairs are the primitive notions of truth(T) and falsity(F).
The dyadic nature of truth functions is a consequence of the dyads: (T T), (F T), (T F), (F F).
Note, there are no triadic truth functions. The dyadic truth functions are all that is needed.

If we take truth(T) and falsity(F) and nand(|) as primitive, we can define all truth functions of any number of propositional variables,
including the monadic truth functions.

(|)nand: (T nand T) =df (F), (F nand T) =df (T), (T nand F)=df (T), (F nand F) =df (T).
Now we can define the truth functions that include propositional variables.

(~)not: ~p =df (p nand p), ~q =df (q nand q) etc..
(&)and: (p & q) =df ~(p nand q).
(v)or: (p v q) =df ~(~p & ~q).
(->)imp: (p -> q) =df (~p v q).
(<->)equ: (p <-> q) =df ((p -> q) & (q -> p)).
etc.

If we use 'nor' as primitive we get...

(nor): (T nor T) =df F, (F nor T) =df F, (T nor F) =df F, (F nor F) =df T.

(~)not: ~p =df (p nor p), ~q =df (q nor q), etc..
(v)or: (p v q) =df ~(p nor q).
(&)and: (p & q) =df ~(~p v ~q).
etc..

[Wyman]

Ah! Thank you for that Owen, wasn't thinking straight.

Still, interesting(to me) that the monadic truth functions produce dyadic tautologies off the bat? Is it because there are two possible states of affairs with one object?

You are very welcome.

Yes, the two states of affairs are the primitive notions of truth(T) and falsity(F).
The dyadic nature of truth functions is a consequence of the dyads: (T T), (F T), (T F), (F F).
Note, there are no triadic truth functions. The dyadic truth functions are all that is needed.

If we take truth(T) and falsity(F) and nand(|) as primitive, we can define all truth functions of any number of propositional variables,
including the monadic truth functions.

(|)nand: (T nand T) =df (F), (F nand T) =df (T), (T nand F)=df (T), (F nand F) =df (T).
Now we can define the truth functions that include propositional variables.

(~)not: ~p =df (p nand p), ~q =df (q nand q) etc..
(&)and: (p & q) =df ~(p nand q).
(v)or: (p v q) =df ~(~p & ~q).
(->)imp: (p -> q) =df (~p v q).
(<->)equ: (p <-> q) =df ((p -> q) & (q -> p)).
etc.

If we use 'nor' as primitive we get...

(nor): (T nor T) =df F, (F nor T) =df F, (T nor F) =df F, (F nor F) =df T.

(~)not: ~p =df (p nor p), ~q =df (q nor q), etc..
(v)or: (p v q) =df ~(p nor q).
(&)and: (p & q) =df ~(~p v ~q).
etc..

Do you find it odd that under the first axiom system (nand), we define 'not p' as 'not both p and p?'

It uses 'not' in the definition of 'not' (as well as 'and' and parentheses). Is it really just the same as using as primitives 'not' and 'and'?

Isn't it more in keeping with the semantics to just use 'not' as a primitive(with another) and to not try and represent the monadic truth functions with dyadic truth functions? - I guess this is similar to my objection above.

Or doesn't it matter as long as everything we want expressed can be expressed?

[Owen]

Ah! Thank you for that Owen, wasn't thinking straight.

Still, interesting(to me) that the monadic truth functions produce dyadic tautologies off the bat? Is it because there are two possible states of affairs with one object?

You are very welcome.

Yes, the two states of affairs are the primitive notions of truth(T) and falsity(F).
The dyadic nature of truth functions is a consequence of the dyads: (T T), (F T), (T F), (F F).
Note, there are no triadic truth functions. The dyadic truth functions are all that is needed.

If we take truth(T) and falsity(F) and nand(|) as primitive, we can define all truth functions of any number of propositional variables,
including the monadic truth functions.

(|)nand: (T nand T) =df (F), (F nand T) =df (T), (T nand F)=df (T), (F nand F) =df (T).
Now we can define the truth functions that include propositional variables.

(~)not: ~p =df (p nand p), ~q =df (q nand q) etc..
(&)and: (p & q) =df ~(p nand q).
(v)or: (p v q) =df ~(~p & ~q).
(->)imp: (p -> q) =df (~p v q).
(<->)equ: (p <-> q) =df ((p -> q) & (q -> p)).
etc.

If we use 'nor' as primitive we get...

(nor): (T nor T) =df F, (F nor T) =df F, (T nor F) =df F, (F nor F) =df T.

(~)not: ~p =df (p nor p), ~q =df (q nor q), etc..
(v)or: (p v q) =df ~(p nor q).
(&)and: (p & q) =df ~(~p v ~q).
etc..

Do you find it odd that under the first axiom system (nand), we define 'not p' as 'not both p and p?'

It uses 'not' in the definition of 'not' (as well as 'and' and parentheses). Is it really just the same as using as primitives 'not' and 'and'?

Isn't it more in keeping with the semantics to just use 'not' as a primitive(with another) and to not try and represent the monadic truth functions with dyadic truth functions? - I guess this is similar to my objection above.

Or doesn't it matter as long as everything we want expressed can be expressed?

Hi Wyman,

Wyman: Do you find it odd that under the first axiom system (nand), we define 'not p' as 'not both p and p?'

No, ~p is not defined as ~(p & p). That ~p <-> ~(p & p), is a consequence of the definitions, and I don't find that odd.

If we use (|) nand as the only primitive (undefined) operator then all operators can be expressed by it alone.

~p =df (p | p).
(p & q) =df ((p | q) | (p | q)).
etc.

Indeed, if the primitive ideas are only: truth(T), falsity(F) and nand(|) we can reduce all propositions functions to functions of the primitive elements alone.

1. (T | T) =df (F), (F | T) =df (T), (T | F)=df (T), (F | F) =df (T).

2. ~T =df (T | T), and (T | T)=F (by 1.). ~F =df (F | F), and (F | F)=T (by 1.).

3. (T & T) =df ((T | T) | (T | T)). ie. (F | F), which is T by 1.

4. (F & T) =df ((F | T) | (F | T)). ie. (T | T), which is F.

5. (T & F) =df ((T | F) | (T | F)). ie. (T | T), which is F.

6. (F & F) =df ((F | F) | (F | F)). ie. (T | T), which is F.
etc.

Using (~, &) as primitives or (~, v) as primitives makes thing much easier, but the point is that
nand and nor are the only single functors that can formally produce all of the other operators.

[thedoc]

Does anyone here remember WFF 'N PROOF, the game, and what is your opinion of it's relationship to logic?

Had never heard of it but thanks for this as it looks a fun and interesting way of teaching the principles of logic and logical thinking.

I had a set, but couldn't interest anyone else in playing. But there was a way that I could do it alone, so that is what I would do.

[A_Seagull]

What do they represent or symbolise?

Some background for those who haven't studied truth-functional logic(TFL).

Think of one object, thing or state of affairs call it 'p'

It has two conditions and they are shown in TFL by T and F or 1 and 0. Normally like this,

p
T
F

or

p
1
0

That is, it is the case or it is not the case.

From this we can produce four truth-functions,


p 1 2 3 4
T T T F F
F T F T F

2. is p itself, i.e. it is true when p is true and false when p is false.
3. is called Negation or NOT, i.e. it is false when p is true and true when p is false.

What are 1. and 4? It's obvious that 1 is a tautology and 4 is a contradiction but what can they be symbolised by? The best I guess is 1. must be (P OR NOT P) and 4. must be (P AND NOT P) so 1. is the OR operator and 4. the AND operator but can one do or think like this with a monadic function?

What you are describing here would seem to have to relation to the 'real world', it would seem to be nothing more than an abstract game.

If there is a relationship with the 'real world', please describe how it can be correlated.

Also, does the 'truth' of tis system have any correlation with what is commonly termed as 'truth' of the 'real world'? And if so what is that correlation or relationship?

[Wyman]

Hi Wyman,

Wyman: Do you find it odd that under the first axiom system (nand), we define 'not p' as 'not both p and p?'

No, ~p is not defined as ~(p & p). That ~p <-> ~(p & p), is a consequence of the definitions, and I don't find that odd.

If we use (|) nand as the only primitive (undefined) operator then all operators can be expressed by it alone.

~p =df (p | p).
(p & q) =df ((p | q) | (p | q)).
etc.

Indeed, if the primitive ideas are only: truth(T), falsity(F) and nand(|) we can reduce all propositions functions to functions of the primitive elements alone.

1. (T | T) =df (F), (F | T) =df (T), (T | F)=df (T), (F | F) =df (T).

2. ~T =df (T | T), and (T | T)=F (by 1.). ~F =df (F | F), and (F | F)=T (by 1.).

3. (T & T) =df ((T | T) | (T | T)). ie. (F | F), which is T by 1.

4. (F & T) =df ((F | T) | (F | T)). ie. (T | T), which is F.

5. (T & F) =df ((T | F) | (T | F)). ie. (T | T), which is F.

6. (F & F) =df ((F | F) | (F | F)). ie. (T | T), which is F.
etc.

Using (~, &) as primitives or (~, v) as primitives makes thing much easier, but the point is that
nand and nor are the only single functors that can formally produce all of the other operators.

That's interesting; I've learned differently. I understand the primitive terms to be p, p', p'', p''', etc. (i.e. an infinite number of propositional variables). The other primitive terms would be the chosen primitive connective(s) nand (or '&, not', etc..).

Truth and Falsity come in, not as primitive terms, but as one possible interpretation of the propositional variables. Arising seems to (I haven't completely figured this out yet) interpret the variables as 'x exists' and 'x does not exist' or something like it - i.e. where we are not talking about the truth or falsity of propositions, but about whether something 'is' or 'isn't'.

[Owen]

Hi Wyman,

Wyman: Do you find it odd that under the first axiom system (nand), we define 'not p' as 'not both p and p?'

No, ~p is not defined as ~(p & p). That ~p <-> ~(p & p), is a consequence of the definitions, and I don't find that odd.

If we use (|) nand as the only primitive (undefined) operator then all operators can be expressed by it alone.

~p =df (p | p).
(p & q) =df ((p | q) | (p | q)).
etc.

Indeed, if the primitive ideas are only: truth(T), falsity(F) and nand(|) we can reduce all propositions functions to functions of the primitive elements alone.

1. (T | T) =df (F), (F | T) =df (T), (T | F)=df (T), (F | F) =df (T).

2. ~T =df (T | T), and (T | T)=F (by 1.). ~F =df (F | F), and (F | F)=T (by 1.).

3. (T & T) =df ((T | T) | (T | T)). ie. (F | F), which is T by 1.

4. (F & T) =df ((F | T) | (F | T)). ie. (T | T), which is F.

5. (T & F) =df ((T | F) | (T | F)). ie. (T | T), which is F.

6. (F & F) =df ((F | F) | (F | F)). ie. (T | T), which is F.
etc.

Using (~, &) as primitives or (~, v) as primitives makes thing much easier, but the point is that
nand and nor are the only single functors that can formally produce all of the other operators.

That's interesting; I've learned differently. I understand the primitive terms to be p, p', p'', p''', etc. (i.e. an infinite number of propositional variables). The other primitive terms would be the chosen primitive connective(s) nand (or '&, not', etc..).

Truth and Falsity come in, not as primitive terms, but as one possible interpretation of the propositional variables. Arising seems to (I haven't completely figured this out yet) interpret the variables as 'x exists' and 'x does not exist' or something like it - i.e. where we are not talking about the truth or falsity of propositions, but about whether something 'is' or 'isn't'.

I agree that we do include propositional variables as a primitive notion, which take on the values T and F. Perhaps we can consider the application of the truth functions without variables as the 'arithmetic' of logic and truth functions with variables as the 'algebra' of logic.

E.g. (p v ~p) assumes that it is true for any value of p. (T v ~T) <-> T, and (F v ~F) <-> T.
Also (p & ~p) assumes that it is false for any value of p. (T & ~T) <-> F, and (F & ~F) <-> F.

We can say (p v ~p) is true for all values of p. That is, (all p)(p v ~p) is a theorem.
We can say (p & ~p) is true for no value of p. That is, ~(some p)(p & ~p) is a theorem.

We can say (p & ~p) is false for all values of p. That is (all p)(p & ~p) is a contradiction.

(all p)(all q)(((p -> q) & p) -> q), is tautologous..a theorem.

That is, (((T -> T) & T) -> T) & (((F -> T) & F) -> T) & (((T -> F) & F) -> F) & (((F -> F) & F) -> T) is tautologous.

More often variables are replaced by specific constant propositions.
eg. If (it is raining) then (the sidewalks are wet)) and (It is raining) therefore (the sidewalks are wet).