This continues Basic Logic - semantics by explaining the fundamentals of extended formal semantics. There are various novelties in the present approach, notably - but not only - that Basic Logic is used to express its own semantics; that two negations are distinguished, while stil binary logic is used; and that a new approach to the paradoxes, such as Russell's paradox of the set of all sets that are not element of themselves results.

1. Extended semantics- fundaments
2. Extended semantics - propositional logic
3. Extended semantics - identities and elements
4. Extended semantics - elements of themselves and paradox
 

1. Extended semantics- fundaments

It is also possible - and indeed, in view of logical paradoxes - desirable to extend and refine the proposed semantics, both truth-functional and domain, and both non-modal and modal.

The extended semantics can be presented and justified in several ways. I will here do so from the point of view of substitution, which is basic for BL.

We have from BL the following as derivable theorems:

T3. |- Ø = |x.~x=x

T4. |- (x)(Fx)   IFF (|x:~Fx) = Ø
T5. |- (Ex)(Fx) IFF (|x: Fx) ≠ Ø

T6. |- a e |x.Fx       IFF a|x.Fx
T7. |- |x.Ax = |x.Bx IFF (x)(Ax iff Bx)

This shows the notation and concepts on the LHS are definable or explained by the same on the RHS. And this in turn shows that substitution into formulas and identity are basic in BL and suffice.

We have assumed already

Substitution rules for equalities

|- T[..x..] & x=y                     IFF T[..y..] & x=y
|- a|b.c = c                          IFF b≠c
|- a|b.c = a                          IFF b=c

Substitution rules for terms

|- x|y.(t₁..t_n)                       IFF (x|y.t₁..x|y.t_n)
|- (x₁..x_m)|(y₁..y_m).(t₁..t_n)      IFF x₁|y₁..x_m|y_m.(t₁..t_n)
|- x₁|y₁ x₂|x₂..x_m|y_m.(t₁..t_n)   IFF x₂|x₂..x_m|y_m.(x₁|y₁.t₁..x₁|y₁.t_n)

Considering substitution, which is fundamental, we can also derive

T8.  |- a|x.Fx |= Fa
T9.  |- y|x.Fx |= ~y|x.~Fx
T10. |- y|x.Fx V y|x.Gx |= y|x.Fx V Gx
T11. |- y|x.Fx & Gx |= y|x.Fx & y|x.Gx

And in particular from T9

T12. |- ~y|x.Fx V ~y|x.~Fx
T13. |- ~y|x.Fx & ~Fx 

These are in fact equivalent (IFF) while T12 can be extended to the fundamental disjunction

T14. |- y|x.Fx & ~y|x.~Fx V y|x.~Fx & ~y|x.Fx V ~y|x.Fx & ~y|x.~Fx

This is a fundamental result, and can be restated with the help of some definitional equivalences

A23. |- y|x.Fx & ~y|x.~Fx   IFF y|x.+Fx
A24. |- y|x.~Fx & ~y|x.Fx   IFF y|x.-Fx 
A25. |- ~y|x.Fx & ~y|x.~Fx IFF y|x.?Fx

as follows

T15. |- y|x.+Fx V y|x.-Fx  V y|x.?Fx

Convenient readings here are as follows

"+Fx" = "Fx is true"         = "Fx is verified"
"-Fx" = "Fx is false"        = "Fx is falsified"
"?Fx" = "Fx is undecided" = "Fx is undetermined"

Note that we obtain these possibilities from the consideration of the possibilities involved in y|x.Fx, y|x.~Fx, ~y|x.Fx, and ~y|x.~Fx, that concern the possible substitutions in formulas.

Also, it is at this point useful to note the following two theorems

T16. |- |x.Fx   = |x.+Fx
T17. |- |x.~Fx = |x.-Fx V ?Fx

If we call ~ 'weak negation' or 'denial' and - 'strong negation' or negation we see that denial amounts to the disjunction of negation and undecidedness. Thus strong negation is a refinement of weak negation, and strong negation and undecidedness can be avoided and replaced by weak negation.

2. Extended semantics - propositional logic

We can also consider this for Propositional Logic and refine ~ by adding - and ?. By writing the three possibilities explicitly as rows we can set up the following fundamental truth-table for Extended Propositional Logic or EPL:

That was involving weak negation. With strong negation there is:

This easily yields theorems of EPL like the following ones

T19. |- +P IFF ~-P & ~?P
T20. |- -P IFF ~+P & ~?P
T21. |- ?P IFF ~-P & ~+P

T22. |- P   IFF +P
T23. |- ~P IFF -P V ?P

and more. The above tables are fundamental, but need some supplementation to take care of the relations between the two kinds of negation and uncertainty, for the latter allows a sharpening of standard PL-equivalence, using this:

A. |- pEq IFF +p&+q V -p&-q V ?p&?q 

This corresponds to the following truth-table that pinpoints the difference and similarities between E and IFF (with the roles of uppercas1 and undercast changed for the moment, for clarity):

	piffq	-piff-q	?piff?q	(pEq&-pE-q)E (-pE-q&?qE?p)	(pEq&?pE?q)E (-pE-q&pEq)	pEq
+p +q	1	1	1	1	1	1
+p -q
+p ?q		1
-p +q	1		1
-p -q	1	1	1	1	1	1
-p ?q
?p +q		1
?p -q	1
?p ?q	1	1	1	1	1	1

There are related theorems about uncertainties, both simple and of statements involving binary logical connectives:

T24. |- ~?P IFF -?P
T25. |- ~?~P
T26. |- ?-P IFF ?+P

T27. |- +(P&Q) IFF +P&+Q
T28. |- -(P&Q) IFF -PV-Q 
T29. |- ?(P&Q) IFF +P&?Q V +Q&?P V ?P&?Q
 
T30. |- +(PVQ) IFF +PV+Q
T31. |- -(PVQ) IFF -P&-Q 
T32. |- ?(PVQ) IFF -P&?Q V -Q&?P V ?P&?Q

Here are the basic tables for conjunctions and disjunctions collected in one table:

	(p&q)	-(p&q)	?(p&q)	(pVq)	-(pVq)	?(pVq)
p  q	1			1
p -q		1		1
p ?q			1	1
-p  q		1		1
-p -q		1			1
-p ?q		1				1
?p  q			1	1
?p -q		1				1
?p ?q			1			1

It is also interesting to note - in the middle two columns - that in the most interesting case of uncertainties of disjunctions and conjunctions of p and q, it does not follow which of the two propositions is uncertain:

	-?(p&q)&-?(pVq)	?(p&q)&-?(pVq)	-?(p&q)&?(pVq)	?(p&q)&?(pVq)
+p+q	1
+p -q	1
+p ?q		1
-p +q	1
-p -q	1
-p ?q			1
?p +q		1
?p -q			1
?p ?q				1

All of this can be developed much further, and be extended to quantification theory and predicates to similar effect. (This has been done to some extend in my M.A.-thesis and also by A.A. Zinoviev in "Logische Sprachregeln", though his approach to semantics differs from the one given here.)

3. Extended semantics - identities and elements

For the moment, I take that for granted, and return to the basics of BL, that are founded on substitution and equalities, and uses these to define being an element, as follows:

Abstractions and elements

|- |x:S[x] = |x:T[x]                      IFF y|x. S[x] IFF T[x]
|- (a₁ .. a_n) e |x₁ .. x_n. S[x₁ .. x_n]  IFF a₁|x₁ .. a_n|x_n. S[x₁ .. x_n]

For having extended logic by distinguishing weak and strong negation, while keeping the valuations binary, immediately involves questions about identities and elements.

Since the problems here are fundamental, I consider them in some more detail. First there is this proof that (Et)(t=Ø) which says in English that there is a void class:

T33. (Et)(t=Ø)
Proof:
1. t=Ø         IFF t=|x: x≠x                          by definition
2. (Et)(t=Ø) IFF |t:(t=Ø) ≠ Ø                      by def of (Et).
                 IFF |t:(t=|x: x≠x) ≠  |x: x≠x      by (1)
3. |x: x≠x = |x: x≠x                                   by = rules
4. |x: x≠x e |t:(t=|x: x≠x)                          by def e and (3)
5. |x: x≠x e |x: x≠x  --> |x: x≠x  ≠ |x: x≠x    by def e and |
6. ~(|x: x≠x e |x: x≠x)                               by (5) and (3)
7. |t: (t=|x: x≠x) ≠  |x: x≠x                        by (6) and (4)
8. (Et)(t=Ø)                                              by (2) and (7). QED.

Next, it is also easy to prove a theorem of standard logic

T34. (x)(x=x)
Proof:
1. |x:(x=x) ≠ |x:(x≠x)  Easy theorem
2. |x:(x=x) ≠ Ø           by (1) and def Ø 
3. |x:(x≠x) = Ø           by (2) and (1)  
4. (x)(x=x)                 by (3) and def (x). QED.

Now we want to know whether identity, that is fundamental in BL, behaves with respect to the two negations we distinguished, and to uncertainty. Here is the answer:

T35. |x:?(x=x) = |x:-(x=x) = Ø
Proof:
1. |x:Fx≠Ø V |x:~Fx≠Ø            Easy theorem
2. |x:FxV~Fx ≠ Ø                    By (1)
3. |x:(Fx&~Fx) = Ø                 By (2)
4. |x:~Fx = |x:-Fx U |x:?Fx      By (2) and definitions
5. |x:~(x=x) = Ø                    By definition of Ø
6. |x:?(x=x) = |x:-(x=x) = Ø     By (2), (3), (4) and (5). Qed.

Hence, if we say that a predicate or statement is sharp if its weak and strong negation coincide, which is the same as the fact that the predicate or statement does not truly admit of uncertainties, we have derived that the fundamental relation of identity is sharp.

4. Extended semantics - elements of themselves and paradox

But we also had elementhood, which is definable in BL using substitution and identity, and which easily and rapidly involves fundamental and paradoxical problems.

For reading 'e' as 'is an element of', it is clear this allows us to formulate statements of the forms 'xey' and '~(xez)', with obvious readings, and abstracts thereof, such as '|y: xey' i.e. the things x is an element of and the like. But a fortiori, we then also have statements of the forms 'xex' and '~(xex)' and abstracts like '|x: ~(xex)', which is read as 'the things that are not element of themselves'.

Well, let's call this abstract in Russell's honor 'r' a.k.a. the Russell set i.e. the set of things that is not element of itself, and ask ourselves this question: Is r an element of itself or not? The problem, also intuitively, is that, on the one hand, one would argue that since r is the set of things that are not their own elements, it should be element of itself, rather like |x:x=x is element of itself, since clearly |x:x=x e |x:x=x IFF (|x:x=x IFF |x:x=x) and the RHS of this is a theorem.

But suppose rer. Then it follows that |x: ~(xex) e |x: ~(xex) and therefore, by the principles of substitution, that ~(|x: ~(xex) e |x: ~(xex)). I.e. if r is an element of itself, it is not an element of itself. That is a good answer, so one may incline that the case is settled, and conclude that ~(rer).

Now suppose ~(rer). Well, it would seem that, in general, y|x.Fx V y|x.~Fx. Thence, since ~(|x: ~(xex)) e (|x: ~(xex)) it should follow that (|x: ~(xex)) e (|x: (xex)). However, if it does we have straightaway by the principles of substitution, that (|x: ~(xex e |x: ~(xex)) and thence we have the direct contradiction we derived above already: ~(|x: ~(xex) e |x: ~(xex)).

Paradox! Contradiction! Here Frege got shipwrecked in his great project of basing mathematics on logic, and here Russell felt forced to invent the theory of types and introduce special assumptions to avoid his own paradox, since a logical system in which a contradiction can be derived has exploded: The fundamental logical sin is inconsistency.

Given the above, the reader will have seen that what we in fact reject is the natural looking assumption that was involved in deriving the second and contradictory part of Russell's paradox, namely y|x.Fx V y|x.~Fx. And indeed, what we have assumed above is the weaker and more comprehensive ~y|x.Fx V ~y|x.~Fx, that allows the possibility of ~y|x.Fx & ~y|x.~Fx i. e. ly: ?Fx. 

This blocks the derivation of the paradox and introduces the possibility of concluding that, in fact, in BL ?(rer) i.e. ?(y|x.~(xex)). That is: The Russell-set is neither element of itself nor element of its complement and therefore the statement that it is is undecided and is neither true nor false. Moreover, it is provably so in BL, and we have in fact given the outline of the proof.

But this poses another problem, namely how to account for undecided classes and the like in BL. I give in the present note only the solution in general terms:

T36. (|x :+(xex) e |x :+(xex)) 
T37. (|x :-(xex) e |x :?(xex))
T38. (|x :?(xex) e |x :-(xex)) 

The class of things that verily belong to themselves belongs to itself; the class of things that verily belong not to themselves belongs to the class of things of which it is uncertain whether they belong to themselves; and the class of things of which it is uncertain whether they belong to themselves belongs to the class of things that verily belong not to themselves.