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@ -3,6 +3,7 @@
#show outline.entry.where(level: 1): it => {
upper(it)
}
#set text(size: 8pt)
#set heading(numbering: "1.")
@ -153,7 +154,7 @@ $[L_1, L_2, L_9, M P]: (A->B)->( not B-> not A)$. What does it mean? It's the
so-called *Contraposition Law*.
Note. The following rule form of Contraposition Law is called *Modus Tollens*:
$[L_1, L_2, L_9, M P]: A->B, not B tack.r not A, or, ((A->B; not B)/(not A)$ // TODO: factcheck
$[L_1, L_2, L_9, M P]: A->B, not B tack.r not A, or, ((A->B; not B)/(not A))$ // TODO: factcheck
=== Theorem 2.4.4
@ -209,14 +210,13 @@ L_11, M P]: (A->B)->(( not A->B)->B)$.
=== Theorem 2.6.3
$[L_1-L_11, M P]: ( not B-> not A)->(A->B)$. Hence, $[L_1-L_11, M P]: (A->B) <-> ( not B-> not A)$.
$[L_1-L_11, M P]: (not B-> not A)->(A->B)$. Hence, $[L_1-L_11, M P]: (A->B) <-> ( not B-> not A)$.
=== Theorem 2.6.3
_(another one with the same number of because numbering error (it seems like it))_
$[L_1-L_9, L_11, M P]: ˫ not (A and B)-> not A or not B$ . Hence, $[L_1-L_9, L_11, M P]: ˫
not (A and B)<-> not A or not B$ .
$[L_1-L_9, L_11, M P]: tack.r not (A and B)-> not A or not B$. Hence, $[L_1-L_9, L_11, M P]: tack.r not (A and B)<-> not A or not B$ .
=== Theorem 2.6.4
@ -242,7 +242,7 @@ L_11, M P]$.
The axiom $L_9$ cannot be proved in $[L_1-L_8, L_10, M P]$.
== Replacement Theorem 1
=== Replacement Theorem 1
Let us consider three formulas: $B$, $B'$, $C$, where $B$ is a sub-formula of
$C$, and $o(B)$ is a propositional occurrence of $B$ in $C$. Let us denote by
@ -259,54 +259,54 @@ formula obtained from $C$ by replacing $o(B)$ by B'. Then, in the minimal logic,
$[L_1-L_9, L_12-L_15, M P, G e n]: B<->B' tack.r C<->C'$.
== Theorem 3.1.1
=== Theorem 3.1.1
$[L_1, L_2, L_12, L_13, M P]: forall x B(x) -> exists x B(x)$. What does it
mean? It prohibits "empty domains".
== Theorem 3.1.2
=== Theorem 3.1.2
+ $[L_1, L_2, L_12, L_14, M P, G e n]: forall x(B->C)->(forall x B -> forall x C)$.
+ $[L_1, L_2, L_12-L_15, M P, G e n]: forall x(B->C)->(exists x B->exists x C)$.
== Theorems 3.1.3
=== Theorems 3.1.3
If $F$ is any formula, then:
+ (U-introduction) $[G e n]: F(x) tack.r forall x F(x)$.
+ (U-elimination) $[L_12, M P, G e n]: forall x F(x) tack.r F(x)$.
+ (E-introduction) $[L_13, M P, G e n]: F(x) tack.r exists z(x+z+1=y).x F(x)$.
+ (E-introduction) $[L_13, M P, G e n]: F(x) tack.r exists x F(x)$.
== Theorems 3.1.4
=== Theorems 3.1.4
If $F$ is any formula, and $G$ is a formula that does not contain free
occurrences of $x$, then:
+ (U2-introduction) $[L_14, M P, G e n] G -> F (x) tack.r G -> forall x F (x)$.
+ (U2-introduction) $[L_14, M P, G e n]: G->F (x) tack.r G->forall x F (x)$.
+ (E2-introduction) $[L_15, M P, G e n]: F(x)->G tack.r exists x F (x)->G$.
== Theorem 3.1.5
=== Theorem 3.1.5
+ $[L_1, L_2, L_5, L_12, L_14, M P, G e n]: forall x forall y B(x,y) <-> forall y forall x B(x,y)$
+ $[L_1, L_2, L_5, L_13, L_15, M P, G e n]: exists x exists y B(x,y) <-> exists y exists x B(x,y)$.
+ $[L_1, L_2, L_12-L_15, M P, G e n]: exists x forall y B(x,y) <-> forall y exists x B(x,y)$.
== Theorem 3.1.6
=== Theorem 3.1.6
If the formula $B$ does not contain free occurrences of $x$, then
$[L_1-L_2, L_12-L_15, M P, G e n]: (forall x B)<->B;(exists x B)<->B$, i.e.,
quantifiers $forall x; exists x$ can be dropped or introduced as needed.
== Theorem 3.2.1
In the classical logic, $[L_1-L_15, M P, G e n]: not x not B forall <-> x B$.
== Theorem 3.3.1
== Formulas Containing Negations and a Single Quantifier
=== Theorem 3.2.1
$[L_1-L_15, M P, G e n]: not x not B forall <-> x B$.
=== Theorem 3.3.1
+ $[L_1-L-5, L_12, L_14, M P, G e n]: forall x(B and C)<-> forall x B and forall x C$.
+ $[L_1, L_2, L_6-L_8, L_12, L_14, M P, G e n]: tack.r forall x B or forall x C -> forall x(B or C)$.
The converse formula $forall x(B or C)-> forall x B or forall x C$ cannot be
true.
== Theorem 3.3.2
=== Theorem 3.3.2
+ $[L_1-L_8, L_12-L_15, M P, G e n]: exists x(B or C)<-> exists x B or exists x C$
+ $[L_1-L_5, L_13-L_15, M P, G e n]: exists x(B and C)-> exists x B and exists C$.
@ -316,22 +316,20 @@ In the classical logic, $[L_1-L_15, M P, G e n]: not x not B forall <-> x B$.
= Three-valued logic
For example, let us consider a kind of "three-valued logic", where 0 means
"`false`", 1 -- "`unknown`" (or `NULL` -- in terms of SQL), and 2 means "true".
"`false`", 1 -- "`unknown`" (or `NULL` -- in terms of SQL), and 2 means "`true`".
Then it would be natural to define "truth values" of conjunction and disjunction
as
$A and B=min ( A, B)$ ;
$A and B=min(A, B)$;
$A or B=max (A , B)$ .
$A or B=max(A, B)$ .
But how should we define "truth values" of implication and negation?
#table(
columns: 5, $A$, $B$, $A and B$, $A or B$, $A->B$, $0$, $0$, $0$, $0$, $i_1$, $0$, $1$, $0$, $1$, $i_2$, $0$, $2$, $0$, $2$, $i_3$, $1$, $0$, $0$, $1$, $i_4$, $1$, $1$, $1$, $1$, $i_5$, $1$, $2$, $1$, $2$, $i_6$, $2$, $0$, $0$, $2$, $i_7$, $2$, $1$, $1$, $2$, $i_8$, $2$, $2$, $2$, $2$, $i_9$,
#grid(
columns: 2, gutter: 2em, table(
columns: 5, $A$, $B$, $A and B$, $A or B$, $A->B$, $0$, $0$, $0$, $0$, $2$, $0$, $1$, $0$, $1$, $2$, $0$, $2$, $0$, $2$, $2$, $1$, $0$, $0$, $1$, $2$, $1$, $1$, $1$, $1$, $2$, $1$, $2$, $1$, $2$, $2$, $2$, $0$, $0$, $2$, $0$, $2$, $1$, $1$, $2$, $1$, $2$, $2$, $2$, $2$, $2$,
), table(columns: 2, $A$, $not A$, $0$, $2$, $1$, $1$, $2$, $0$),
)
#table(columns: 2, $A$, $not A$, $0$, $i_10$, $1$, $i_11$, $2$, $i_12$)
= Model interpreation
== Interpretation of a language