EQ 7: Leibniz Axiom and Theorems

(1) Axiom, Leibniz: (e = f) $\Rightarrow$ $E_e^z = E_f^z$

Note 1: difference between Leibniz Inference Rule and the Leibniz Axiom.

The inference rule states that if X = Y in all states then $E_X^z = E_Y^z$ in all states.

The axiom states that if (x = y) in one state then $E_x^z = E_y^z$ in that state.

Note 2: the implication is one way. iow $E_e^z = E_f^z$ $\nRightarrow$ (e = f)

e.g: Let E $\equiv$ false $\wedge$ zee . Here $E_e^z = E_f^z$ but e $\neq$ f

Substitution Theorems:

If an equality conjuncts or implies, or a conjunction with an equality as a conjunct implies , an Expression E containing one side of the equality, you can replace that part of E with the other side of the equality.

(2) Theorem: Substitution 1: (e = f) $\wedge$ $E_e^z \equiv$ (e = f) $\wedge$ $E_f^z$

(3) Theorem: Substitution 2: (e = f) $\Rightarrow$ $E_e^z \equiv$ (e = f) $\Rightarrow$ $E_f^z$

(4) Theorem: Substitution 3: q $\wedge$ (e = f) $\Rightarrow$ $E_e^z \equiv$ q $\wedge$ (e = f) $\Rightarrow$ $E_f^z$

Replace By True Theorems:

If a variable conjuncts or implies, or a conjunction with a variable as a conjunct implies an expression E containing that variable, the occurences of the variable in the expression can be replaced by true

(5) Theorem Replace by true 1: p $\wedge$ $E_p^z \equiv$ p $\wedge$ $E_{true}^z$

(6) Theorem: Replace by true 2: p $\Rightarrow$ $E_p^z \equiv$ p $\Rightarrow$ $E_{true}^z$

(7) Theorem: Replace by true 3: q $\wedge$ p $\Rightarrow$ $E_p^z \equiv$ q $\wedge$ p $\Rightarrow$ $E_{true}^z$

Example:

Prove p $\wedge$ q $\Rightarrow$ ( p $\equiv$ q)

$\hspace{2 cm }$ p $\wedge$ q $\Rightarrow$ ( p $\equiv$ q)

=$\hspace{0.5 cm }$ recognize pattern. conjunction with a variable (p) on one side implying an expression containing the same variable
$\hspace{0.5 cm }$ invoke Theorem (6) above, replace p in expression by true

$\hspace{2 cm }$ p $\wedge$ q $\Rightarrow$ ( true $\equiv$ q)

=$\hspace{0.5 cm }$ recognize pattern. conjunction with a variable (q) on one side implying an expression containing the same variable
$\hspace{0.5 cm }$ invoke Theorem (6) above, replace q in expression by true

$\hspace{2 cm }$ p $\wedge$ q $\Rightarrow$ ( true $\equiv$ true)

=$\hspace{0.5 cm }$ Theorem: true is the identity of $\equiv$, m $\equiv$ m $\equiv$ true

$\hspace{2 cm }$ p $\wedge$ q $\Rightarrow$ true

=$\hspace{0.5 cm }$ Theorem: true is the Right Zero of $\Rightarrow$, m $\Rightarrow$ true $\equiv$ true

$\hspace{2 cm }$ true

Replace By False Theorems:

If an Expression E containing a variable p disjuncts p, implies p or implies a disjunction with p on one side,p can be replaced by false in the expression

(8) Theorem: Replace by False 1: $E_p^z$ $\vee$ p $\equiv$ $E_{false}^z$ $\vee$ p

(9) Theorem: Replace by False 2: $E_p^z$ $\Rightarrow$ p $\equiv$ $E_{false}^z$ $\Rightarrow$ p

(10) Theorem: Replace by False 2: $E_p^z$ $\Rightarrow$ p $\vee$ q $\equiv$ $E_{false}^z$ $\Rightarrow$ p $\vee$ q

(11) Theorem: Shannon: $E_p^z$ $\equiv$ ( p $\wedge$ $E_{true}^z$ ) $\vee$ ( $\neg$ p $\wedge$ $E_{false}^z$ )