Case Study: Corporate Financial Aspect of Twitter Assignment | Online Assignment

Details are on attachments. Please send me the result as both word and pdf fileClassical logic is based on the principle of bivalence (􀳋􀮋􀝪􀽧): Every proposition has exactly one truth
value, either true or false. The laws of excluded middle (φ _ :φ) (􀴉􀮤􀽯), double negation (::φ ! φ) (􀳋
􀥏􀵱􀰆􀽯), etc. are veried from this principle.
Intuitionistic logic, which does not adopt the principle of bivalence, is formulated by natural deduction
without the RAA rule. Note that the laws of excluded middle and double negation are provable using the
RAA rule, as follows.
[:(φ _ :φ)]2
[:(φ _ :φ)]2
[φ]1
φ _ :φ (_I)
? (:E)
:φ (:I)1
φ _ :φ (_I)
? (:E)
φ _ :φ
(RAA)2
[::φ]2 [:φ]1
? (:E)2
φ (RAA)1
::φ ! φ (!I)2
Intuitionistic logic is a logic for constructive proofs.
(Wikipedia) Constructive Proof.
In mathematics, a constructive proof is a method of proof that demonstrates the existence of
a mathematical object by creating or providing a method for creating the object. This is in
contrast to a non-constructive proof, which proves the existence of a particular kind of object
without providing an example.
The following is a famous example of a non-constructive proof of \there exist irrational numbers x and y
such that xy is rational”.
We assume there do not exist irrational numbers x and y such that xy is rational. Then
p
2
p
2
is irrational. Now take x =
p
2
p
2
, and y =
p
2; we have xy = (
p
2
p
2
)
p
2
= 2. This contradicts
the assumption.
This is a proof by contradiction, which is formalized by the RAA rule in natural deduction and cannot be
used in intuitionistic logic.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Natural deduction and sequent calculus for intuitionistic logic)
Denition 6.1.
By \RAA-free derivations” we mean derivations in natural deduction that do not contain the RAA rule.
We say that φ is RAA-free derivable from 􀀀, written as
􀀀 ⊢
RAA-free φ
if and only if there is an RAA-free derivation for 􀀀 ⊢ φ. We say that a formula φ is RAA-free provable in
natural deduction if and only if ⊢
RAA-free φ, that is, there is an RAA-free derivation of φ without hypotheses.
Denition 6.2.
We dene the sequent calculus LJ by the following axioms and rules.
φ; 􀀀 ) φ
(axiom) ?; 􀀀 ) (axiom)
􀀀 ) φ Δ )
􀀀;Δ ) φ ^
(^R)
φ; ; 􀀀 ) Δ
φ ^ ; 􀀀 ) Δ
(^L)
􀀀 ) φ
􀀀 ) φ _
(_R)
􀀀 )
􀀀 ) φ _
(_R)
φ; 􀀀 ) Δ ; ) Δ
φ _ ; 􀀀; ) Δ
(_L)
1
φ; 􀀀 )
􀀀 ) φ !
(!R)
􀀀 ) φ ;Δ )
φ ! ; 􀀀;Δ )
(!L)
φ; 􀀀 )
􀀀 ) :φ
(:R)
􀀀 ) φ
:φ; 􀀀 ) (:L)
􀀀 )
􀀀 ) φ
(weakening R)
􀀀 ) φ φ;Δ )
􀀀;Δ )
(cut)
The notion of LJ-derivations, cut-free LJ-derivations, provability in LJ, and cut-free provability in LJ are
similarly dened as LK (Denitions 4.2 and 4.6).
Roughly speaking LJ is obtained from LK (with certain modication) by imposing restriction that the
right-hand side of each sequent contains none or only one formula.
Exercise 6.3. Let φ = (p _ q) ! :(:p ^ :q) and = :(:p ^ :q) ! (p _ q) (converse of φ).
(1) Write an RAA-free derivation for ⊢
RAA-free φ and an LJ-derivation of ) φ.
(2) Write a derivation for ⊢ with RAA rule (difficult!) and an LK-derivation of ) .
Observation. In intuitionistic logic, p _ q and :(:p ^ :q) are not equivalent.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Kripke semantics for intuitionistic logic)
Denition 6.4 (cf. Def 5.3.1 in [van Dalen], which is for predicate logic).
Let P be the power set of the set of all the propositional variables (i.e., each element of P is a set of
propositional variables). A Kripke model is a triple K = ⟨K;; V ⟩ where ⟨K; ⟩ is a non-empty partially
ordered set (see Note 6.5) and V is a function from K to P such that the condition
(8k; ℓ 2 K)(k ℓ ) V (k) V (ℓ))
holds (this condition is called monotonicity). Elements of K are called nodes. We inductively dene a
relation ⊩ between nodes and formulas. We read \k ⊩ φ” as \the formula φ holds at the node k.”
k ⊩ p () p 2 V (k). (p is a propositional variable)
k ̸⊩ ?.
k ⊩ φ ^ () k ⊩ φ and k ⊩ .
k ⊩ φ _ () k ⊩ φ or k ⊩ .
k ⊩ φ ! () for all ℓ k (ℓ ⊩ φ ) ℓ ⊩ ).
k ⊩ :φ () for all ℓ k (ℓ ̸⊩ φ).
Note 6.5.
⟨K; ⟩ is said to be a partially ordered set if and only if x x (re
exive), (x y and y z imply
x z)(transitive), and (x y and y x imply x = y)(antisymmetric) hold for any x; y; z 2 K.
Example 6.6.
Consider a Kripke model ⟨fa; bg;; V ⟩ where aa, ab, b ̸ a, bb, V (a)=fg, and V (b)=fpg. Then we
have:
b ⊩ p and a ̸⊩ p.
b ̸⊩ :p and a ̸⊩ :p.
b ⊩ ::p and a ⊩ ::p.
Therefore, a ⊩ p ! p, a ⊩ p ! ::p, a ̸⊩ ::p ! p, and a ̸⊩ p _ :p.
2
Exercise 6.7.
(1) Show that a ̸⊩ :(:p ^ :q) ! (p _ q) where a is the node in the model in Example 6.6. (cf. Exercise 6.3
(2))
(2) Find a Kripke model K and a node k 2 K such that k ̸⊩ (p ! q) _ (q ! p).
Note 6.8.
If V is a constant function (i.e., V (k) = V (k′) for all k; k′), then Kripke model ⟨K;; V ⟩ can be considered to
be a classical valuation. Precisely speaking, if a valuation v (see Def. 2.2) is dened by v(p) = 1 , p 2 V (k)
(for some k), then we have
[[φ]]v = 1 if and only if k ⊩ φ
for any formula φ and any node k (this is shown by induction on the length of φ).
Informally Kripke models can be interpreted as follows. Each node is our knowledge or the set of facts
we know. The relation is the passage of time. The monotonicity represents that our knowledge increases
as time goes on. \k ⊩ φ” is read as \we have a proof of φ based on the knowledge k”. For example, the
denition of k ⊩ φ ! re
ects a situation such as follows.
Suppose we have a proof of a theorem \if P ̸= NP, then a cryptosystem X is secure”. This
means that if we can prove P ̸= NP some day in the future, then the security of X is also proved
on that day.
Moreover, due to this interpretation and the above Note 6.8, classical logic can be considered to be the logic
of a person whose knowledge is saturated and has no room for increase.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In the last class, we show:
Theorem 5.4 (Fundamental of classical logic).
The following four conditions are equivalent.
(1) 􀀀 j= φ.
(2) 􀀀 ⊢ φ.
(3) 􀀀 ) φ is provable in LK.
(4) 􀀀 ) φ is cut-free provable in LK.
Especially (if 􀀀 is empty) the following four conditions are equivalent.
(C1) φ is a tautology.
(C2) φ is provable in natural deduction.
(C3) ) φ is provable in LK.
(C4) ) φ is cut-free provable in LK.
If the conditions (C1){(C4) hold, we say that \φ is provable in classical logic” or \φ is a theorem of classical
logic”. Similar result for intuitionistic logic is:
3
Theorem 6.9 (Fundamental of intuitionistic logic).
The following four conditions are equivalent.
(1) (8K : Kripke model)(8k : node in K)
[(
(8 2 􀀀)(k ⊩ )
)
=) k ⊩ φ
]
(2) 􀀀 ⊢
RAA-free φ.
(3) 􀀀 ) φ is provable in LJ.
(4) 􀀀 ) φ is cut-free provable in LJ.
Especially (if 􀀀 is empty) the following four conditions are equivalent.
(I1) k ⊩ φ for any Kripke model K and any node k in K.
(I2) φ is RAA-free provable in natural deduction.
(I3) ) φ is provable in LJ.
(I4) ) φ is cut-free provable in LJ.
Proof (Sketch)
(2 ) 1) is proved similarly to Theorem 3.12, using a basic property of Kripke models: If k ℓ and k ⊩ φ,
then ℓ ⊩ φ (this is shown by induction on the length of φ). (2 , 3) is proved similarly to Theorem 4.5.
(3 , 4) is proved similarly to Theorem 4.7. A proof of (1 ) 3), which is an extension of the proof of
Theorem 5.3, is somewhat complicated. ■
If the above condition (I1){(I4) hold, we say that \φ is provable in intuitionistic logic” or \φ is a theorem
of intuitionistic logic”.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Classifying formulas)
From the denition (particularly the difference between the conditions (C2) and (I2)), it is obvious that
if φ is a theorem of intuitionistic logic, then φ is a theorem of classical logic. Thus formulas are classied
into three classes I, II, and III below.
I: Theorem of intuitionistic logic.
II: Theorem of classical logic, but not a theorem of intuitionistic logic.
III: Not a theorem of classical logic.
Denition 6.10
For each class X (= I, II, III) above, we dene a witness of φ 2 X as follows.
X Witness of φ 2 X
I An RAA-free derivation of φ without hypotheses, or an LJ-derivation of ) φ.
II Pair of the following two:
(IIa) A derivation of φ without hypotheses, or an LK-derivation of ) φ.
(IIb) A Kripke model K and a node k 2 K such that k ̸⊩ φ.
III A valuation v such that [[φ]]v = 0.
Example 6.11.
4
(1) :::p ! :p 2 I. A witness is:
p ) p (axiom)
:p; p ) (:L)
p ) ::p (:R)
:::p; p ) (:L)
:::p ) :p (:R)
) :::p ! :p (!R)
(2) ::p ! p 2 II. A witness is pair of the following.

p ) p (axiom)
) p; :p (:R)
::p ) p (:L)
) ::p ! p (!R)
The Kripke model in Example 6.6 and the node a.
(3) :p ! p 2 III. A witness is a valuation v with v(p) = 0.
5

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