What is...Linear Logic? Introduction Jonathan Skowera
... relation between ⊗ and ⊕ is U ⊗ (V ⊕ W ) ∼ = U ⊗ V ⊕ U ⊗ W . The choice of symbols reflects this relation, but proving it requires a complete list of rules of inference, and we aren’t quite there yet. Implication Using the negative, implication can be defined in analogy with the classical case where ...
... relation between ⊗ and ⊕ is U ⊗ (V ⊕ W ) ∼ = U ⊗ V ⊕ U ⊗ W . The choice of symbols reflects this relation, but proving it requires a complete list of rules of inference, and we aren’t quite there yet. Implication Using the negative, implication can be defined in analogy with the classical case where ...
Discrete Structures & Algorithms Propositional Logic
... In day-to-day speech, sometimes we use “or” as an “exclusive or”. ...
... In day-to-day speech, sometimes we use “or” as an “exclusive or”. ...
Natural Deduction Calculus for Quantified Propositional Linear
... In this paper we continue our investigation of natural deduction framework for non-classical setting, this time tackling propositional linear-time temporal logic extended with propositional quantification [Sistla (1983)]. We follow the notation adopted in [French and Reynolds (2002)] calling this lo ...
... In this paper we continue our investigation of natural deduction framework for non-classical setting, this time tackling propositional linear-time temporal logic extended with propositional quantification [Sistla (1983)]. We follow the notation adopted in [French and Reynolds (2002)] calling this lo ...
Propositional logic, I (Lógica Proposicional, I)
... Validity: A wff is said to be valid if it has the value True under all possible interpretations. Ex. T,T∨P,¬P∨P,P⇒P,P⇒(Q⇒P),((P⇒ Q)⇒P)⇒P » A valid wff is a tautology (it is devoid of meaning about the world). Metatheorem 1: if ¬w is unsatisfiable, then the wff w is valid, and viceversa. ...
... Validity: A wff is said to be valid if it has the value True under all possible interpretations. Ex. T,T∨P,¬P∨P,P⇒P,P⇒(Q⇒P),((P⇒ Q)⇒P)⇒P » A valid wff is a tautology (it is devoid of meaning about the world). Metatheorem 1: if ¬w is unsatisfiable, then the wff w is valid, and viceversa. ...
IS IT EASY TO LEARN THE LOGIC
... it is not generally emphasized the importance of truth tables as the foundation of the basic rules of elementary logic, because logical operations in the process of formulas transformation are artificial and mechanical. Likewise the use of logical rules in general, seems to be more involved in the c ...
... it is not generally emphasized the importance of truth tables as the foundation of the basic rules of elementary logic, because logical operations in the process of formulas transformation are artificial and mechanical. Likewise the use of logical rules in general, seems to be more involved in the c ...
Sequent calculus for predicate logic
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
Slide 1
... is there exist integers p and q with q0 such that r=p/q. A real number that is not rational is called irrational. • Ex.7: Prove that the sum of two rational numbers is rational. • Ex.8: Prove that if n is an integer and n2 is odd, then n is odd. ...
... is there exist integers p and q with q0 such that r=p/q. A real number that is not rational is called irrational. • Ex.7: Prove that the sum of two rational numbers is rational. • Ex.8: Prove that if n is an integer and n2 is odd, then n is odd. ...
Logic and Proof - Collaboratory for Advanced Computing and
... Methods of Proving Theorems Proving implications p → q: Direct proof: Assume p is T, and use rules of inference to prove that q is T Indirect proof: Prove its contrapositive; assume ¬q, and prove ¬p Proof by cases: Prove (p1 ∨ p2) → q by proving (p1 → q) and (p1 → q) • Based on [(p1 ∨ p2) → q ...
... Methods of Proving Theorems Proving implications p → q: Direct proof: Assume p is T, and use rules of inference to prove that q is T Indirect proof: Prove its contrapositive; assume ¬q, and prove ¬p Proof by cases: Prove (p1 ∨ p2) → q by proving (p1 → q) and (p1 → q) • Based on [(p1 ∨ p2) → q ...
Resources - CSE, IIT Bombay
... B is result of MP (contd) If it can be shown that A1, A2, A3,… An-1 |- An Ei and A1, A2, A3,… An-1 |- (An (EiB)) ...
... B is result of MP (contd) If it can be shown that A1, A2, A3,… An-1 |- An Ei and A1, A2, A3,… An-1 |- (An (EiB)) ...
the common rules of binary connectives are finitely based
... τ (p, q, r, s) = qq 2 (s2 s2 )p3 r3 (qq 2 (s2 s2 )p3 )3 as is shown by straight-forward calculation. Theorem 2. If `1 , . . . , `n are independent and f.b. then `1 ∩ . . . ∩ `n is f.b. Example 2. As is well known, |=→ , |=← , |=↔ , |=↑ are f.b. Since these logics are independent according to the abo ...
... τ (p, q, r, s) = qq 2 (s2 s2 )p3 r3 (qq 2 (s2 s2 )p3 )3 as is shown by straight-forward calculation. Theorem 2. If `1 , . . . , `n are independent and f.b. then `1 ∩ . . . ∩ `n is f.b. Example 2. As is well known, |=→ , |=← , |=↔ , |=↑ are f.b. Since these logics are independent according to the abo ...
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... 1 Intuitionistic Logic and Constructive Mathematics It turns out that there is there is a deep connection between the type systems we have been exploring for the lambda calculus, and proof systems for a variety of logic known as intuitionistic logic. Intuitionistic logic is the basis of constructive ...
... 1 Intuitionistic Logic and Constructive Mathematics It turns out that there is there is a deep connection between the type systems we have been exploring for the lambda calculus, and proof systems for a variety of logic known as intuitionistic logic. Intuitionistic logic is the basis of constructive ...