logical axiom
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
Normalised and Cut-free Logic of Proofs
... The old question discussed by Gödel in 1933/38 concerning the intended provability semantics of the classical modal logic S4 and intuitionistic logic IPC was finally settled by the logic of proofs introduced by Artemov [2001]. The logic of proofs Lp is a natural extension of classical propositional ...
... The old question discussed by Gödel in 1933/38 concerning the intended provability semantics of the classical modal logic S4 and intuitionistic logic IPC was finally settled by the logic of proofs introduced by Artemov [2001]. The logic of proofs Lp is a natural extension of classical propositional ...
Discrete Computational Structures (CS 225) Definition of Formal Proof
... 2. A result of applying one of the logical equivalency rules (text, p. 35) to a previous statement in the proof. 3. A result of applying one of the valid argument forms (text, p. 61) to one or more previous statements in the proof. ...
... 2. A result of applying one of the logical equivalency rules (text, p. 35) to a previous statement in the proof. 3. A result of applying one of the valid argument forms (text, p. 61) to one or more previous statements in the proof. ...
INTLOGS16 Test 2
... Note: Here, in keeping with the new notation introduced in class, φ(y) is a formula in which y is free. In addition, we stipulate that x is not free in φ(y). Q2 As you know, we have introduced the following numerical quantifiers: ∃=k , ∃≤k , ∃≥k , where of course k ∈ Z + . This allows us for instanc ...
... Note: Here, in keeping with the new notation introduced in class, φ(y) is a formula in which y is free. In addition, we stipulate that x is not free in φ(y). Q2 As you know, we have introduced the following numerical quantifiers: ∃=k , ∃≤k , ∃≥k , where of course k ∈ Z + . This allows us for instanc ...
Normal Forms
... Input: a formula F Output: an equisatisfiable, rectified, closed formula in Skolem form ∀y1 . . . ∀yk G where G is quantifier-free 1. Rectify F by systematic renaming of bound variables. The result is a formula F1 equivalent to F . 2. Let y1 , y2 , . . . , yn be the variables occurring free in F1 . ...
... Input: a formula F Output: an equisatisfiable, rectified, closed formula in Skolem form ∀y1 . . . ∀yk G where G is quantifier-free 1. Rectify F by systematic renaming of bound variables. The result is a formula F1 equivalent to F . 2. Let y1 , y2 , . . . , yn be the variables occurring free in F1 . ...
Howework 8
... P rov be a provability predicate for the theory Q and X and Y be formulas in the Q. Assume |=Q P rov(dXe) ⊃ Y and |=Q P rov(dY e) ⊃ X ...
... P rov be a provability predicate for the theory Q and X and Y be formulas in the Q. Assume |=Q P rov(dXe) ⊃ Y and |=Q P rov(dY e) ⊃ X ...
EECS 203-1 – Winter 2002 Definitions review sheet
... • Propositional variable and propositional expression: A propositional variable is just a name, like p, q, . . .. A propositional expression is either a propositional variable, or a formula in one of the forms P ∧ Q, P ∨ Q, ¬P , P → Q, or P ↔ Q, where P and Q are themselves propositional expressions ...
... • Propositional variable and propositional expression: A propositional variable is just a name, like p, q, . . .. A propositional expression is either a propositional variable, or a formula in one of the forms P ∧ Q, P ∨ Q, ¬P , P → Q, or P ↔ Q, where P and Q are themselves propositional expressions ...
1 Quantifier Complexity and Bounded Quantifiers
... So far we have used ordinary quantifiers ∀ and ∃. In order to study quantifier complexity, we now introduce bounded versions, defined here: (∀y ≤ t)A(y) ↔ (∀y)(y ≤ t → A(y)) (∃y ≤ t)A(y) ↔ (∃y)(y ≤ t ∧ A(y)) where t is a term not involving y. Define a formula to be ∆0 if all of its quantifiers are bounde ...
... So far we have used ordinary quantifiers ∀ and ∃. In order to study quantifier complexity, we now introduce bounded versions, defined here: (∀y ≤ t)A(y) ↔ (∀y)(y ≤ t → A(y)) (∃y ≤ t)A(y) ↔ (∃y)(y ≤ t ∧ A(y)) where t is a term not involving y. Define a formula to be ∆0 if all of its quantifiers are bounde ...
The Origin of Proof Theory and its Evolution
... last item, i.e. there cannot be two lists that agree on all but the last item and disagree on the last item. A relation is an arbitrary set of lists. A collection of objects satisfies a relation if and only if the list of those objects is a member of this set. Logical connectives are { , } for the p ...
... last item, i.e. there cannot be two lists that agree on all but the last item and disagree on the last item. A relation is an arbitrary set of lists. A collection of objects satisfies a relation if and only if the list of those objects is a member of this set. Logical connectives are { , } for the p ...
271HWPropLogic
... Homework Prop Logic – Inference 1. Consider the statement “The car is either at John’s house or at Fred’s house. If the car is not at John’s house then it must be at Fred’s house”. Describe a set of propositional letters which can be used to represent this statement. Describe the statement using pro ...
... Homework Prop Logic – Inference 1. Consider the statement “The car is either at John’s house or at Fred’s house. If the car is not at John’s house then it must be at Fred’s house”. Describe a set of propositional letters which can be used to represent this statement. Describe the statement using pro ...
Propositional Calculus - Syntax
... → Implication, if then ↔ Double implication, if and only if ⊗ Exclusive Or ...
... → Implication, if then ↔ Double implication, if and only if ⊗ Exclusive Or ...