MATH 311W Wksht 1 • A logical statement is a phrase that is
... both. For example in everyday English, “He will have Coke or Pepsi” usually does not include the possibility of having both, but in mathematics: the following statements are true: The number 7 is positive or prime. (True, it is both). The number 2 is even or prime. (True, it is both). • Truth Tables ...
... both. For example in everyday English, “He will have Coke or Pepsi” usually does not include the possibility of having both, but in mathematics: the following statements are true: The number 7 is positive or prime. (True, it is both). The number 2 is even or prime. (True, it is both). • Truth Tables ...
Classicality as a Property of Predicate Symbols
... It is clear from the proof of the weak subformula property that formula F is derivable in L(D) if and only if A⊃F is derivable intuitionistically; where A is conjunction of ∀a1…∀an(R(a1,…,an)∨¬R(a1,…,an)) for all symbols R∈D that are bipolar in F. This relationship can be viewed as translation of L( ...
... It is clear from the proof of the weak subformula property that formula F is derivable in L(D) if and only if A⊃F is derivable intuitionistically; where A is conjunction of ∀a1…∀an(R(a1,…,an)∨¬R(a1,…,an)) for all symbols R∈D that are bipolar in F. This relationship can be viewed as translation of L( ...
Propositional Logic: Normal Forms
... assign true to all marked atoms, and false to the others. If φ is not true under ν, it means that there exists a conjunct P1 ∧ . . . ∧ Pki → P 0 of φ that is false. By the semantics, this can only mean that P1 ∧ . . . ∧ Pki is true but P 0 is false. However, by the definition of ν, all Pi s are mark ...
... assign true to all marked atoms, and false to the others. If φ is not true under ν, it means that there exists a conjunct P1 ∧ . . . ∧ Pki → P 0 of φ that is false. By the semantics, this can only mean that P1 ∧ . . . ∧ Pki is true but P 0 is false. However, by the definition of ν, all Pi s are mark ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... the constants and predicates in I occur in both A and C. More precisely: Lemma 2 (Soundness). If an interpolating QPA+UP sequent Γ Δ I is provable in the calculus, then it is valid. In particular, the sequent ΓL , ΓR ΔL , ΔR is valid in this case. As shown in [2], Lem. 2 holds for the calculus ...
... the constants and predicates in I occur in both A and C. More precisely: Lemma 2 (Soundness). If an interpolating QPA+UP sequent Γ Δ I is provable in the calculus, then it is valid. In particular, the sequent ΓL , ΓR ΔL , ΔR is valid in this case. As shown in [2], Lem. 2 holds for the calculus ...
First-Order Predicate Logic (2) - Department of Computer Science
... the case for finite F coincides with querying relational database instances and can be done very efficiently. It is also the underlying problem of model checking approaches to program verification: F is a representation of a program and one wants to know whether a property expressed by G is true. • ...
... the case for finite F coincides with querying relational database instances and can be done very efficiently. It is also the underlying problem of model checking approaches to program verification: F is a representation of a program and one wants to know whether a property expressed by G is true. • ...
A course in Mathematical Logic
... To know the truth value of the propositional formula (A ∧ B) → C, we need to know the truth values of the propositional variables A, B, C. For example, if A, C are true and B is false, then (A ∧ B) → C is true. The formula A ∧ B is false because B is false, and any implication, whose premises is fa ...
... To know the truth value of the propositional formula (A ∧ B) → C, we need to know the truth values of the propositional variables A, B, C. For example, if A, C are true and B is false, then (A ∧ B) → C is true. The formula A ∧ B is false because B is false, and any implication, whose premises is fa ...
Chapter 4. Logical Notions This chapter introduces various logical
... even be denied that there are any legitimate inductive arguments on the grounds that, once all of the implicit premisses are made explicit, such arguments will be seen to be deductive in nature. But these are not questions that we shall explore. The intuitive notion of valid (deductive) argument is ...
... even be denied that there are any legitimate inductive arguments on the grounds that, once all of the implicit premisses are made explicit, such arguments will be seen to be deductive in nature. But these are not questions that we shall explore. The intuitive notion of valid (deductive) argument is ...
Formal deduction in propositional logic
... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...
... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...
The Natural Number System: Induction and Counting
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
Definability properties and the congruence closure
... property is Lo,,~itself, by LindstrSm's argument coding partial isomorphisms. [] Compare this result with Theorem 4.8 in Krinicki [K]. There are many interesting congruence closed logics, which by the first part of the Corollary can not be sublogics of L,o,o(Th). For example L~o~withK > co1 (these a ...
... property is Lo,,~itself, by LindstrSm's argument coding partial isomorphisms. [] Compare this result with Theorem 4.8 in Krinicki [K]. There are many interesting congruence closed logics, which by the first part of the Corollary can not be sublogics of L,o,o(Th). For example L~o~withK > co1 (these a ...
logica and critical thinking
... Self-defeating paradox: The concept of “all powerful” God Zeno’s paradox: An apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises Semantic paradox: The liar’s paradox The lawyer’s paradox Prisoner’s dilemma Russell’s paradox (Barber’s para ...
... Self-defeating paradox: The concept of “all powerful” God Zeno’s paradox: An apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises Semantic paradox: The liar’s paradox The lawyer’s paradox Prisoner’s dilemma Russell’s paradox (Barber’s para ...
On Action Logic
... of Γ, ϕ, ∆ ` ψ, (3) on the rank of Φ ` ϕ. (The complexity of a formula is the total number of occurrences of logical connectives in this formula.) Then, we can identify ACTω with the system FL plus (*L), (*R). As a consequence, ACTω is a conservative supersystem of all its fragments, defined by a re ...
... of Γ, ϕ, ∆ ` ψ, (3) on the rank of Φ ` ϕ. (The complexity of a formula is the total number of occurrences of logical connectives in this formula.) Then, we can identify ACTω with the system FL plus (*L), (*R). As a consequence, ACTω is a conservative supersystem of all its fragments, defined by a re ...
CSE 452: Programming Languages
... expresses programs in the form of symbolic logic uses a logical inferencing process for reasoning ...
... expresses programs in the form of symbolic logic uses a logical inferencing process for reasoning ...
Propositional Logic
... proposition is a possible “condition'” of the world about which we want to say something. The condition need not be true in order for us to talk about it. In fact, we might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntac ...
... proposition is a possible “condition'” of the world about which we want to say something. The condition need not be true in order for us to talk about it. In fact, we might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntac ...
Techniques for proving the completeness of a proof system
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof system achieves. With a completeness result, a paper about a proof system has more chances to get ...
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof system achieves. With a completeness result, a paper about a proof system has more chances to get ...