page 135 LOGIC IN WHITEHEAD`S UNIVERSAL ALGEBRA
... Continuing with logical structures and logical forms, Whitehead recalls that in Principia Mathematica, logic amounts to the study of propositional forms like a∨b and a·b whose arguments are propositions, and to the investigation of the mingling of these forms. The propositions that appear in these f ...
... Continuing with logical structures and logical forms, Whitehead recalls that in Principia Mathematica, logic amounts to the study of propositional forms like a∨b and a·b whose arguments are propositions, and to the investigation of the mingling of these forms. The propositions that appear in these f ...
Conditional Statements
... Example 1: Vertical angles are congruent. can be written as... Conditional Statement: If two angles are vertical, then they are congruent. ...
... Example 1: Vertical angles are congruent. can be written as... Conditional Statement: If two angles are vertical, then they are congruent. ...
John Nolt – Logics, chp 11-12
... The operators '•' and ' 0 ' are thus akin, respectively, to universal and existential quantifiers over a domain of possible worlds. So, for example, to say that it is necessary that 2 + 2 = 4 is to say that in all possible worlds 2 + 2 = 4; and to say that it is possible for the earth to be destroye ...
... The operators '•' and ' 0 ' are thus akin, respectively, to universal and existential quantifiers over a domain of possible worlds. So, for example, to say that it is necessary that 2 + 2 = 4 is to say that in all possible worlds 2 + 2 = 4; and to say that it is possible for the earth to be destroye ...
Design and Analysis of Cryptographic Protocols
... The major problems with BAN logic are: 4.1 Problems with protocol idealization The BAN logic requires the protocol description to be rewritten in the language of the logic. The problem is that there are no formal rules how to achieve this. In the process of translation from one expression language i ...
... The major problems with BAN logic are: 4.1 Problems with protocol idealization The BAN logic requires the protocol description to be rewritten in the language of the logic. The problem is that there are no formal rules how to achieve this. In the process of translation from one expression language i ...
An Interpolating Theorem Prover
... leftmost global term in σ, and let σ • stand for the rightmost global term in σ (with respect to (A, B)). We observe that A implies x = • σ and y = σ • , since all the equalities to the left of • σ and to the right of σ • must come from A. Thus, A gives us solutions for x and y as global terms. More ...
... leftmost global term in σ, and let σ • stand for the rightmost global term in σ (with respect to (A, B)). We observe that A implies x = • σ and y = σ • , since all the equalities to the left of • σ and to the right of σ • must come from A. Thus, A gives us solutions for x and y as global terms. More ...
pdf
... rooted at [>'] contained in'M' such that for all the frontier nodes t of N, qeL'(t) (resp. and for all interior nodes u of N, peL'(u)). Proof. We give the proof for AFq. The proof for A(p U q) is similar. We first assume that in the original structure M, each node has a finite number of successors. ...
... rooted at [>'] contained in'M' such that for all the frontier nodes t of N, qeL'(t) (resp. and for all interior nodes u of N, peL'(u)). Proof. We give the proof for AFq. The proof for A(p U q) is similar. We first assume that in the original structure M, each node has a finite number of successors. ...
A Conditional Logical Framework *
... could be hardly captured by a rigid type discipline, where bad terms and hypotheses are ruled out a priori, see e.g. [NPP08]. In this paper we develop all the metatheory of LFK . In particular, we prove subject reduction, strong normalization, confluence; this latter under the sole assumption that t ...
... could be hardly captured by a rigid type discipline, where bad terms and hypotheses are ruled out a priori, see e.g. [NPP08]. In this paper we develop all the metatheory of LFK . In particular, we prove subject reduction, strong normalization, confluence; this latter under the sole assumption that t ...
PARADOX AND INTUITION
... What does it really mean that some consequences of a given theorem are counterintuitive? Does it imply that there is something wrong with the axioms we have accepted? Or may be, there is something hidden in the arguments (rules of inference) applied in the proof of (allegedly) counterintuitive theor ...
... What does it really mean that some consequences of a given theorem are counterintuitive? Does it imply that there is something wrong with the axioms we have accepted? Or may be, there is something hidden in the arguments (rules of inference) applied in the proof of (allegedly) counterintuitive theor ...
Logic and the Axiomatic Method
... assurance that results are correct. In many cases they also give more general results. For example, the Egyptians and Hindus knew by experiment that if a triangle has sides of lengths 3, 4, and 5, it is then a right triangle. But the Greeks proved that if a triangle has sides of lengths a, b, ...
... assurance that results are correct. In many cases they also give more general results. For example, the Egyptians and Hindus knew by experiment that if a triangle has sides of lengths 3, 4, and 5, it is then a right triangle. But the Greeks proved that if a triangle has sides of lengths a, b, ...
? A Unified Semantic Framework for Fully
... Various sequent calculi that seem to have completely different natures belong to the family of basic systems. For example, this includes standard sequent calculi for modal logics, as well as the usual multiple-conclusion systems for intuitionistic logic, its dual, and bi-intuitionistic logic. On the ...
... Various sequent calculi that seem to have completely different natures belong to the family of basic systems. For example, this includes standard sequent calculi for modal logics, as well as the usual multiple-conclusion systems for intuitionistic logic, its dual, and bi-intuitionistic logic. On the ...
vmcai - of Philipp Ruemmer
... ΓL =def {φ | bφcL ∈ Γ } and ΓR =def {φ | bφcR ∈ Γ }, which extract the L/Rparts of a set Γ of labelled formulae. A sequent Γ ` ∆ I I is valid if (i) the sequent ΓL ` ∆L , I is valid, (ii) the sequent ΓR , I ` ∆R is valid, and (iii) the constants and uninterpreted predicate/functions in I occur in bo ...
... ΓL =def {φ | bφcL ∈ Γ } and ΓR =def {φ | bφcR ∈ Γ }, which extract the L/Rparts of a set Γ of labelled formulae. A sequent Γ ` ∆ I I is valid if (i) the sequent ΓL ` ∆L , I is valid, (ii) the sequent ΓR , I ` ∆R is valid, and (iii) the constants and uninterpreted predicate/functions in I occur in bo ...
Classical first-order predicate logic This is a powerful extension
... A formula with free variables is neither true nor false in a structure M , because the free variables have no meaning in M . It’s like asking ‘is x = 7 true?’ We get stuck trying to evaluate a predicate formula in a structure in the same way as a propositional one, because the structure does not fix ...
... A formula with free variables is neither true nor false in a structure M , because the free variables have no meaning in M . It’s like asking ‘is x = 7 true?’ We get stuck trying to evaluate a predicate formula in a structure in the same way as a propositional one, because the structure does not fix ...
Kripke Models Built from Models of Arithmetic
... also uses the Gödel–Carnap Fixed Point Lemma). The proof of PrL(PA) ⊆ GL (arithmetical completeness) is due to Robert Solovay [9]. Given a modal formula A with 0GL A, we need a realization ∗ with 0PA A∗ . The idea of Solovay’s proof is to simulate in PA a Kripke model M = h{1, . . . , n}, R, V i fo ...
... also uses the Gödel–Carnap Fixed Point Lemma). The proof of PrL(PA) ⊆ GL (arithmetical completeness) is due to Robert Solovay [9]. Given a modal formula A with 0GL A, we need a realization ∗ with 0PA A∗ . The idea of Solovay’s proof is to simulate in PA a Kripke model M = h{1, . . . , n}, R, V i fo ...
Logic and Proof Jeremy Avigad Robert Y. Lewis Floris van Doorn
... Aristotle observed that the correctness of this inference has nothing to do with the truth or falsity of the individual statements, but, rather, the general pattern: Every A is B. Every B is C. Therefore every A is C. We can substitute various properties for A, B, and C; try substituting the propert ...
... Aristotle observed that the correctness of this inference has nothing to do with the truth or falsity of the individual statements, but, rather, the general pattern: Every A is B. Every B is C. Therefore every A is C. We can substitute various properties for A, B, and C; try substituting the propert ...
The logic of negationless mathematics
... In principle logical theorems can be dispensed with. Their purpose is merely to enable abbreviations in the mathematical process. Instead of a large quantity of applications of the logical axioms one application of a logical theorem may be used. The mathematician will perhaps say that he is not reas ...
... In principle logical theorems can be dispensed with. Their purpose is merely to enable abbreviations in the mathematical process. Instead of a large quantity of applications of the logical axioms one application of a logical theorem may be used. The mathematician will perhaps say that he is not reas ...
