The Diagonal Lemma Fails in Aristotelian Logic
... are not true if ~(∃x)Fx. (Nor are they false.) The author of truth-relevant logic probably never realized that his system was a propositional counterpart of the traditional Aristotelian logic! He arrived at it from a different angle, the angle of relevance. But truth-relevant logic can be extended n ...
... are not true if ~(∃x)Fx. (Nor are they false.) The author of truth-relevant logic probably never realized that his system was a propositional counterpart of the traditional Aristotelian logic! He arrived at it from a different angle, the angle of relevance. But truth-relevant logic can be extended n ...
MATH 4110: Advanced Logic
... An excellent student has a clear comprehension of the details of an intricate, non‐trivial mathema cal result: the completeness of first‐order logic with iden ty. They can give a clear and comprehensive outline of the major steps in the proof using their own words and without notes. They have a clea ...
... An excellent student has a clear comprehension of the details of an intricate, non‐trivial mathema cal result: the completeness of first‐order logic with iden ty. They can give a clear and comprehensive outline of the major steps in the proof using their own words and without notes. They have a clea ...
Lesson 1
... This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second). Circumstances according to ...
... This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second). Circumstances according to ...
Tools-Slides-3 - Michael Johnson`s Homepage
... According to logic, ‘logic’ can’t both have 5 letters and not have 5 letters. I learned logic in logic class but I learned ‘logic’ in English class. Using ‘logic’ doesn’t require logic. ...
... According to logic, ‘logic’ can’t both have 5 letters and not have 5 letters. I learned logic in logic class but I learned ‘logic’ in English class. Using ‘logic’ doesn’t require logic. ...
Lesson 2
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
Lesson 2
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
A short article for the Encyclopedia of Artificial Intelligence: Second
... theory and implementation and can detract from the clarity of such implementations. Using a higher-order version of logic programming (Nadathur & Miller, 1990), for example, can remove some of the need for these encodings. Computational higher-order logics have also been used as a kind of meta-langu ...
... theory and implementation and can detract from the clarity of such implementations. Using a higher-order version of logic programming (Nadathur & Miller, 1990), for example, can remove some of the need for these encodings. Computational higher-order logics have also been used as a kind of meta-langu ...
Analysis of the paraconsistency in some logics
... 1. We will say that a theory Γ is contradictory, with respect to ¬, if there exists a formula A such that Γ ` A y Γ ` ¬A; 2. We say that a theory Γ is trivial if ∀A : Γ ` A; 3. We say that a theory is explosive if, when adding to it any couple of contradictory formulas, the theory becomes trivial; 4 ...
... 1. We will say that a theory Γ is contradictory, with respect to ¬, if there exists a formula A such that Γ ` A y Γ ` ¬A; 2. We say that a theory Γ is trivial if ∀A : Γ ` A; 3. We say that a theory is explosive if, when adding to it any couple of contradictory formulas, the theory becomes trivial; 4 ...
Lindenbaum lemma for infinitary logics
... Theorem 1 (Lindenbaum lemma). Let ` be a finitary logic. Then meet-irreducible theories form a basis of Th(`). Finitarity is crucial for the proof of the lemma: it entails that the union of a chain of sets of formulae not proving a formula ϕ still does not prove ϕ, a crucial step in the proof based ...
... Theorem 1 (Lindenbaum lemma). Let ` be a finitary logic. Then meet-irreducible theories form a basis of Th(`). Finitarity is crucial for the proof of the lemma: it entails that the union of a chain of sets of formulae not proving a formula ϕ still does not prove ϕ, a crucial step in the proof based ...
Available on-line - Gert
... R O is inadequate as a system of deontic logic. For example, D2 is intuitively acceptable while D20 (“everything that is the case ought to be permitted”; “if there is slavery, it is forbidden to forbid slavery”) is unacceptable: yet D2 yields D20 by D1. This means that D1 has to be rejected. It is b ...
... R O is inadequate as a system of deontic logic. For example, D2 is intuitively acceptable while D20 (“everything that is the case ought to be permitted”; “if there is slavery, it is forbidden to forbid slavery”) is unacceptable: yet D2 yields D20 by D1. This means that D1 has to be rejected. It is b ...
Assumption Sets for Extended Logic Programs
... worlds h and t are reflexive, and h ≤ t. Simplifying, we can also regard an N 2-model simply as a pair hH, T i, where H is the set of literals verified at world h and T is the set of literal verified at world t. Note that for any such model hH, T i, we always have H ⊆ T . We now consider the relatio ...
... worlds h and t are reflexive, and h ≤ t. Simplifying, we can also regard an N 2-model simply as a pair hH, T i, where H is the set of literals verified at world h and T is the set of literal verified at world t. Note that for any such model hH, T i, we always have H ⊆ T . We now consider the relatio ...
slides
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
on fuzzy intuitionistic logic
... T h e s t a r t i n g point in Fuzzy Intuitionistic Logic is to fuzzify t r u t h . We accept formulae t h a t have different t r u t h values. This corresponds to t h e use of sentences in everyday life; they m a y be t r u e 'in different ways'. By accepting different t r u t h values, we also bre ...
... T h e s t a r t i n g point in Fuzzy Intuitionistic Logic is to fuzzify t r u t h . We accept formulae t h a t have different t r u t h values. This corresponds to t h e use of sentences in everyday life; they m a y be t r u e 'in different ways'. By accepting different t r u t h values, we also bre ...
Logic: Introduction - Department of information engineering and
... Modern Logic teaches us that one claim is a logical consequence of another if there is no way the latter could be true without the former also being true. It is also used to disconfirm a theory if a particular claim is a logical consequence of a theory, and we discover that the claim is false, then ...
... Modern Logic teaches us that one claim is a logical consequence of another if there is no way the latter could be true without the former also being true. It is also used to disconfirm a theory if a particular claim is a logical consequence of a theory, and we discover that the claim is false, then ...
Relational Predicate Logic
... Proofs The rules for predicate logic proofs outlined in Chapter Nine were devised to handle relational predicate logic as well. However, relational predicate logic is more complex as we can encounter lines with more than one quantifier and more than one type of variable. ...
... Proofs The rules for predicate logic proofs outlined in Chapter Nine were devised to handle relational predicate logic as well. However, relational predicate logic is more complex as we can encounter lines with more than one quantifier and more than one type of variable. ...
Natural Deduction Calculus for Quantified Propositional Linear
... While the propositional quantification does not add any expressiveness to the classical logic QPTL is more expressive than PLTL presenting the same potential of expressiveness as linear-time µ-calculus (linear-time propositional temporal fixpoint logic) [Kaivola (1997)], ETL (propositional linear-ti ...
... While the propositional quantification does not add any expressiveness to the classical logic QPTL is more expressive than PLTL presenting the same potential of expressiveness as linear-time µ-calculus (linear-time propositional temporal fixpoint logic) [Kaivola (1997)], ETL (propositional linear-ti ...
Predicate Logic
... • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain of interest consisting of just two people, john and mary, and we know that tall(mary) and tall(john) are true, we can say that X tall(X) is true. ...
... • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain of interest consisting of just two people, john and mary, and we know that tall(mary) and tall(john) are true, we can say that X tall(X) is true. ...
To What Type of Logic Does the "Tetralemma" Belong?
... function φ that expresses affirmation or denial explicitly. Given a proposition A, we can write φ(A) = 1 in order to affirm A, and φ(A) = 0 in order to deny it. The distinction between a proposition per se and its affirmation or denial is closely related to the distinction between what, using a differen ...
... function φ that expresses affirmation or denial explicitly. Given a proposition A, we can write φ(A) = 1 in order to affirm A, and φ(A) = 0 in order to deny it. The distinction between a proposition per se and its affirmation or denial is closely related to the distinction between what, using a differen ...
CLASSICAL LOGIC and FUZZY LOGIC
... matrix describing the membership values of the relation R, i.e., χR(x, y) That is, the matrix R represents the rule IF A, THEN B as a matrix of characteristic (crisp membership) values. ...
... matrix describing the membership values of the relation R, i.e., χR(x, y) That is, the matrix R represents the rule IF A, THEN B as a matrix of characteristic (crisp membership) values. ...
THE HISTORY OF LOGIC
... formula is satisfiable at all, then it is satisfiable in a countable (or finite) domain. He was firmly rooted in the algebraic school, using techiques developed there. Skolem went on the generalize that result in several ways, and to produce more enlightening proofs of them. The results are known as ...
... formula is satisfiable at all, then it is satisfiable in a countable (or finite) domain. He was firmly rooted in the algebraic school, using techiques developed there. Skolem went on the generalize that result in several ways, and to produce more enlightening proofs of them. The results are known as ...
Jean Van Heijenoort`s View of Modern Logic
... results that you obtained. For example, my paper entitled “Über formal unentscheidbare Sätze etc.” also provides a contribution to the set-theoretical relativism held by you. This is because, as shown by a simple calculation*, the consistent, but not wconsistent systems examined on page 190 indicate ...
... results that you obtained. For example, my paper entitled “Über formal unentscheidbare Sätze etc.” also provides a contribution to the set-theoretical relativism held by you. This is because, as shown by a simple calculation*, the consistent, but not wconsistent systems examined on page 190 indicate ...
Logic - Mathematical Institute SANU
... deduction where the premises are true, or acceptable in some sense. A more distant relative is argument, because an argument may, but need not, be deductive. The study of argumentation in general belongs more to rhetoric than to logic, and is far less systematic and exact. Even further removed from ...
... deduction where the premises are true, or acceptable in some sense. A more distant relative is argument, because an argument may, but need not, be deductive. The study of argumentation in general belongs more to rhetoric than to logic, and is far less systematic and exact. Even further removed from ...
Knowledge Representation
... • There is a precise meaning to expressions in predicate logic. • Like in propositional logic, it is all about determining whether something is true or false. • X P(X) means that P(X) must be true for every object X in the domain of interest. • X P(X) means that P(X) must be true for at least on ...
... • There is a precise meaning to expressions in predicate logic. • Like in propositional logic, it is all about determining whether something is true or false. • X P(X) means that P(X) must be true for every object X in the domain of interest. • X P(X) means that P(X) must be true for at least on ...
valid - Informatik Uni Leipzig
... Theorem. If T (4, 5, B, D) is valid in a frame F, then F is a T-Frame (4-, 5-, B-, or D-frame, respectively). Proof for T and T. Assume that F is not a T-frame. We will construct an interpretation based on F that falsifies T . Because F is not a T-frame, there is a world w such that not wRw. Constru ...
... Theorem. If T (4, 5, B, D) is valid in a frame F, then F is a T-Frame (4-, 5-, B-, or D-frame, respectively). Proof for T and T. Assume that F is not a T-frame. We will construct an interpretation based on F that falsifies T . Because F is not a T-frame, there is a world w such that not wRw. Constru ...