Chapter 15 Logic Name Date Objective: Students will use
... The inclusive disjunction is true when one or both propositions are true, since in this case p or q means p or q, or both p and q. i.e. p V q = p or q or both p and q The exclusive disjunction is true when only one of the propositions is true, since in this case p or q means p or q but not both. i.e ...
... The inclusive disjunction is true when one or both propositions are true, since in this case p or q means p or q, or both p and q. i.e. p V q = p or q or both p and q The exclusive disjunction is true when only one of the propositions is true, since in this case p or q means p or q but not both. i.e ...
LOGICAL CONSEQUENCE AS TRUTH-PRESERVATION STEPHEN READ Abstract
... could be reduced to modality: rejecting the truth-functional analysis as permitting irrelevance, they both sought to capture the essence of implication by insisting on a necessary connection of truth-values. But that cannot be done. The paradoxes of strict implication show that Lewis’ and MacColl’s ...
... could be reduced to modality: rejecting the truth-functional analysis as permitting irrelevance, they both sought to capture the essence of implication by insisting on a necessary connection of truth-values. But that cannot be done. The paradoxes of strict implication show that Lewis’ and MacColl’s ...
1 Deductive Reasoning and Logical Connectives
... 2. x and y are people and x likes y, but y doesn’t like x. Note that in the previous examples, one can not describe the statement as being true of false without evaluating the value of variables! For example, if p(x) means “x is prime”, then p(x) would be true if x is 3 but false if x is 4. Before w ...
... 2. x and y are people and x likes y, but y doesn’t like x. Note that in the previous examples, one can not describe the statement as being true of false without evaluating the value of variables! For example, if p(x) means “x is prime”, then p(x) would be true if x is 3 but false if x is 4. Before w ...
The Surprise Examination Paradox and the Second Incompleteness
... The second incompleteness theorem follows directly from Gödel’s original proof for the first incompleteness theorem. As described above, Gödel expressed the statement “this statement has no proof” and showed that, if the theory is consistent, this is a true statement (over N) that has no proof. In ...
... The second incompleteness theorem follows directly from Gödel’s original proof for the first incompleteness theorem. As described above, Gödel expressed the statement “this statement has no proof” and showed that, if the theory is consistent, this is a true statement (over N) that has no proof. In ...
Document
... A formal language for expressing statements. An inference mechanism (a collection of rules) to reason about valid arguments. ...
... A formal language for expressing statements. An inference mechanism (a collection of rules) to reason about valid arguments. ...
STANDARD COMPLETENESS THEOREM FOR ΠMTL 1
... (1) if x, y ∈ F , then x ∗ y ∈ F , (2) if x ∈ F , x ≤ y, then y ∈ F . LEMMA 2.6. For any filter F in a ΠMTL-algebra L, let us define the following equivalence relation in L: x ∼F y iff x → y ∈ F and y → x ∈ F . Then ∼F is a congruence and the quotient L/F is a ΠMTL-algebra. We will denote the equiva ...
... (1) if x, y ∈ F , then x ∗ y ∈ F , (2) if x ∈ F , x ≤ y, then y ∈ F . LEMMA 2.6. For any filter F in a ΠMTL-algebra L, let us define the following equivalence relation in L: x ∼F y iff x → y ∈ F and y → x ∈ F . Then ∼F is a congruence and the quotient L/F is a ΠMTL-algebra. We will denote the equiva ...
Logic is a discipline that studies the principles and methods used in
... A formal language for expressing statements. An inference mechanism (a collection of rules) to reason about valid arguments. ...
... A formal language for expressing statements. An inference mechanism (a collection of rules) to reason about valid arguments. ...
Intuitionistic Logic
... Negation is also defined by means of proofs: p : ¬A says that each proof a of A can be converted by the construction p into a proof of an absurdity, say 0 = 1. A proof of ¬A thus tells us that A has no proof! The most interesting propositional connective is the implication. The classical solution, i ...
... Negation is also defined by means of proofs: p : ¬A says that each proof a of A can be converted by the construction p into a proof of an absurdity, say 0 = 1. A proof of ¬A thus tells us that A has no proof! The most interesting propositional connective is the implication. The classical solution, i ...
Chapter 5: Methods of Proof for Boolean Logic
... The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from ...
... The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from ...
Nonmonotonic Logic - Default Logic
... Given a closed default theory T = hW , Di, its semantics is defined by means of a set of closed formulae (extension) E . I given a set of formulae E , the closure operator ΓT (E ) is the smallest set S s.t. I I I ...
... Given a closed default theory T = hW , Di, its semantics is defined by means of a set of closed formulae (extension) E . I given a set of formulae E , the closure operator ΓT (E ) is the smallest set S s.t. I I I ...
Carnap and Quine on the analytic-synthetic - Philsci
... view of Carnap’s empirical theory of meaning, i.e. meaning as verification of testability. Carnap also rejected Kant’s synthetic a priori, so that all a priori statements are analytic and vice versa. Nevertheless, in one of the later sections the argumentation will hinge on a distinction between the ...
... view of Carnap’s empirical theory of meaning, i.e. meaning as verification of testability. Carnap also rejected Kant’s synthetic a priori, so that all a priori statements are analytic and vice versa. Nevertheless, in one of the later sections the argumentation will hinge on a distinction between the ...
Sense and denotation as algorithm and value
... approach we would expect both of them to have sense, and in fact different senses, since we understand them differently. A natural way to read this version of the liar (6) as an algorithm for computing its truth value leads to the single instruction Step (1). Do step (1); if the value t is returned ...
... approach we would expect both of them to have sense, and in fact different senses, since we understand them differently. A natural way to read this version of the liar (6) as an algorithm for computing its truth value leads to the single instruction Step (1). Do step (1); if the value t is returned ...
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... e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elimination ...
... e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elimination ...
A Concise Introduction to Mathematical Logic
... The way of arguing about formal languages and theories is traditionally called the metatheory. An important task of a metatheoretic analysis is to specify procedures of logical inference by so-called logical calculi, which operate purely syntactically. There are many different logical calculi. The c ...
... The way of arguing about formal languages and theories is traditionally called the metatheory. An important task of a metatheoretic analysis is to specify procedures of logical inference by so-called logical calculi, which operate purely syntactically. There are many different logical calculi. The c ...
Infinity 1. Introduction
... many facts about the world in a single statement. Quantifiers were invented by Frege in 1879 and Peirce and Mitchell in 1883. Peirce and Mitchell devised the notation Σ i xi , meaning that xi is true for some value of i, and Π i xi , meaning that xi is true for all values of i, by a conscious analog ...
... many facts about the world in a single statement. Quantifiers were invented by Frege in 1879 and Peirce and Mitchell in 1883. Peirce and Mitchell devised the notation Σ i xi , meaning that xi is true for some value of i, and Π i xi , meaning that xi is true for all values of i, by a conscious analog ...
HOARE`S LOGIC AND PEANO`S ARITHMETIC
... T t- p if, and only if, Mod(T) I=:p. As far as the proof theory ~)f a data type axioma.tisation T is concerned, the semantics of the specification is ModiTL Before looking at Peano arithmetic and the special problems at hand, consider the algebraic specification methods for data types where one inva ...
... T t- p if, and only if, Mod(T) I=:p. As far as the proof theory ~)f a data type axioma.tisation T is concerned, the semantics of the specification is ModiTL Before looking at Peano arithmetic and the special problems at hand, consider the algebraic specification methods for data types where one inva ...
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. ...
ch1_1
... are definitions: Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
... are definitions: Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
Modal_Logics_Eyal_Ariel_151107
... These propositions describe facts about the system as “the system is deadlocked” or “the value of variable x is 5”. An interpreted system is a tuple (S,V), where ...
... These propositions describe facts about the system as “the system is deadlocked” or “the value of variable x is 5”. An interpreted system is a tuple (S,V), where ...
Lecture notes from 5860
... of symbol manipulation. Believing that is the mistake of formalism.” The first challenge for understanding modern type theory is to understand these higher-order recursive functions. We see here that such an understanding is important even for arithmetic. An interesting course project would be to gi ...
... of symbol manipulation. Believing that is the mistake of formalism.” The first challenge for understanding modern type theory is to understand these higher-order recursive functions. We see here that such an understanding is important even for arithmetic. An interesting course project would be to gi ...
Logic and the Axiomatic Method
... I wish to persuade you that a certain statement is true or false by pure reasoning. I could do this by showing you that the statement follows logically from some other statement that you may already believe. I may have to convince you that that statement is also true, and follo ...
... I wish to persuade you that a certain statement is true or false by pure reasoning. I could do this by showing you that the statement follows logically from some other statement that you may already believe. I may have to convince you that that statement is also true, and follo ...
Logic - UNM Computer Science
... one could certainly build a table that describes all possible input and output to the addition function. From our experiences with cardinality, we know that this will be an infinite table with countably number of entries. Input ...
... one could certainly build a table that describes all possible input and output to the addition function. From our experiences with cardinality, we know that this will be an infinite table with countably number of entries. Input ...
Definability in Boolean bunched logic
... Proof. In each case we build models M and M 0 such that there is a bounded morphism from M to M 0 , but M has the property ...
... Proof. In each case we build models M and M 0 such that there is a bounded morphism from M to M 0 , but M has the property ...
In defence of an argument against truthmaker maximalism
... in the case of the Liar Sentence, which would mean again its outright inconsistency. But if S ′ is not F , S ′ is again simply true (exactly as in the previous example of ‘consisting of no more than 5 words’) and does not ‘establish (the negation of) just about anything you please’ as well. It follo ...
... in the case of the Liar Sentence, which would mean again its outright inconsistency. But if S ′ is not F , S ′ is again simply true (exactly as in the previous example of ‘consisting of no more than 5 words’) and does not ‘establish (the negation of) just about anything you please’ as well. It follo ...
Introduction to Logic
... Formal Language • Formal logic replaces the ordinary language of argument with a symbolic language. • This language is meant to be free of all ambiguity and vagueness. • The language is meant to wear its logical structure on its face. • Our formal languages: SL and QL. ...
... Formal Language • Formal logic replaces the ordinary language of argument with a symbolic language. • This language is meant to be free of all ambiguity and vagueness. • The language is meant to wear its logical structure on its face. • Our formal languages: SL and QL. ...