Basic Knowledge and Conditions on Knowledge
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Section 1.4: Predicate Logic
Section 1.4: Predicate Logic January 22, 2021 Abstract We now consider the logic associated with predicate wffs, including a new set of derivation rules for demonstrating validity (the analogue of tautology in the propositional calculus) – that is, for proving theorems! 1 Derivation rules • First of all, all the rules of propositional logic still hold. Whew! Propositional wffs are simply boring, variable-less predicate wffs. • Our author suggests the following “general plan of attack”: – strip off the quantifiers – work with the separate wffs – insert quantifiers as necessary Now, how may we legitimately do so? Consider the classic syllo- gism: a. (All) Humans are mortal. b. Socrates is human. Therefore Socrates is mortal. The way we reason is that the rule “Humans are mortal” applies to the specific example “Socrates”; hence this Socrates is mortal. We might write this as a. (∀x)(H(x) → M(x)) b. H(s) Therefore M(s). It seems so obvious! But how do we justify that in a proof se- quence? • New rules for predicate logic: in the following, you should un- derstand by the symbol x in P (x) an expression with free variable x, possibly containing other (quantified) variables: e.g. P (x) ≡ (∀y)(∃z)Q(x,y,z) (1) – Universal Instantiation: from (∀x)P (x) deduce P (t). Caveat: t must not already appear as a variable in the ex- pression for P (x): in the equation above, (1), it would not do to deduce P (y) or P (z), as those variables appear in the expression (in a quantified fashion) already. -
United Nations Development Programme United Nations
UNITED NATIONS DEVELOPMENT PROGRAMME UNITED NATIONS DEVELOPMENT PROGRAMME GLOBAL ENVIRONMENT FACILITY GOVERNMENT OF THE REPUBLIC OF POLAND Project Brief Number: POL/03/G3X PIMS number: 1623 Title: Biodiversity Conservation and Management in the Barycz Valley Country: Poland Duration: 3 years UNDP and Cost Sharing ACC/UNDP (Sub) Sector: G3: Environment (in US$) GEF Focal Area: Multiple Focal Area: BD/IW UNDP Managed Funds GEF Operational Programme: OP 12 UNDP/GEF Implementing Agency: PTPP “Pro Natura” Project: 964,350 Executing Agency: PTPP “Pro Natura” PDF: 23,968 Sub-total GEF 988,318 Estimated Starting Date: April 2004 Co financing: 10,237,351 Operational Programme OP 12 Total Project Costs: 11,225,669 Strategic priority EM1 Summary The objective of this project is to implement the Barycz Valley’s “Regional Sustainable Development Strategy” (RSDS) on a pilot demonstration basis. With technical and financial support from the PDF-A, the municipalities of the Barycz Valley defined priorities and actions that integrate resource use and biodiversity protection into social and economic development of the Barycz Valley. These agreed priorities and actions constitute the “Regional Sustainable Development Strategy”, a planning document that puts in place an integrated ecosystem management approach to the use of land, water and biodiversity resources in the Barycz Valley. The MSP will co-finance the execution of priority activities of the RSDS in the areas of (i) nature tourism; (ii) decreasing pollution loads into international water systems; (iii) nature-friendly fish farming; (iv) conservation of globally significant meadows and (v) public support for biodiversity conservation. These activities show clear global benefits in the area of biodiversity and international waters. -
Conditions of Fish Farming in Natura 2000 Areas, Based on the Example of the Catchment of Barycz
Journal of Ecological Engineering Volume 17, Issue 3, July 2016, pages 185–192 DOI: 10.12911/22998993/63322 Research Article CONDITIONS OF FISH FARMING IN NATURA 2000 AREAS, BASED ON THE EXAMPLE OF THE CATCHMENT OF BARYCZ Katarzyna Tokarczyk-Dorociak1, Andrzej Drabiński1, Szymon Szewrański2, Sławomir Mazurek3, Wanda Kraśniewska3 1 Institute of Landscape Architecture, Wrocław Universiy of Environmental and Life Sciences, pl. Grunwaldzki 24a, 50-363 Wrocław, Poland, e-mail: [email protected] 2 Department of Spatial Economy, Wroclaw University of Environmental and Life Sciences, Grunwaldzka 53, 50-357 Wrocław, Poland 3 Stawy Milickie SA, Ruda Sułowska 20, 56-300 Milicz, Poland Received: 2016.03.31 ABSTRACT Accepted: 2016.06.01 One of the factors that contributed to the construction of approx.77 km2 offish ponds in Published: 2016.07.01 the catchment of Barycz starting from the 13th century, which in turn transformed the woods into a mosaic of waters, forests and arable land, were the advantageous physi- ographic conditions. Fish farming operations conducted in this area led to the creation of a cultural landscape characterised by unique natural values, similar to the natural landscape. Approx. 240 species of birds are observed here, of which 170 are nesting species. Due to its natural values, this area has been subject to natural reserve protec- tion as part of the Landscape Park “Dolina Baryczy” (the Barycz Valley). It was entered in the “Living Lakes” list and it is protected under the Ramsar Convention as well as under the European nature protection network Natura 2000. The established forms of nature protection mean the introduction of a certain binding regime, pursuant to which the economic activity conducted in protected areas must take into account the prohibi- tions and orders introduced by documents that establish the said forms of protection. -
Logic and Proof
CS2209A 2017 Applied Logic for Computer Science Lecture 11, 12 Logic and Proof Instructor: Yu Zhen Xie 1 Proofs • What is a theorem? – Lemma, claim, etc • What is a proof? – Where do we start? – Where do we stop? – What steps do we take? – How much detail is needed? 2 The truth 3 Theories and theorems • Theory: axioms + everything derived from them using rules of inference – Euclidean geometry, set theory, theory of reals, theory of integers, Boolean algebra… – In verification: theory of arrays. • Theorem: a true statement in a theory – Proved from axioms (usually, from already proven theorems) Pythagorean theorem • A statement can be a theorem in one theory and false in another! – Between any two numbers there is another number. • A theorem for real numbers. False for integers! 4 Axioms example: Euclid’s postulates I. Through 2 points a line segment can be drawn II. A line segment can be extended to a straight line indefinitely III. Given a line segment, a circle can be drawn with it as a radius and one endpoint as a centre IV. All right angles are congruent V. Parallel postulate 5 Some axioms for propositional logic • For any formulas A, B, C: – A ∨ ¬ – . – . – A • Also, like in arithmetic (with as +, as *) – , – Same holds for ∧. – Also, • And unlike arithmetic – ( ) 6 Counterexamples • To disprove a statement, enough to give a counterexample: a scenario where it is false – To disprove that • Take = ", = #, • Then is false, but B is true. – To disprove that if %& '( ) &, ( , then '( %& ) &, ( , • Set the domain of x and y to be {0,1} • Set P(0,0) and P(1,1) to true, and P(0,1), P(1,0) to false. -
Philosophy 109, Modern Logic Russell Marcus
Philosophy 240: Symbolic Logic Hamilton College Fall 2014 Russell Marcus Reference Sheeet for What Follows Names of Languages PL: Propositional Logic M: Monadic (First-Order) Predicate Logic F: Full (First-Order) Predicate Logic FF: Full (First-Order) Predicate Logic with functors S: Second-Order Predicate Logic Basic Truth Tables - á á @ â á w â á e â á / â 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 Rules of Inference Modus Ponens (MP) Conjunction (Conj) á e â á á / â â / á A â Modus Tollens (MT) Addition (Add) á e â á / á w â -â / -á Simplification (Simp) Disjunctive Syllogism (DS) á A â / á á w â -á / â Constructive Dilemma (CD) (á e â) Hypothetical Syllogism (HS) (ã e ä) á e â á w ã / â w ä â e ã / á e ã Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 2 Rules of Equivalence DeMorgan’s Laws (DM) Contraposition (Cont) -(á A â) W -á w -â á e â W -â e -á -(á w â) W -á A -â Material Implication (Impl) Association (Assoc) á e â W -á w â á w (â w ã) W (á w â) w ã á A (â A ã) W (á A â) A ã Material Equivalence (Equiv) á / â W (á e â) A (â e á) Distribution (Dist) á / â W (á A â) w (-á A -â) á A (â w ã) W (á A â) w (á A ã) á w (â A ã) W (á w â) A (á w ã) Exportation (Exp) á e (â e ã) W (á A â) e ã Commutativity (Com) á w â W â w á Tautology (Taut) á A â W â A á á W á A á á W á w á Double Negation (DN) á W --á Six Derived Rules for the Biconditional Rules of Inference Rules of Equivalence Biconditional Modus Ponens (BMP) Biconditional DeMorgan’s Law (BDM) á / â -(á / â) W -á / â á / â Biconditional Modus Tollens (BMT) Biconditional Commutativity (BCom) á / â á / â W â / á -á / -â Biconditional Hypothetical Syllogism (BHS) Biconditional Contraposition (BCont) á / â á / â W -á / -â â / ã / á / ã Philosophy 240: Symbolic Logic, Prof. -
“Volunteering: Step Forward to Entrepreneurship and Employability”
“Volunteering: Step Forward to Entrepreneurship and Employability” Poland, Milicz 25.09.2018 - 03.10.2018 Main aim of the project "Volunteering: Step forward to Entrepreneurship and Employability" is to develop capacities of EVS and volunteering organization, increase quality of services provided by EVS organizations and decrease level of youth unemployment of ex – EVS volunteers . Partner countries on this project are : Poland, Republic of Macedonia, Italy, Hungary, Estonia, Romania, Germany, Finland, Spain, Turkey, Latvia, Slovenia. *Each country should send 2 participants* The project is consisted of two mobilities: Training Course in Milicz, Poland (25.09-03.10.2018) and Seminar in Struga, Republic of Macedonia (20-28.11.2018). Participants are obliged to attend both activities. Key objectives of the project: - To increase competencies of youth workers, leaders, mentors/tutors working with EVS to identify and to explore the needs and interests of EVS volunteers related to employability and entrepreneurship; - To develop competencies of youth workers, leaders, mentors/tutors to develop tailor learning support for EVS volunteers; - To strengthen knowledge of youth workers, leaders, mentors/tutors on tools capturing learning and increasing employability and entrepreneurial mindset of EVS volunteers; - To increase capacities of participating organizations to design quality services supporting development of competencies essential for the job market and/or development of businesses; - To support networking and brainstorming ideas for new projects increasing the quality of voluntary and EVS services. - How effectively support EVS volunteers in development of their personal and professional competencies and to reduce the level of unemployment of ex - EVS volunteers. Target group of this training is youth leaders, workers, mentors and tutors motivated to increase quality of EVS services in their organizations. -
Title of the Article in 16 Pt Times Fonts, Bold, with a Line Spacing Of
Int. J. Environ. Res. Public Health 2005, 2(2), 194–203 International Journal of Environmental Research and Public Health ISSN 1660-4601 www.ijerph.org © 2005 by MDPI Conceptual Modeling for Adaptive Environmental Assessment and Management in the Barycz Valley, Lower Silesia, Poland Piotr Magnuszewski1*, Jan Sendzimir2, and Jakub Kronenberg3 1Institute of Physics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland 2International Institute of Applied Systems Analysis, Schlossplatz 1, A-2361 Laxenburg, Austria 3Dept of International Economic Relations, University of Lodz, POW 3/5, 90-255 Lodz, Poland *Correspondence to Dr. Piotr Magnuszewski. E-mail: [email protected] Received: 10 January 2005 / Accepted: 10 April 2005 / Published: 14 August 2005 Abstract: The complexity of interactions in socio-ecological systems makes it very difficult to plan and implement policies successfully. Traditional environmental management and assessment techniques produce unsatisfactory results because they often ignore facets of system structure that underlie complexity: delays, feedbacks, and non-linearities. Assuming that causes are linked in a linear chain, they concentrate on technological developments (“hard path”) as the only solutions to environmental problems. Adaptive Management is recognized as a promising alternative approach directly addressing links between social and ecological systems and involving stakeholders in the analysis and decision process. This “soft path” requires special tools to facilitate -
Predicate Logic
Predicate Logic Andreas Klappenecker Predicates A function P from a set D to the set Prop of propositions is called a predicate. The set D is called the domain of P. Example Let D=Z be the set of integers. Let a predicate P: Z -> Prop be given by P(x) = x>3. The predicate itself is neither true or false. However, for any given integer the predicate evaluates to a truth value. For example, P(4) is true and P(2) is false Universal Quantifier (1) Let P be a predicate with domain D. The statement “P(x) holds for all x in D” can be written shortly as ∀xP(x). Universal Quantifier (2) Suppose that P(x) is a predicate over a finite domain, say D={1,2,3}. Then ∀xP(x) is equivalent to P(1)⋀P(2)⋀P(3). Universal Quantifier (3) Let P be a predicate with domain D. ∀xP(x) is true if and only if P(x) is true for all x in D. Put differently, ∀xP(x) is false if and only if P(x) is false for some x in D. Existential Quantifier The statement P(x) holds for some x in the domain D can be written as ∃x P(x) Example: ∃x (x>0 ⋀ x2 = 2) is true if the domain is the real numbers but false if the domain is the rational numbers. Logical Equivalence (1) Two statements involving quantifiers and predicates are logically equivalent if and only if they have the same truth values no matter which predicates are substituted into these statements and which domain is used. -
Chapter 4, Propositional Calculus
ECS 20 Chapter 4, Logic using Propositional Calculus 0. Introduction to Discrete Mathematics. 0.1. Discrete = Individually separate and distinct as opposed to continuous and capable of infinitesimal change. Integers vs. real numbers, or digital sound vs. analog sound. 0.2. Discrete structures include sets, permutations, graphs, trees, variables in computer programs, and finite-state machines. 0.3. Five themes: logic and proofs, discrete structures, combinatorial analysis, induction and recursion, algorithmic thinking, and applications and modeling. 1. Introduction to Logic using Propositional Calculus and Proof 1.1. “Logic” is “the study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning.” (thefreedictionary.com) 2. Propositions and Compound Propositions 2.1. Proposition (or statement) = a declarative statement (in contrast to a command, a question, or an exclamation) which is true or false, but not both. 2.1.1. Examples: “Obama is president.” is a proposition. “Obama will be re-elected.” is not a proposition. 2.1.2. “This statement is false.” is a paradox that many would not consider a proposition. 2.1.3. For ECS 20, we will use p, q, and r to symbolize propositions, subpropositions, or logical variables which take true or false values. 2.2. Compound Proposition = a proposition that has its truth value completely determined by the truth values of two or more subpropositions and the operators (also called connectives) connecting them. 3. Basic Logical Operations 3.1. Conjunction = = “and.” If both p and q are true, then p q is true, otherwise p q is false. -
Flood Risk of Lower Silesia Voivodship
CIVIL AND ENVIRONMENTAL ENGINEERING REPORTS No. 10 2013 FLOOD RISK OF LOWER SILESIA VOIVODSHIP Mariusz ADYNKIEWICZ-PIRAGAS*, Iwona LEJCUŚ Institute of Meteorology and Water Mangement – National Resarch Instytute Wrocław Branch, Regional Research Department, Parkowa St. 30, 51-616 Wrocław, Poland Floods are natural events of a random nature that cause damage in property, agriculture, and industry. Floods in the upper and middle Odra basin, particularly on a large scale, are characterized by their specificity of arising and shaping. Analyses of historical material prove that the largest floods are during the summer season, especially in July and August. Those events are caused by wide and intensive precipitation lasting 2-3 days. Moreover spatial ranges in the Odra basin and runoff sequence are also the important reasons. Other important factors for the flood risk scale in a region is knowledge of the flood risk index established on the basis of observed floods or that of Maximum Probability Flood. In this paper flood risk in the territory of Lower Silesia Province was evaluated on the basis of chosen indices of flood risk. Keywords: Odra catchment, flood risk zones 1. INTRODUCTION Flood is a natural disaster that threatens the safety of people and animals, and cause damage to human property, and losses in the national economy. It is natural and random phenomenon. It can cause torrential rains, short thunderstorms, rapid melting of snow, strong winds on the coast from the sea towards the land and the freezing of rivers. Flood is a high water, during which water overflows the level of embankment crown and flood river valleys or depressed areas, thereby causing damage and financial and non-economic (social, moral, etc.) losses [Dubicki, Malinowska-Małek 1999]. -
Formal Logic: Quantifiers, Predicates, and Validity
Formal Logic: Quantifiers, Predicates, and Validity CS 130 – Discrete Structures Variables and Statements • Variables: A variable is a symbol that stands for an individual in a collection or set. For example, the variable x may stand for one of the days. We may let x = Monday, x = Tuesday, etc. • We normally use letters at the end of the alphabet as variables, such as x, y, z. • A collection of objects is called the domain of objects. For the above example, the days in the week is the domain of variable x. CS 130 – Discrete Structures 55 Quantifiers • Propositional wffs have rather limited expressive power. E.g., “For every x, x > 0”. • Quantifiers: Quantifiers are phrases that refer to given quantities, such as "for some" or "for all" or "for every", indicating how many objects have a certain property. • Two kinds of quantifiers: – Universal Quantifier: represented by , “for all”, “for every”, “for each”, or “for any”. – Existential Quantifier: represented by , “for some”, “there exists”, “there is a”, or “for at least one”. CS 130 – Discrete Structures 56 Predicates • Predicate: It is the verbal statement which describes the property of a variable. Usually represented by the letter P, the notation P(x) is used to represent some unspecified property or predicate that x may have. – P(x) = x has 30 days. – P(April) = April has 30 days. – What is the truth value of (x)P(x) where x is all the months and P(x) = x has less than 32 days • Combining the quantifier and the predicate, we get a complete statement of the form (x)P(x) or (x)P(x) • The collection of objects is called the domain of interpretation, and it must contain at least one object. -
How to Adopt a Logic
How to adopt a logic Daniel Cohnitz1 & Carlo Nicolai2 1Utrecht University 2King’s College London Abstract What is commonly referred to as the Adoption Problem is a challenge to the idea that the principles of logic can be rationally revised. The argument is based on a reconstruction of unpublished work by Saul Kripke. As the reconstruction has it, Kripke essentially extends the scope of Willard van Orman Quine’s regress argument against conventionalism to the possibility of adopting new logical principles. In this paper we want to discuss the scope of this challenge. Are all revisions of logic subject to the regress problem? If not, are there interesting cases of logical revision that are subject to the regress problem? We will argue that both questions should be answered negatively. 1 Introduction What is commonly referred to as the Adoption Problem1 is often considered a challenge to the idea that the principles for logic can be rationally revised. The argument is based on a reconstruction of unpublished work by Saul Kripke.2 As the reconstruction has it, Kripke essentially extends the scope of William van Orman Quine’s regress argument (Quine, 1976) against conventionalism to the possibility of adopting new logical principles or rules. According to the reconstruction, the Adoption Problem is that new logical rules cannot be adopted unless one already can infer with these rules, in which case the adoption of the rules is unnecessary (Padro, 2015, 18). In this paper we want to discuss the scope of this challenge. Are all revisions of logic subject to the regress problem? If not, are there interesting cases of logical revision that are subject to the regress problem? We will argue that both questions should be answered negatively.