slides - Computer and Information Science
... • We can summarise the operation of in a truth table. The idea of a truth table for some formula is that it describes the behavior of a formula under all possible interpretations of the primitive propositions that are included in the formula. • If there are n different atomic propositions in some fo ...
... • We can summarise the operation of in a truth table. The idea of a truth table for some formula is that it describes the behavior of a formula under all possible interpretations of the primitive propositions that are included in the formula. • If there are n different atomic propositions in some fo ...
6. Truth and Possible Worlds
... is possible for B to be false here, then that’s bad news. For then even a perfect thinker, starting with truth, could end up in error. Fortunately we shall see that this is not possible. Logical entailment cannot lead from truth to something false. Entailment, we say, is truth preserving. ...
... is possible for B to be false here, then that’s bad news. For then even a perfect thinker, starting with truth, could end up in error. Fortunately we shall see that this is not possible. Logical entailment cannot lead from truth to something false. Entailment, we say, is truth preserving. ...
What is Logic?
... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. ...
... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. ...
Propositional Logic: Part I - Semantics
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
The Foundations: Logic and Proofs - UTH e
... Propositional Variables: p, q, r, s, … The proposition that is always true is denoted by T and the proposition that is always false is denoted by F. Compound Propositions; constructed from logical connectives and other propositions ...
... Propositional Variables: p, q, r, s, … The proposition that is always true is denoted by T and the proposition that is always false is denoted by F. Compound Propositions; constructed from logical connectives and other propositions ...
Valid Arguments and Logical Fallacies
... When making an argument, the writer must make a point and support that point. As readers, we must recognize the point and determine whether or not that point is supported by relevant and verifiable proof. To evaluate an argument, the reader needs to ask these two questions: ...
... When making an argument, the writer must make a point and support that point. As readers, we must recognize the point and determine whether or not that point is supported by relevant and verifiable proof. To evaluate an argument, the reader needs to ask these two questions: ...
Philosophy as Logical Analysis of Science: Carnap, Schlick, Gödel
... the Circle, found inspiration in the Tractatus. As a former physicist, he took space-time points and objects or events occupying them to be fundamental physical entities. Following the Tractatus he also took the statements of physics to be truth functions of atomic statements about these entities. B ...
... the Circle, found inspiration in the Tractatus. As a former physicist, he took space-time points and objects or events occupying them to be fundamental physical entities. Following the Tractatus he also took the statements of physics to be truth functions of atomic statements about these entities. B ...
CA320 - Computability & Complexity Overview
... We will be interested in a special kind of relation, called an equivalence relation. A relation R on a set A is an equivalence relation if it satisfies the following conditions: 1. R is reflexive, i.e. ∀x ∈ A, xRx 2. R is symmetric, i.e. ∀x, y ∈ A, if xRy , then yRx 3. R is transitive, i.e. ∀x, y , ...
... We will be interested in a special kind of relation, called an equivalence relation. A relation R on a set A is an equivalence relation if it satisfies the following conditions: 1. R is reflexive, i.e. ∀x ∈ A, xRx 2. R is symmetric, i.e. ∀x, y ∈ A, if xRy , then yRx 3. R is transitive, i.e. ∀x, y , ...
Lecture01 - Mathematics
... a) Propositional logic is the study of propositions (true or false statements) and ways of combining them (logical operators) to get new propositions. It is effectively an algebra of propositions. In this algebra, the variables stand for unknown propositions (instead of unknown real numbers) and the ...
... a) Propositional logic is the study of propositions (true or false statements) and ways of combining them (logical operators) to get new propositions. It is effectively an algebra of propositions. In this algebra, the variables stand for unknown propositions (instead of unknown real numbers) and the ...
Logic and Proof - Collaboratory for Advanced Computing and
... Methods of Proving Theorems Proving implications p → q: Direct proof: Assume p is T, and use rules of inference to prove that q is T Indirect proof: Prove its contrapositive; assume ¬q, and prove ¬p Proof by cases: Prove (p1 ∨ p2) → q by proving (p1 → q) and (p1 → q) • Based on [(p1 ∨ p2) → q ...
... Methods of Proving Theorems Proving implications p → q: Direct proof: Assume p is T, and use rules of inference to prove that q is T Indirect proof: Prove its contrapositive; assume ¬q, and prove ¬p Proof by cases: Prove (p1 ∨ p2) → q by proving (p1 → q) and (p1 → q) • Based on [(p1 ∨ p2) → q ...
Solutions to Workbook Exercises Unit 16: Categorical Propositions
... Mx: x meows Fx: x likes canned food Wx: x wags its tail (a) Some dogs howl. ∃x (Dx • Hx) x ...
... Mx: x meows Fx: x likes canned food Wx: x wags its tail (a) Some dogs howl. ∃x (Dx • Hx) x ...
1 Chapter 9: Deductive Reasoning
... however, that there is nothing about the particular premises that makes the argument valid. Any argument of the same form p or q not-p ∴q will also be valid. This illustrates that validity is a property of the form of the argument, and not its content, i.e., validity is independent of the content of ...
... however, that there is nothing about the particular premises that makes the argument valid. Any argument of the same form p or q not-p ∴q will also be valid. This illustrates that validity is a property of the form of the argument, and not its content, i.e., validity is independent of the content of ...
the theory of form logic - University College Freiburg
... as combinations of predicate and individual terms: “The atomic proposition consists of names. It is a connexion, a concatenation, of names” (Wittgenstein 1922, §4.22). By calling all non-logical constants ‘names’, Wittgenstein does not, however, reject the idea of incomplete terms. On a plausible in ...
... as combinations of predicate and individual terms: “The atomic proposition consists of names. It is a connexion, a concatenation, of names” (Wittgenstein 1922, §4.22). By calling all non-logical constants ‘names’, Wittgenstein does not, however, reject the idea of incomplete terms. On a plausible in ...
A puzzle about de rebus beliefs
... are critics who admire only one another’ as asserting that Ralph believes one of the de rebus or plural propositions that can be obtained by supplying some critics as argument to the propositional function described by the English open sentence ‘that they admire only one another’; there are some ind ...
... are critics who admire only one another’ as asserting that Ralph believes one of the de rebus or plural propositions that can be obtained by supplying some critics as argument to the propositional function described by the English open sentence ‘that they admire only one another’; there are some ind ...
Logical Implications
... The three values of x lead to different pairs of truth values for x > 2 and x2 > 4 but in all three cases, the implication (x > 2) → (x2 > 4) is true according to the definition of p → q. It is important to observe that the only way the implication could be false is if we could find some real number ...
... The three values of x lead to different pairs of truth values for x > 2 and x2 > 4 but in all three cases, the implication (x > 2) → (x2 > 4) is true according to the definition of p → q. It is important to observe that the only way the implication could be false is if we could find some real number ...
Slide 1
... is there exist integers p and q with q0 such that r=p/q. A real number that is not rational is called irrational. • Ex.7: Prove that the sum of two rational numbers is rational. • Ex.8: Prove that if n is an integer and n2 is odd, then n is odd. ...
... is there exist integers p and q with q0 such that r=p/q. A real number that is not rational is called irrational. • Ex.7: Prove that the sum of two rational numbers is rational. • Ex.8: Prove that if n is an integer and n2 is odd, then n is odd. ...
Introduction to proposition
... not a freshman.” Solution: There are many ways to translate this sentence into a logical expression. Although it is possible to represent the sentence by a single propositional variable, such as p, this would not be useful when analyzing its meaning or reasoning with it. Instead, we will use proposi ...
... not a freshman.” Solution: There are many ways to translate this sentence into a logical expression. Although it is possible to represent the sentence by a single propositional variable, such as p, this would not be useful when analyzing its meaning or reasoning with it. Instead, we will use proposi ...
logical axiom
... is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above set, we are actually looking at three smaller sets, each set containing form ...
... is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above set, we are actually looking at three smaller sets, each set containing form ...
Chapter1_Parts2
... Solution: Construct the truth table for the proposition. Then an equivalent proposition is the disjunction with n disjuncts (where n is the number of rows for which the formula evaluates to T). Each disjunct has m conjuncts where m is the number of distinct propositional variables. Each conjunct inc ...
... Solution: Construct the truth table for the proposition. Then an equivalent proposition is the disjunction with n disjuncts (where n is the number of rows for which the formula evaluates to T). Each disjunct has m conjuncts where m is the number of distinct propositional variables. Each conjunct inc ...
Tractatus Logico-Philosophicus
The Tractatus Logico-Philosophicus (Latin for ""Logico-Philosophical Treatise"") is the only book-length philosophical work published by the German-Austrian philosopher Ludwig Wittgenstein in his lifetime. The project had a broad aim – to identify the relationship between language and reality and to define the limits of science – and is recognized as a significant philosophical work of the twentieth century. G. E. Moore originally suggested the work's Latin title as homage to the Tractatus Theologico-Politicus by Baruch Spinoza.Wittgenstein wrote the notes for the Tractatus while he was a soldier during World War I and completed it when a prisoner of war at Como and later Cassino in August 1918. It was first published in German in 1921 as Logisch-Philosophische Abhandlung. The Tractatus was influential chiefly amongst the logical positivists of the Vienna Circle, such as Rudolf Carnap and Friedrich Waismann. Bertrand Russell's article ""The Philosophy of Logical Atomism"" is presented as a working out of ideas that he had learned from Wittgenstein.The Tractatus employs a notoriously austere and succinct literary style. The work contains almost no arguments as such, but rather consists of declarative statements that are meant to be self-evident. The statements are hierarchically numbered, with seven basic propositions at the primary level (numbered 1–7), with each sub-level being a comment on or elaboration of the statement at the next higher level (e.g., 1, 1.1, 1.11, 1.12).Wittgenstein's later works, notably the posthumously published Philosophical Investigations, criticised many of the ideas in the Tractatus.