Mathematical Logic Fall 2004 Professor R. Moosa Contents
... m = 0 then we are also done, so we may assume that m > 0. Suppose that l > 0 and the lemma is true for all words with smaller length. So the first letter of t1 and v1 is the same. There are two possibilities, either t1 is a variable, or t1 = F a1 , . . . , ak for some F , ai ’s, and k. If t1 = F a1 ...
... m = 0 then we are also done, so we may assume that m > 0. Suppose that l > 0 and the lemma is true for all words with smaller length. So the first letter of t1 and v1 is the same. There are two possibilities, either t1 is a variable, or t1 = F a1 , . . . , ak for some F , ai ’s, and k. If t1 = F a1 ...
Separation Logic with One Quantified Variable
... first-order quantifiers can be found in [11, 4]. However, these known results crucially rely on the memory model addressing cells with two record fields (undecidability of 2SL in [6] is by reduction to the first-order theory of a finite binary relation). In order to study decidability or complexity ...
... first-order quantifiers can be found in [11, 4]. However, these known results crucially rely on the memory model addressing cells with two record fields (undecidability of 2SL in [6] is by reduction to the first-order theory of a finite binary relation). In order to study decidability or complexity ...
Basic Proof Techniques
... Theorem 5. Let a,b,c,d be integers. If a > c and b > c, then M AX(a, b) − c is always positive. Proof. Assume that a > c and b > c. We know that a > c and b > c, but we cannot say for certain if a > b or b > a. Therefore we proceed by cases. 1. Case 1: Assume that a > b. Because a > b we know that ...
... Theorem 5. Let a,b,c,d be integers. If a > c and b > c, then M AX(a, b) − c is always positive. Proof. Assume that a > c and b > c. We know that a > c and b > c, but we cannot say for certain if a > b or b > a. Therefore we proceed by cases. 1. Case 1: Assume that a > b. Because a > b we know that ...
CPCS202 - The Lab Note
... Activity Outcomes: The students will learn about the application of Boolean Algebra in circuit designing. They will also know how to minimize the Boolean expressions using the axioms and postulates of Boolean algebra. This will ultimately help them in designing logical circuits with as few logic gat ...
... Activity Outcomes: The students will learn about the application of Boolean Algebra in circuit designing. They will also know how to minimize the Boolean expressions using the axioms and postulates of Boolean algebra. This will ultimately help them in designing logical circuits with as few logic gat ...
logic for computer science - Institute for Computing and Information
... Gottlob Frege, a German mathematician working in relative obscurity. Frege aimed to derive all of mathematics from logical principles, in other words pure reason, together with some self-evident truths about sets. (Such as 'sets are identical if they have the same members' or 'every property determi ...
... Gottlob Frege, a German mathematician working in relative obscurity. Frege aimed to derive all of mathematics from logical principles, in other words pure reason, together with some self-evident truths about sets. (Such as 'sets are identical if they have the same members' or 'every property determi ...
An Introduction to Proof Theory - UCSD Mathematics
... propositional logic is sketched in 1.1.5 below. Thus Γ may, without loss of generality, be assumed to be a finite set of formulas, say Γ = {B1 , . . . , Bk }. Secondly, note that Γ ² A implies that B1 ⊃ B2 ⊃ · · · ⊃ A is a tautology. So, by part (1), the latter formula has an F -proof, and by k addi ...
... propositional logic is sketched in 1.1.5 below. Thus Γ may, without loss of generality, be assumed to be a finite set of formulas, say Γ = {B1 , . . . , Bk }. Secondly, note that Γ ² A implies that B1 ⊃ B2 ⊃ · · · ⊃ A is a tautology. So, by part (1), the latter formula has an F -proof, and by k addi ...
The disjunction introduction rule: Syntactic and semantics
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
Ways Things Can`t Be
... set of worlds at which the proposition is true). Then the logical operations of conjunction, disjunction, and negation are modeled by the set theoretic operations of intersection, union, and negation. How are propositions modeled once we consider impossible worlds as well as possible worlds? Take a ...
... set of worlds at which the proposition is true). Then the logical operations of conjunction, disjunction, and negation are modeled by the set theoretic operations of intersection, union, and negation. How are propositions modeled once we consider impossible worlds as well as possible worlds? Take a ...
Exam 2 Sample
... 3. (10 pts) (a) How many different 5-card poker hands are there? (b) How many different 5-card poker hands make a "full house" (3 cards have one value, and the other two cards have another value -- for example, 3 kings and 2 tens)? For possible partial credit, explain your reasoning. (c) How many di ...
... 3. (10 pts) (a) How many different 5-card poker hands are there? (b) How many different 5-card poker hands make a "full house" (3 cards have one value, and the other two cards have another value -- for example, 3 kings and 2 tens)? For possible partial credit, explain your reasoning. (c) How many di ...
The Project Gutenberg EBook of The Algebra of Logic, by Louis
... and Lambert ; but their labors remained little known, and it was Boole and De Morgan, about the middle of the nineteenth century, to whom a mathematicalthough of course non-quantitativeway of regarding logic was due. By this, not only was the traditional or Aristotelian doctrine of logic reformed ...
... and Lambert ; but their labors remained little known, and it was Boole and De Morgan, about the middle of the nineteenth century, to whom a mathematicalthough of course non-quantitativeway of regarding logic was due. By this, not only was the traditional or Aristotelian doctrine of logic reformed ...
Inference in First
... • Truth Tabling is not complete for FOL because truth table size may be infinite • Natural Deduction is complete for FOL but is not practical because the “branching factor” in the search is too large (swe would have to potentially try every inference rule in every possible way using the set of known ...
... • Truth Tabling is not complete for FOL because truth table size may be infinite • Natural Deduction is complete for FOL but is not practical because the “branching factor” in the search is too large (swe would have to potentially try every inference rule in every possible way using the set of known ...
Intuitionistic Logic
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
Defending a Dialetheist Response to the Liar`s Paradox
... Hartry Field’s recent book, Saving Truth from Paradox. Any rational dialetheist solution will endorse a para-consistent logic else accept triviality, as they accept that some sentences and their negations are true, which would entail triviality if ex contradictione quodlibet was valid. A dialetheist ...
... Hartry Field’s recent book, Saving Truth from Paradox. Any rational dialetheist solution will endorse a para-consistent logic else accept triviality, as they accept that some sentences and their negations are true, which would entail triviality if ex contradictione quodlibet was valid. A dialetheist ...
http://homes.dsi.unimi.it/ ghilardi/allegati/dispcesena.pdf
... Such an algebra is denoted F(X)/Γ and called the Lindenbaum algebra over X and Γ. 2 In case Γ is empty, we shall write ` ϕ, ψ ` ϕ, ∼, F(X), etc. instead of `∅ ϕ, ψ `∅ ϕ, ∼∅ , F(X)/∅, respectively. For an alphabet X, we have an (injective) set-theoretic map ηX : X −→ U(F(X)) associating with x ∈ X th ...
... Such an algebra is denoted F(X)/Γ and called the Lindenbaum algebra over X and Γ. 2 In case Γ is empty, we shall write ` ϕ, ψ ` ϕ, ∼, F(X), etc. instead of `∅ ϕ, ψ `∅ ϕ, ∼∅ , F(X)/∅, respectively. For an alphabet X, we have an (injective) set-theoretic map ηX : X −→ U(F(X)) associating with x ∈ X th ...