Sections 1.7 and 1.8
... Conditional Statement- “if p, then q” p and q are declarative statements. The “if” part of the sentence is called the hypothesis. The “then” part of the sentence is called the conclusion. If “p” the “q” can also be written p=>q (p implies q). Converse- the converse of p=>q is q=>p. To write the conv ...
... Conditional Statement- “if p, then q” p and q are declarative statements. The “if” part of the sentence is called the hypothesis. The “then” part of the sentence is called the conclusion. If “p” the “q” can also be written p=>q (p implies q). Converse- the converse of p=>q is q=>p. To write the conv ...
A Simple Exposition of Gödel`s Theorem
... To give a proper definition of a formal proof, Gödel needed to specify the axioms, and to formulate precisely the requirement that each well-formed formula was either an axiom of followed from earlier members of the sequence in virtue of one of the rules of inference. And having done this in meta-lo ...
... To give a proper definition of a formal proof, Gödel needed to specify the axioms, and to formulate precisely the requirement that each well-formed formula was either an axiom of followed from earlier members of the sequence in virtue of one of the rules of inference. And having done this in meta-lo ...
Universally true assertions
... In using universal generalization to prove a statement about an arbitrary object c, we are not allowed to make any special assumptions about c except that it satisfies the hypothesis. On the other hand, if we suspected that the theorem were false, we could prove that it is false merely by finding a ...
... In using universal generalization to prove a statement about an arbitrary object c, we are not allowed to make any special assumptions about c except that it satisfies the hypothesis. On the other hand, if we suspected that the theorem were false, we could prove that it is false merely by finding a ...
Introduction to logic
... not only deductively valid. They also have true premises. These arguments are called sound. Argument (2.4) above is valid, but not sound. Another example: (2.5) All fish fly. Anything which flies talks. So, all fish talk. This is a deductively valid argument but unlikely to persuade anyone. ...
... not only deductively valid. They also have true premises. These arguments are called sound. Argument (2.4) above is valid, but not sound. Another example: (2.5) All fish fly. Anything which flies talks. So, all fish talk. This is a deductively valid argument but unlikely to persuade anyone. ...
The initial question: “What is the meaning of a first
... extension of alpha equivalence to the entire first-order language under consideration. Occasionally, Fine's text tend to be admittedly vague. The following can be perceived as shortcomings. There is neither (1) mature semantics nor (2) the proof theory for FOL under the principle of the alphabetic i ...
... extension of alpha equivalence to the entire first-order language under consideration. Occasionally, Fine's text tend to be admittedly vague. The following can be perceived as shortcomings. There is neither (1) mature semantics nor (2) the proof theory for FOL under the principle of the alphabetic i ...
Introduction to Theoretical Computer Science, lesson 3
... First, determine the elements of the universe denoted by terms, then determine the truth-values of atomic formulas, and finally, determine the truth-value of the (composed) formula Evaluation of terms: Let v be a valuation that associates each variable x with an element of the universe: v(x) U. By ...
... First, determine the elements of the universe denoted by terms, then determine the truth-values of atomic formulas, and finally, determine the truth-value of the (composed) formula Evaluation of terms: Let v be a valuation that associates each variable x with an element of the universe: v(x) U. By ...
Introduction to Logic What is Logic? Simple Statements Which one is
... proposition, P1 , P2 , … Pn , called Premises and other proposition Q, called the Conclusion. An argument is denoted by P1 , P2 , … Pn | Q An argument is said to be Validif the premises yield the conclusion. An argument is said to be Fallacy if that is not ...
... proposition, P1 , P2 , … Pn , called Premises and other proposition Q, called the Conclusion. An argument is denoted by P1 , P2 , … Pn | Q An argument is said to be Validif the premises yield the conclusion. An argument is said to be Fallacy if that is not ...
MATH 312H–FOUNDATIONS
... This principle has several variants known as recursive definition, inductive definition, finite induction... their meaning is clear whenever they appear in the course. To make it clear at this point by an example: We can say: If n is an even integer, then it can be written in the form 2m where m is ...
... This principle has several variants known as recursive definition, inductive definition, finite induction... their meaning is clear whenever they appear in the course. To make it clear at this point by an example: We can say: If n is an even integer, then it can be written in the form 2m where m is ...
Chapter 1 Logic and Set Theory
... R ∧ (R → S) and Q = S. Then, the truth of the statement P → Q depends only on the truth of external propositions R and S. The notion of implication can be rigorously defined as follows, P implies Q if the statement P → Q is a tautology. We abbreviate P implies Q by writing P ⇒ Q. It is important to ...
... R ∧ (R → S) and Q = S. Then, the truth of the statement P → Q depends only on the truth of external propositions R and S. The notion of implication can be rigorously defined as follows, P implies Q if the statement P → Q is a tautology. We abbreviate P implies Q by writing P ⇒ Q. It is important to ...
Chapter 1 Logic and Set Theory
... P ⇔ Q holds if and only if P ⇒ Q and Q ⇒ P are both true. Being able to recognize that two statements are equivalent will become handy. It is sometime possible to demonstrate a result by finding an alternative, equivalent form of the statement that is easier to prove than the original form. A list o ...
... P ⇔ Q holds if and only if P ⇒ Q and Q ⇒ P are both true. Being able to recognize that two statements are equivalent will become handy. It is sometime possible to demonstrate a result by finding an alternative, equivalent form of the statement that is easier to prove than the original form. A list o ...
Mathematics for Computer Science/Software Engineering
... Heuristic justification for this definition: if p is true and q is false, then the statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at ...
... Heuristic justification for this definition: if p is true and q is false, then the statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at ...
Supplement: Conditional statements and basic methods of proof
... First note that there is only one set of circumstances under which a conditional statement is false: The hypothesis is true and the conclusion is false. Therefore to establish that a conditional statement is false, it suffices to produce a specific counterexample; that is, a specific situation for w ...
... First note that there is only one set of circumstances under which a conditional statement is false: The hypothesis is true and the conclusion is false. Therefore to establish that a conditional statement is false, it suffices to produce a specific counterexample; that is, a specific situation for w ...
MODULE I
... 4) Show that RPQ) is a valid conclusion from PQ, QR, PM, ┐M. 5) Show that SR is a valid conclusion from the premises PQ, PR, QS. 6) Derive P→(Q→S) from the premises P→(Q→R), Q→(R→S). Indirect method of proof The technique of indirect method of proof is as follows. 1. Introduce the neg ...
... 4) Show that RPQ) is a valid conclusion from PQ, QR, PM, ┐M. 5) Show that SR is a valid conclusion from the premises PQ, PR, QS. 6) Derive P→(Q→S) from the premises P→(Q→R), Q→(R→S). Indirect method of proof The technique of indirect method of proof is as follows. 1. Introduce the neg ...
Identity and Philosophical Problems of Symbolic Logic
... logic. But it has been argued that most natural language sentences do not have two truth-values. ...
... logic. But it has been argued that most natural language sentences do not have two truth-values. ...
Modal Reasoning
... Remember that, with stacked modal operators, we move from the outside in; for example, Example 5. M, w1 |= 33p Interpreted as, ‘From w1 , all reachable worlds have 33p as true. From each of these worlds, there is a world where 3p is true. Finally, from each of these worlds, there is an accessible w ...
... Remember that, with stacked modal operators, we move from the outside in; for example, Example 5. M, w1 |= 33p Interpreted as, ‘From w1 , all reachable worlds have 33p as true. From each of these worlds, there is a world where 3p is true. Finally, from each of these worlds, there is an accessible w ...
2.2 Conditional Statements
... operations is that → is performed last Construct a truth table for the statement form p ∨ ∼q → ∼p. Solution By the order of operations given above, the following two expressions are equivalent: p ∨ ∼q → ∼p and (p ∨ (∼q)) → (∼p), and this order governs the construction of the truth table. Construct i ...
... operations is that → is performed last Construct a truth table for the statement form p ∨ ∼q → ∼p. Solution By the order of operations given above, the following two expressions are equivalent: p ∨ ∼q → ∼p and (p ∨ (∼q)) → (∼p), and this order governs the construction of the truth table. Construct i ...