Exam 2: propositional logic

What is the main operator of a WFF (well formed formula)?

It is the operator (connective) in a compound statement that has as it’s scope everything else in the statement

What are the rules of inference?

A rule by means of which the conclusion of an argument is derived from the premises. There are 18 rules: eight rules of implication, ten rules of replacement.

What is a rule of implication? Give an example

the first eight rules of inference; consist of simple, valid argument forms whose premises imply their conclusions. they’re one-directional and apply to whole lines only

ex. Modus ponens (MP)

p > q
p
q

What is a rule of replacement? Give an example?

A rule consisting of two logically equivalent statements forms by means of which the conclsuion of an argument is dervived from the premises. There are ten rules, and they apply to a whole line or any part of a line


ex… double negation (DN):

p:: ~~p

what are the differences between rules of implication and rules of replacement?

Implication rules are valid argument forms that work on entire lines only, whereas replacement rules are logical equivaleneces that work on any part of a line

What is a natural deduction?

A method of derving the conclusion of valid arguments expressed in symbolism of propositional logic

What is a simple statement? Give an example,
What is a compound statement? Give an example?

Simple: A statement that does not contain any other statement as a component

Example: Fast food tend to be unhealthy (F)

A compound statement: contains at least one simple statements

It is not the case that al-Qadea is a humanitarian organization (~A)

Define the 5 logical operators?

Symbol	Name	Statement formed	Components called	Translates	True when
~	tilde	negation	—	not; it is not the case that	reverses the truth value
•	dot	conjunction	conjuncts	and, also, moreover, but, yet	both conjuncts are true
∨	wedge	disjunction	disjuncts	or, unless	at least one disjunct is true — false only when both are false
⊃	horseshoe	conditional (expresses material implication)	antecedent, consequent	if…then, only if, implies, provided that	false only when antecedent is true and consequent false
≡	triple bar	biconditional (expresses material equivalence)	—	if and only if	both components have the same truth value

Define and give examples of contingent statements, tautology, and self contridition

Contingent Statement: Truth value varies based on the truth value of its components

example: p ∨ ~p. main column all T

Tautology: a statement that is necessairly true

example: p ∨ ~p main column all T

Self Contradiction: is a statement that is necessairly false
example : p • ~p. main column all F

Explain what it means to say that two or more statements are

logically equivalent:

consistent:

inconsistent:

contradictory:

logically equivalent: The expressions on either side have the same truth value regardless of the truth value of their components.

consistent: Statements for which there is at least one line on their truth tables in which all of them are true, constant

inconsistent: Statements such that there is no line on their truth tables in which all of them are true

contradictory: Statemnts that necessairly have opposite truth values

What is a statement varaiable ?

A lowercase letter, such as p or q, that can represent any statement

Explain what it means to say that a statement in proposistional logic is a truth function?

A statement is a truth function when its truth value is completely determined by the truth values of its components.

What is an argument form in propositional logic?

an arrangement of statement varaibles and operators such that the uniform substitution of statement varaibles results in an argument

How does natural deduction rely on truth tables?

Natural deduction relies on truth tables because each of the eighteen rules is either a valid argument form (the implication rules) or a logical equivalence (the replacement rules). Each rule was established by a truth table — that is where they come from, and why we are entitled to use them. Natural deduction never builds a truth table for the argument in front of you; it applies results that truth tables have already proved. It depends on truth tables entirely while using none.

What is a formal fallacy? Give an example

A formal fallacy is an argument that is invalid because of its form, and the two in propositional logic are AC (affirming the consequent) and DA ( denying the antecedent

example: AC (conclusion asserting the antecedent) and DA (denying the antecedent)

what is a statement form in propositional logic? Give an example

An arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables result in a statement.


example: b • c , c v e, r ⊃ q

what is a substitution instance in propositional logic?

A substitution instance is the statement or argument that results from uniformly substituting statements for the variables in a statement form or an argument form.

ex.
form: p ⊃ p uniform: A ⊃ A ✓

What is a well formed formula (WWFs. What are the syntactical rules governing what counts as a WWF?

A syntactically correct arrangment of symbols

the rules are
1. statements cannot be combined without an opertor occuring between them

2. a tilde cannot immedietly follow a statement, but it can immediately precede any stateemnt

3. a tilde cannor immedietly percede any other operator

4. A dot, wedge, horseshoe or triple bar must go immedietlt between statements

5. parentheses, brackets and braces must be inserted to prevent ambiguity.