M2: Prop logic and Logical Equivalence Flashcards
__ is useful for evaluating whether an argument is deductively valid as opposed to inductively valid, which is the way that people typically reason in everyday life
Propositional Logic
Three kinds of symbols in formalizing propositional logic:
Sentence letters (ex: P, Q, R, a, b, c)
Operators/Connectives (~^v, ->, <–>)
Parenthesis
___ is used for organization purposes, similar to parentheses in algebra and arithmetic.
Parentheses
___ is one that cannot be separated into further sentences.
Atomical Sentence
IMPORTANT NOTE:
1. The standard procedure is to define ___with the sentences they stand for.
letters
IMPORTANT NOTE:
2. When symbolizing, one should try to break each natural language sentence into an ___
atomical sentence
My sister and I like coffee” could be defined:
m = My sister and I like coffee.
But also:
n = My sister likes coffee.
o = I like coffee.
IMPORTANT NOTES
We have an operation to express “negation”. So a negative sentence will be better expressed as a ____, and later we can add the correct symbol
Positive sentence
____it is a formula which is always true for every value of its propositional variables
Tautology
A ___ in the English language is an expression or phrase that says the same thing twice, just in a different way.
tautology
A statement, or set of statements is logically consistent when it involves no ____.
logical contradiction
_____is a formula which is always false for every value of its propositional variables
Contradiction
_____is a formula which is always false for every value of its propositional variables
Contradiction
It is a formula which has both some true and some false values for every value of its propositional variables.
Contingency
It is a formula which has both some true and some false values for every value of its propositional variables.
Contingency
A formula is valid if it is____.
true for all values of its terms
___ refers to the existence of a combination of values to make the expression true.
satisfiability
A propositional logic is said to be satisfiable if its either a ___
tautology or contingency
Hence if a logic is a ___ then it is said to be unsatisfiable.
contradiction
Steps in knowing validity using truth table
- Assign a letter to represent each component statement in the argument
- Express each remises and conclusion symbolically
- Form the symbolic statement of the entire argument by writing the conj of all premises as the antecedent of the conditional statement and the conclusion as the consequence
- Complete the truth table for the conditional statement formed in step 3. If a tautology, then the arg is valid, otherwise, it is invalid
VALID ARG FORMS
___ is also called “affirming the antecedent”
Modus Ponens
VALID ARG FORMS
___ is also called “affirming the antecedent” because one of its premises affirms that the antecedent of the conditional is true
Modus Ponens
WHAT ARG FORM?
If p is a sufficient condition of q, and p is true, then q must be true
P->Q_If there is no God, then life is meaningless.
P_____There is no god
——–
q_____Therefore, life is meaningless
Modus ponens
VALID ARG FORM
___ also means “denying the consequent” because one of its premises denies that the consequent of the conditional is true
Modus Tollens
WHAT ARG FORM
If q is a necessary condition of p, and q is false, then p must be false
p->q_If I’m a mom, then I’m a woman
~q___I’m not a woman
————
~p___Therefore, I’m not a mom.
Modus Tollens
VALID ARG FORM
The basic form ___ gets its name from the feature from that one of the two premises is a disjunction. If one is false, then the other is true
disjunctive syllogism
WHAT ARG FORM
If p is false, then q must be true.
p v q__Either I will watch a concert or I will read a novel
~p_____I will not watch a concert.
——–
q______Therefore, I will read a novel.
disjunctive Syllogism
VALID ARG FORM
The arg consists of three conditionals
Reasoning by Transitivity (Hypothetical Syllogism)
WHAT ARG FORM
The first conditional says that p is a sufficient condition for q.
The second one says that q in turn is a sufficient condition for r
It would then follow that p is a sufficient condition for r
p->q__If 1<2 then 2<3
q->r__If 2<3 then 2>1
____
p->r__Therefore, if 1<2 then 2>1
Reasoning by Transitivity (Hypothetical Syllogism)
___are those arguments that if the arguments fall under one of these forms, there is no need to construct the truth table because it means that the arguments are invalid.
invalid argument forms
INVALID ARFG FORMS
___ also called converse error and it resembles modus ponens
p->q_If I’m a mom, then I’m a woman.
q____I am a woman
——–
p____Therefore, I am a mom.
Fallacy of the Converse
INVALID ARG FORMS
___ is also called inverse error. It appears very similar to a valid arg form- the contrapositive
Ex
p->q__If I’m a mom, then I’m a woman
~q____I am not a mom.
———
~q____Therefore, I am not a woman.
Fallacy of the inverse
All but the final proposition are called ___.
premises
The last proposition is the ___
conclusion
W
