Philosophy 103 - Introduction to Logic: Critical Thinking

PROPOSITIONAL LOGIC

 

While Categorical logic studies the relationship between categories; Propositional logic studies propositions themselves and their relationships. Hence, the symbols which we use in this type of logic are used as variables for propositions, not for categories. Hence, "P" is a propositional-variable. (I'll use "proposition", "statement" and sometimes "claim" interchangeably).

 

 

I. TYPES OF PROPOSITIONS

There are two sorts of claims we will be studying: A simple proposition does not contain any other proposition as a component part. An example of a simple proposition is: Socrates is a man. A compound proposition contains another proposition as a component part: Socrates is a dog or Socrates is not a dog.

Every proposition is either true or false: we can thus define the truth-value of a claim by saying that the truth-value of a true proposition is "true" and that of a false proposition is "false". Thus the truth-value of a proposition is its truth or falsity. We can display the truth-value of propositions by using a technique called truth-tables. A truth-table displays the conditions under which a claim is true. For any given proposition, P:

P
T
F

This says that P is either true of false. Given that fact, we can rigorously define the relationship between any two propositions as follows:

P Q
T T
T F
F T
F F

 

II. COMPOUND PROPOSITIONS: Definitions and Truth-Tables

A truth-functional proposition is a compound proposition whose truth-value depends on the truth-value of it component propositions.

 

1. NEGATION: A negation is a compound truth-functional proposition because it is the denial of a claim. In ordinary language, negation is often indicated by the word "not".

P ~P
T F
F T

Thus, for any claim P, the negation of P, that is ~P, has the opposite truth value.

 

2. CONJUNCTION: a compound proposition made up of two simpler claims. These simpler propositions are called conjuncts. A conjunction is true if and only if both of its conjuncts are true.

P

Q

P

Q


T

T

 

T

 

T

F

 

F

 

F

T

 

F

 

F

F

 

F

 

Thus, if either part of a conjunct is false, then the conjunction itself is false. In English, conjunctions are indicated by "and" or "but". Or "even though. Thus, John is in class even though Jane is outside is symbolized as P and Q.

 

 

3. DISJUNCTION: a compound made of two simpler propositions, called disjuncts. There are two types of disjunctions, or two ways to interpret a disjunction. A exclusive disjunction is one which says either P or Q, but not both. An inclusive, or weak disjunction is one which says that either P or Q, possibly both (and/or is often used here). Logicians are interested in inclusive disjunctions. Thus, for our purposes, a disjunction is true if and only if at least one of its disjuncts are true.

 

P

Q

P

v

Q


T

T

 

T

 

T

F

 

T

 

F

T

 

T

 

F

F

 

F

 

 

4. CONDITIONALS are compound propositions which take the form of "if/then" statements. An example is: If the train is late, then we will miss our flight. We all the claim before the "then" but after the "if" the antecedent of the conditional. The claim after the "then" is called the consequent.

A conditional does not state either that its antecedent or that its consequent is true; rather, it states that if the antecedent is true, then the consequent is true. Therefore, logicians define conditionals this way: A conditional is false if and only if its antecedent is true and its consequent is false; otherwise, a conditional is true.

P

Q

P

Q


T

T

 

T

 

T

F

 

F

 

F

T

 

T

 

F

F

 

T

 

This looks quite odd at first. But keep these examples in mind: Where "<" means "less than":

If 1<2 then 1<4
If 3<2 then 3<4
If 4<2 then 4<4

Why are all of these true? Because for any value of x, If x<2 then x<4 is true. And the only way that could be false is if some number could be less than two but not less that 4.