1/9 1. Propositions and Truth Values

Builds a foundation in propositional logic by distinguishing propositions from nonpropositions, assigning truth values, analyzing compound statements, and using truth tables to connect individual claims with argument validity.

Claims and the Scope of Logic

Logic is the systematic study of reasoning. It helps represent claims clearly, determine whether individual claims are true or false, and assess whether conclusions follow from premises.

A is a declarative claim with exactly one . For example:

“The Pacific Ocean is larger than the Atlantic Ocean.” is true.
“ $7$ is an even number.” is false.
“There is life elsewhere in the universe.” is still a even if its is not currently known.

The key point is that a must make a definite claim; we do not need to know which it has in order for it to qualify as a .

Recognizing nonpropositions

Questions, commands, and many exclamations do not assert claims that are true or false:

“Is the library open?” is a question.
“Close the door.” is a command.
“What a beautiful day!” is usually an expression rather than a .
“Let $x$ be a positive integer.” is an instruction.

An such as $x+2=5$ is not a by itself because its truth depends on the value of $x$ . Supplying a value or binding the variable with a quantifier can make the resulting claim definite.

Takeaway: Start logical analysis by asking whether a sentence makes one definite declarative claim.

Truth Values and Context

A records whether a is true or false. Classical propositional logic has two possible truth values, commonly written as $\mathrm{T}$ and $\mathrm{F}$ , or as $1$ and $0$ .

For example, if $p$ is the “ $10$ is greater than $4$ ,” then:

\operatorname{TV}(p)=\mathrm{T}

If $q$ is the “ $10$ is less than $4$ ,” then:

\operatorname{TV}(q)=\mathrm{F}

Truth is not the same as belief. Someone may believe a false or doubt a true one; logic focuses on the ’s rather than on a person’s psychological state.

also depends on the claim, not merely on whether the sentence is grammatically complete. “The number $12$ is prime” is a complete sentence with false , while “The number $12$ is not prime” has true .

Context may need to be fixed before a sentence expresses one definite . For example, “It is cold” may be true in one place or at one time and false in another. Precise logical analysis replaces unresolved context with a clear specification when necessary.

Takeaway: Determine whether a claim has a definite , and keep that assessment separate from personal belief or grammatical form.

Building Compound Propositions

A simple , also called an atomic , contains no smaller joined by a connective. For example:

$p$ : “The train arrives at noon.”
$q$ : “The meeting is online.”

A combines propositions or modifies one . Common connectives include:

Negation: “not”
Conjunction: “and”
Disjunction: “or”
Conditional: “if ... then”
Biconditional: “if and only if”

A results when propositions are connected. If $p$ means “The train arrives at noon” and $q$ means “The meeting is online,” then the statement “The train arrives at noon and the meeting is online” can be represented as:

p\land q

The word “but” often functions logically like “and.” Thus, “The package arrived, but the recipient was not home” has the same basic conjunction structure as “The package arrived and the recipient was not home,” even though “but” adds a contrast in ordinary language.

The truth of a is determined by the truth values of its components together with the meaning of the connective. Propositional logic concentrates on this structure rather than on every subject-specific detail in the claims.

Takeaway: Break a compound statement into component propositions, identify the connective, and then analyze how the connective combines their truth values.

Using Truth Tables

A lists all possible truth-value combinations for component propositions and gives the resulting value of a compound expression. One has $2$ possible truth values. Two independent propositions have $2^2=4$ possible combinations. More generally, $n$ independent propositions produce $2^n$ combinations.

For conjunction, represented by $p\land q$ , the result is true only when both component propositions are true. The four possible rows can be read as follows:

When $p$ is $\mathrm{T}$ and $q$ is $\mathrm{T}$ , $p\land q$ is $\mathrm{T}$ .
When $p$ is $\mathrm{T}$ and $q$ is $\mathrm{F}$ , $p\land q$ is $\mathrm{F}$ .
When $p$ is $\mathrm{F}$ and $q$ is $\mathrm{T}$ , $p\land q$ is $\mathrm{F}$ .
When $p$ is $\mathrm{F}$ and $q$ is $\mathrm{F}$ , $p\land q$ is $\mathrm{F}$ .

This pattern captures the logical behavior of “and”: the conjunction is true exactly when both components are true. It does not determine whether the real-world claims represented by $p$ and $q$ are actually true. Truth tables are later used to study compound statements, logical equivalence, and arguments.

Takeaway: Truth tables separate the possible truth values of component claims from the rule used to combine them.

From Propositions to Arguments

The truth of a and the of an argument are different issues. Truth concerns an individual statement; concerns the relationship between premises and a conclusion.

An argument uses premises to support a conclusion. Consider:

Premise: The Earth is larger than the Moon.
Premise: Dogs are mammals.
Conclusion: Therefore, triangles have three sides.

Each statement is true, but the premises do not support the conclusion. This illustrates why evaluating individual truth values is not enough to evaluate an argument’s reasoning.

A useful progression for introductory logic is:

Identify which sentences are propositions.
Determine or represent their truth values.
Translate compound propositions into symbols and connectives.
Construct and interpret truth tables.
Identify premises and conclusions in arguments.
Test whether the premises guarantee the conclusion.
Extend the analysis with quantifiers and the study of common fallacies.

Takeaway: True statements do not automatically form a good argument; logical support must also be evaluated.