Math as a Language

This set of vocabulary flashcards covers fundamental concepts of Propositional Logic and Predicate Calculus, including logical operators, truth tables, laws of logic, and quantifiers.

Proposition

A declarative statement that is either true or false, but not both.

Propositional variable

A variable used to represent a proposition with an undetermined value.

Negation symbols

The symbols $¬$ or $\sim$.

Conjunction

A compound proposition using AND, denoted by $P \wedge Q$.

Disjunction ($P \vee Q$) False Condition

Only when both propositions $P$ and $Q$ are false.

Conditional statement symbol

The symbol $\Rightarrow$ (IF and THEN).

Exclusive Or ($\oplus$) Truth Condition

It is true if only one proposition is true, but not both.

Biconditional symbol ($\Leftrightarrow$)

It means ‘IF and ONLY IF’.

Hypothesis (or antecedent)

The proposition $P$ in the conditional $P \Rightarrow Q$.

Conclusion (or consequent)

The proposition $Q$ in the conditional $P \Rightarrow Q$.

Converse

The expression $Q \Rightarrow P$ derived from the conditional $P \Rightarrow Q$.

Inverse

The expression $\neg P \Rightarrow \neg Q$ derived from the conditional $P \Rightarrow Q$.

Contrapositive

The expression $\neg Q \Rightarrow \neg P$ derived from the conditional $P \Rightarrow Q$.

Truth table rows formula

$2^n$ where $n$ is the number of distinct variables.

4-variable truth table row count

$16$ rows.

Tautology

A proposition that is true under all circumstances.

Contradiction

A proposition that is false under all circumstances.

Contingency

A propositional form that is neither a tautology nor a contradiction.

Commutativity

The rule stating that order does not matter for $\wedge$ and $\vee$.

Rule of Idempotence

A proposition combined with itself is a tautology (or just itself).

Rule of Associativity

The rule that grouping doesn’t matter if all operators are the same.

De Morgan’s Law (Negation of a Conjunction)

$\neg(P \wedge Q)$ is equivalent to $\neg P \vee \neg Q$.

Rule of Involution

Negating a negation cancels it out.

Material Implication

$P \Rightarrow Q$ is equivalent to $\neg P \vee Q$.

Deductive Argument

An argument where one proposition is a conclusion and others are premises.

Valid deductive argument

An argument considered valid if the propositional form is a tautology.

Fallacy

Incorrect reasoning based on contingencies.

Propositional Function (predicate)

A statement $P(x)$ about a variable $x$.

Categorical Propositions

Propositions containing quantifiers like ‘all’, ‘some’, or ‘none’.

Universal Quantifier

Specifies that a property is true for ALL members of a set.

Existential Quantifier

Specifies that a property is true for SOME members of a set.

$$\forall x \in A, P(x)$$

For all $x$ in set $A$, $P(x)$ is true.

$$\exists x \in A, P(x)$$

There exists at least one $x$ in set $A$ such that $P(x)$ is true.

Conjunction (AND) truth condition

True only when both values are true.

Disjunction (OR) false condition

False only when both values are false.

Material Equivalence

A rule involving combinations of biconditionals.

Negating a Tautology

The result is a Contradiction.

Conditional vs. Converse equivalence

A conditional is not equivalent to its converse ($P \Rightarrow Q \not\equiv Q \Rightarrow P$).

Equivalent of the Inverse

The Converse.

$\wedge$ symbol

Represents Conjunction (AND).

$\vee$ symbol

Represents Disjunction (OR).

Negation of ‘All $x$ are $P$’

‘Some $x$ are not $P$’.

Negation of ‘Some $x$ are $P$’

‘No $x$ are $P$’ (or ‘All $x$ are not $P$’).

Exclusivity of truth value

A proposition cannot be both True and False.

Associativity grouping rule

Allows changing the grouping in $(P \wedge Q) \wedge R$.

Logical meaning of ‘exclusive or’

One or the other, but not both.

Truth value of $F \Rightarrow T$

True.

Propositional variable (Truth Value representation)

A variable representing an unknown truth value.

$\neg$ symbol

Stands for Not.

Predicate characteristics

Describes a statement whose truth depends on its variables.