[Discrete Mathematics III] Propositional Logic - Propositional Symbols and Connectives

propositions and truth values

Proposition : A declarative sentence that can be judged true or false is a proposition, and the judgment result is unique.
Truth value : A proposition can take on a "value", called a truth value. There are only two truth values, "True" and "False", represented by "T" (or "1") and "F" (or "0") respectively.
The central problem of mathematical logic research is reasoning, and the premise and conclusion of reasoning are propositions. Propositions are thus the basic unit of reasoning.

Judgment proposition : declarative sentence --> whether the truth value is unique
(it has nothing to do with whether people know it or not; what happens in the future with only the only truth value is also a proposition; in short, there must be a unique truth value objectively, but the truth value cannot be determined) Paradox
: Paradox On the surface, there are two opposite conclusions implied in the same proposition or reasoning , and these two conclusions can be justified. The abstract formula of the paradox is: if event A occurs, then non-A is deduced, and non-A occurs, then A is deduced. "I'm lying" is not a proposition.
Uncertain truth value : x+5>3 is a propositional variable . When x and y are given, they can constitute a part of a propositional formula, but they are not a proposition.
Ambiguous declarative sentences : not a paradox, not a proposition, set a rule that indicates which parse tree (or syntax tree) is correct in each case of ambiguity. Such rules are called disambiguating rules.

Classification of Propositions

Atomic proposition: A proposition composed of simple declarative sentences, which cannot be further divided.
Compound propositions: connect simple propositions with logical connectives.

True Proposition: A proposition whose truth value is true (1).
False proposition: A proposition whose true value is false (0).

Propositional constants: declarative sentence and the truth value is determined and unchanged --> simple proposition.
Propositional variable: statement with uncertain truth value --> not a proposition.

proposition symbolization

Indicated by lowercase letters.

propositional connectives

Negative formula
1. Definition: Let $p$ is a proposition, compound proposition is not$p$ is called   p\p The negation of$p$$\neg p$;$\neg$ becomes a negative conjunction;
2. Truth table:$\neg p$ is true when$p$ is false.

$p$	$\neg p$
0	1
1	0

Conjunction
1. Definition: Let $p,q$ is two propositions, compound proposition "$p$ and$q$ ” called$p,$The conjunction of$q$$p\wedge q$;$\wedge$ becomes a conjunction;
2. Truth table:$p\wedge q$ is true if and only if$p$ and$q$ is the same as true.

$p$	$q$	$p\wedge q$
0	0	0
0	1	0
1	0	0
1	1	1

Disjunction
1. Definition: Let $p,q$ is two propositions, compound proposition "$p$ or$q$ ” called$p,$The disjunctive formula of$q$$p\vee q$;$\vee$ becomes a disjunctive connective;
2. Truth table:$p\vee q$ is true if and only if$p$ and$q$ at least one of the two is true.

$p$	$q$	$p\vee q$
0	0	0
0	1	1
1	0	1
1	1	1

Note compatible or and exclusive or .
"2 and 5 are prime numbers" - compatible or, $p\vee q$
"Li Ming was born in 1987 or 1988" - exclusion or,$(p\wedge \neg q)\vee (\neg p\vee q)$

Implication
1. Definition: Let $p,q$ is two propositions, compound proposition "if$p$ , then$q$ ” called$p,$The implication of$q$$p\to q$;$\to$ Become an implied connective. where$p$ is the antecedent of this implication,$q$ is the consequent of this implication.
2. Truth table:$p\to q$ is false if and only if$p$ true and$q$ is false; everything else is true.

$p$	$q$	$p\wedge q$
0	0	1
0	1	1
1	0	0
1	1	1

The antecedent is the cause and the latter is the effect.

Equivalence
1. Definition: Let $p,q$ is two propositions, compound proposition "$p$ if and only if$q$ ” called$p,$The equivalent formula of$q$$p\leftrightarrow q$;$\leftrightarrow$ becomes an equivalent connective;
2. Truth table:$p\leftrightarrow q$ is true if and only if$p$ and$q$ are both true or false.

$p$	$q$	$p\wedge q$
0	0	1
0	1	0
1	0	0
1	1	1

Note the equivalence ( $\leftrightarrow$ ), equivalent calculus ($\iff$ ) and derivation ($\Rightarrow$ ) difference.

Operation precedence: " $\neg$ " highest, followed by "$\wedge$”、“$\vee$”、“$\to$”、“$\leftrightarrow$ ". If there are parentheses, the parentheses take precedence.

The above five kinds of connectives are also called truth connectives or logical connectives.