Home » Articole » Articles » Society » Philosophy » Logic » Semantics of propositional logic

# Semantics of propositional logic

posted in: Logic

Semantics determines the rules for interpreting propositions. Assigning truth values to each of the elementary propositions intervening in a proposition amounts to choosing a model of this proposition.

More precisely, if we place ourselves in classical logic, the interpretation of a proposition A is the fact of assigning a truth value (0 or 1) to the propositional variables and explaining how the connectives behave vis-à-vis with respect to truth values. This behavior is given by a truth table. In this way we can give a value 0 or 1 to each proposition which depends on the values ​​taken by the truth tables. There are the following cases of interpretation:

• When the proposition always takes the value 0 regardless of the values ​​of the propositional variables, the proposition is said to be an antilogy or a contradiction. It is also said to be unsatisfactory.
• When proposition A always takes the value 1, A is a tautology. We also say that A is valid and we denote this assertion ╞ A .
• If the proposition takes at least once the value 1, we say that we can satisfy A, or that A is satisfyable.
• If the proposition takes at least once the value 1 and at least once the value 0, it is a synthetic or contingent proposition.

### Boolean interpretation of connectors

We explain the behavior, then give the associated truth table

• P ˅ Q will take the value 1 if and only if at least one of the two propositions P or Q takes the value 1.
 P Q P ˅ Q 0 0 0 0 1 1 1 0 1 1 1 1
• P ˄ Q will take the value 1 if and only if the two propositions P and Q simultaneously take the value 1.
 P Q P ˄ Q 0 0 0 0 1 0 1 0 0 1 1 1
• P → Q will take the value 0 if and only if P takes the value 1 and Q the value 0.
 P Q P → Q 0 0 1 0 1 1 1 0 0 1 1 1
• ¬ P takes the value 1 if and only if P takes the value 0.
 P ¬ P 0 1 1 0
• P ↔ Q will take the value 1 if and only if P and Q have the same value.
 P Q P ↔ Q 0 0 1 0 1 0 1 0 0 1 1 1
• ⊥ takes the value 0.

Example 1:

(¬A → A) → A is a tautology.

Indeed, if we attribute to A the value 0, then ¬A takes the value 1, (¬A → A) takes the value 0, and (¬A → A) → A takes the value 1. Similarly, if we assigns A the value 1, then ¬A takes the value 0, (¬A → A) takes the value 1, and (¬A → A) → A takes the value 1.

Example 2:

A → (A → ¬A) is not a tautology.

Indeed, if we attribute to A the value 1, then ¬A takes the value 0, (A → ¬A) takes the value 0, and A → (A → ¬A) takes the value 0.

### Complete connector systems

An n-input truth table defines an n-ary connector. A set of propositional connectors is said to be complete if any connector can be defined by means of the connectors of the set. Any truth table is described using conjunctions of disjunctions and negation, for example we fully describe the truth table of equivalence above by “p ↔ q takes the value true if and only if p and q evaluate to false or p and q evaluate to true”, i.e. p ↔ q ≡ (¬p ∧ ¬q) ∨ (p ∧ q). This method is general and allows to show that the system {¬, ∧, ∨} is a complete system of connectors. We deduce that {¬, ∧}, {¬, ∨} are also complete systems (because of Morgan’s laws, A∨B ≡ ¬ (¬A ∧ ¬B), A ∧ B ≡ ¬ (¬A ∨ ¬B)). The set {¬, →} is also complete: A ∨ B ≡ ¬A → B.

The set consisting of the single NAND connector (denoted “|” by Henry Sheffer and therefore called the Sheffer bar) is also complete, ¬P being equivalent to P|P and P∨Q to ( P|P) | (Q|Q). This feature is used for the construction of logic circuits, a single logic gate is then sufficient to design all existing circuits.

(Includes texts from Wikipedia translated and adapted by Nicolae Sfetcu)

##### Last Thoughts

by Henri Poincaré Translated by Nicolae Sfetcu Henri Poincaré is a mathematician, physicist, philosopher and engineer, born April 29, 1854 in Nancy and died July 17, 1912 in Paris. He has carried out works of major importance in optics and … Read More

not rated \$4.99
##### Causal Loops in Time Travel

About the possibility of time traveling based on several specialized works, including those of Nicholas J. J. Smith (“Time Travel“), William Grey (”Troubles with Time Travel”), Ulrich Meyer (”Explaining causal loops”), Simon Keller and Michael Nelson (”Presentists should believe in … Read More

not rated \$0.00

A short introduction about Easter from the perspective of various religions, traditions and cultures, including Easter season, Easter bread, Easter eggs, greetings, etc. Easter is the most important religious holiday of the Christian liturgical year, observed in March, April, or … Read More

not rated \$0.00