Question

Translate the following expressions of propositional logic into words using the following translation of the proposition letters: $p=$ "All the world is apple pie." $q=$ "All the seas are ink." $r=$ "All the trees are bread and cheese." $s=$ There is nothing to drink." $t=$ "Socrates was a man." $u=$ "All men are mortal." $v="$ Socrates was mortal." (a) $(p\wedge q\wedge r)\to s$ (b) $(t\wedge u)\to v$ (c) $\mathrm{\neg }s\to \mathrm{\neg }v$ (d) $p\wedge (q\wedge r)\vee (t\wedge u)\vee (\mathrm{\neg }s\vee \mathrm{\neg }v)$ $(e)((p\vee t)\wedge (q\vee u))\leftrightarrow (s\wedge v)$ One must sometimes be a bit creative in using language to make the results comprehensible

Short answer

(a) If apple pie, ink, and cheese, then nothing to drink. (b) If Socrates and men are mortal, then Socrates was mortal. (c) If something to drink, then Socrates was not mortal. (d) Either conditions (apple pie and ink and cheese; Socrates and mortality; or drink or immortality). (e) Either apple pie or Socrates, and either ink or mortality iff nothing to drink and mortal.

Step-by-step solution

Translate the Logical Expression (a)

The expression $(p\wedge q\wedge r)\to s$ translates to "If all the world is apple pie, and all the seas are ink, and all the trees are bread and cheese, then there is nothing to drink." This is because the conjunction on the left forms the condition under which the implication leads to the consequence (s).

Translate the Logical Expression (b)

The expression $(t\wedge u)\to v$ translates to "If Socrates was a man, and all men are mortal, then Socrates was mortal." This uses the conjunction of being a man and all men being mortal as the premises leading to Socrates' mortality.

Translate the Logical Expression (c)

The expression $egs\to egv$ translates to "If there is something to drink, then Socrates was not mortal." The negations change the original statements to their opposites and are related by the implication.

Translate the Logical Expression (d)

The expression $p\wedge (q\wedge r)\vee (t\wedge u)\vee (egs\vee egv)$ can be translated as "Either all the world is apple pie and all the seas are ink and all the trees are bread and cheese, or Socrates was a man and all men are mortal, or there is something to drink or Socrates was not mortal." This uses a combination of conjunctions and disjunctions.

Translate the Logical Expression (e)

The expression $((p\vee t)\wedge (q\vee u))\leftrightarrow (s\wedge v)$ translates as "Either all the world is apple pie or Socrates was a man, and either all the seas are ink or all men are mortal, if and only if there is nothing to drink and Socrates was mortal." This is a biconditional statement with both sides conditioned by conjunctions and disjunctions.

Key concepts

Logical Expressions

Propositional logic primarily deals with logical expressions, which are combinations of propositions linked by logical connectors. Each proposition indicates a basic statement that can either be true or false. By connecting these propositions, logical expressions form more complex statements that reveal relationships, conditions, or outcomes.

Here are some key aspects of logical expressions:

• **Atomic Propositions**: These are the simplest form of logical expressions, like our individual letters $p,q,r,s,t,u,v$.
• **Logical Connectors**: They connect propositions to form expressions, such as "and" ($\wedge$), "or" ($\vee$), "not" ($eg$), "implies" ($\to$), and "if and only if" ($\leftrightarrow$).

Understanding these symbols and how they come together is essential to effectively translate and manipulate logical expressions. They form the foundation for more sophisticated logical reasoning processes.

Translations in Logic

The translation of logical expressions involves converting symbolic representations into natural language. This process enhances understanding by framing abstract symbols within understandable statements.

To translate logical expressions:

• **Identify each proposition** with its meaning, such as "$p$ = all the world is apple pie."
• **Interpret logical operators**, translating $\wedge$ as "and," $\vee$ as "or," $eg$ as "not," $\to$ as "if...then," and $\leftrightarrow$ as "if and only if."
• **Combine** the translated propositions according to their logical operators to form coherent sentences, such as our examples "If $(p\wedge q\wedge r)\to s$", which translates to "If all the world is apple pie, all the seas are ink, and all trees are bread and cheese, then there is nothing to drink."

Through translation, propositional logic bridges abstract reasoning with verbal clarity, enhancing comprehension.

Conjunctions and Disjunctions

Conjunctions and disjunctions in logic determine how multiple propositions come together within expressions.

**Conjunctions** use the "and" operator, represented by $\wedge$, combining propositions that must all be true for the entire statement to be true. For instance, $(p\wedge q\wedge r)$ implies all three parts must hold true simultaneously.

**Disjunctions** employ the "or" operator, represented by $\vee$. In this case, if any single proposition within the disjunction holds true, the entire statement is considered true. For example, in the expression $p\wedge (q\wedge r)\vee (t\wedge u)$, only one of these groups of propositions needs to be true.

Understanding these concepts is vital in determining conditions and outcomes within logical scenarios, affecting how propositions contribute to the truth-value of expressions.

Logical Implications

Logical implication is a critical aspect of logical reasoning, often expressed with the "if...then" structure using $\to$ in propositions. This relationship represents a conditional statement where the truth of one proposition leads to the truth of another.

Consider the expression $(t\wedge u)\to v$. Here, if Socrates was a man ($t$) and all men are mortal ($u$), then it logically follows that Socrates was mortal ($v$).

Logical implications check the dependency between conditions and outcomes, ensuring clarity in argument structures. They are foundational for deductive reasoning, validating conclusions based on given premises.

This form of logical connectiveness allows for the creation of reliable computational models, proofs, and algorithmic processes that systematically derive results based on initial conditions.