Analysing Propositional Reasoning Questions

The connection between specific sentences & specific truth assignments is known as satisfaction. In logic, qualities and relationships of sentences that hold across all truth assignments are usually of more interest. The validity, contingency, and unsatisfiability of individual sentences (as opposed to relationships among phrases) are the topics of this chapter. Then we’ll look at logical entailment, logical equivalency, and logical consistency, which are three different sorts of logical relationships between sentences. We wrap up with a study of the links between the logical features of individual phrases and the logical relationships that exist between them.

Logical Characteristics

We observed in the last chapter that some sentences are correct in certain truth assignments but incorrect in others. This isn’t always the case, though. There are sentences which are always true and always false, as well as sentences that are true and untrue at different times. As a result, sentences are divided into three distinct groups.

A sentence is legitimate if and only if every truth assignment agrees with it. For instance, the statement (p ˄ p) is correct. If p is true as a result of a truth assignment, the first disjunct is true, and the disjunction as a whole is true. If p is false as a result of a truth assignment, the second disjunct is true, and the disjunction as a whole is true.

Validity, contingency, & unsatisfiability can be divided into two groups for a variety of purposes. If and only if a sentence is valid or dependent, we call it satisfiable. In other words, at least some truth assignment satisfies the phrase. If and only if a sentence is unsatisfiable or contingent, we call it falsifiable. In other words, at least 1 truth assignment falsifies the phrase.

Equivalence of Logical Terms

Intuitively, we consider two sentences to be equal if they say exactly the same thing and are true in the same reality. In more technical terms, a sentence is logically equal to a sentence if and only if and only if every truth assignment that satisfies satisfies and every truth assignment that satisfies satisfies.

The sentence (p ˅ q) and the sentence (p ˄ q) are logically identical. Both phrases are false if both p and q are true. If either p or q is true, the first sentence’s disjunction is true, but the statement as a whole is false. Similarly, if either p or q is true, one of conjuncts with in second sentence is false, and so the statement is false as a whole. Both sentences are logically equivalent since they are satisfied by the same truth assignments.

The sentences (p ˅ q) and (p ˄ q) on the other hand, are not logically comparable. One is false if p is true & q is false, yet the disjunction is true in this case. As a result, they are not equal logically.

Checking the truth table for the proposition constants in the language is one approach to see if two sentences are logically equivalent. The truth table technique is what it’s called. 

(1) To begin, we create a truth table for the proposition constants, with a column for each statement. 

(2) The two expressions are then evaluated.

(3) Finally, the findings are compared. If the results again for two sentences are same in all cases, they are logically comparable; otherwise, they are not.

Let’s utilise this strategy to demonstrate that (p ˅ q) and (p ˄ q) are logically equal. We create our truth table, adding a column for each one of the two phrases, and assessing them for each true assignment. After that, we can see every row that answers yes to the first phrase also answers yes to the second. As a result, the phrases are equal logically.

Property and Relationship Interconnections

Before we wrap up this chapter, it’s important remembering that logical characteristics like validity and satisfiability have strong ties to the logical relationships discussed in the previous three parts.

Equivalence Theorem: So, if the sentence () is valid, a sentence and a sentence are logically equal.

The validity of the associated inference has a similar relationship to logical entailment between two phrases. Also, situations of logical entailment containing finite collections of phrases have a natural extension. These findings are summed up in the following theorem.

Theorem of Deduction: If and only if (ϕ) is true, a sentence logically entails another sentence (ϕ => Ψ). More generally, a finite collection of sentences {ϕ1…ϕn} logically implies only if the complex sentence (ϕ1˄…˄ϕn) is true.

Unsatisfiability Theorem: If and only if a set of sentences is unsatisfiable, the set of sentences logically entails a sentence.

Consistency Theorem: It states that a ∆ sentence is logically consistent with another sentence ϕ if and only if (∆ U ¬φ) is satisfiable. A statement is logical and consistent with a finite collection of sentences {ϕ1 ˄…˄ϕn} if but only if the complex sentence (ϕ1…ϕn) is satisfiable.

Conclusion

A sentence is legitimate if and only if every truth assignment agrees with it. A phrase is unsatisfiable if no truth assignment can satisfy it. A sentence is dependent if and only if it is satisfiable and falsifiable, that is, if it is neither legitimate nor unsatisfiable. If and only if a sentence is valid or dependent, it is satisfiable. If and only if a sentence is unsatisfiable or contingent, it is falsifiable

If & only if there is a truth assignment that satisfies both ∆ and ϕ, a sentence is consistent with a set of sentences. The Equivalence Theorem argues that if and only if the sentence (ϕ => Ψ) is valid, then ϕ sentence and a Ψ sentence are logically comparable.