Determine whether or not the following are correct or erroneous applications of

Question

Determine whether or not the following are correct or erroneous applications of the conjunction rules. starting with the given premise(s), is the use of the conjunction rule correct or not?

1.
1) P & ( Q& (R& S)) PREMISE
2) R & S                     1 &E

2.
1) P& (Q & (R&S)) PREMISE
2) P&Q                     1 &E

3.
1) P& (Q & (R&S)) PREMISE
2) Q & (R&S)          1 &E

4.
1) P                          Premise
2) Q&R                    Premise
3) P & Q                  1,2&I

Explanation / Answer

Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement. The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both. However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another.

Table of Contents

1. Introduction

A statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false. So, for example, the following are statements:

Sometimes, a statement can contain one or more other statements as parts. Consider for example, the following statement:

While the above compound sentence is itself a statement, because it is true, the two parts, "Ganymede is a moon of Jupiter" and "Ganymede is a moon of Saturn", are themselves statements, because the first is true and the second is false.

The term proposition is sometimes used synonymously with statement. However, it is sometimes used to name something abstract that two different statements with the same meaning are both said to "express". In this usage, the English sentence, "It is raining", and the French sentence "Il pleut", would be considered to express the same proposition; similarly, the two English sentences, "Callisto orbits Jupiter" and "Jupiter is orbitted by Callisto" would also be considered to express the same proposition. However, the nature or existence of propositions as abstract meanings is still a matter of philosophical controversy, and for the purposes of this article, the phrases "statement" and "proposition" are used interchangeably.

Propositional logic, also known as sentential logic, is that branch of logic that studies ways of combining or altering statements or propositions to form more complicated statements or propositions. Joining two simpler propositions with the word "and" is one common way of combining statements. When two statements are joined together with "and", the complex statement formed by them is true if and only if both the component statements are true. Because of this, an argument of the following form is logically valid:

Paris is the capital of France and Paris has a population of over two million.
Therefore, Paris has a population of over two million.

Propositional logic largely involves studying logical connectives such as the words "and" and "or" and the rules determining the truth-values of the propositions they are used to join, as well as what these rules mean for the validity of arguments, and such logical relationships between statements as being consistent or inconsistent with one another, as well as logical properties of propositions, such as being tautologically true, being contingent, and being self-contradictory. (These notions are defined below.)

Propositional logic also studies way of modifying statements, such as the addition of the word "not" that is used to change an affirmative statement into a negative statement. Here, the fundamental logical principle involved is that if a given affirmative statement is true, the negation of that statement is false, and if a given affirmative statement is false, the negation of that statement is true.

What is distinctive about propositional logic as opposed to other (typically more complicated) branches of logic is that propositional logic does not deal with logical relationships and properties that involve the parts of a statement smaller than the simple statements making it up. Therefore, propositional logic does not study those logical characteristics of the propositions below in virtue of which they constitute a valid argument:

The recognition that the above argument is valid requires one to recognize that the subject in the first premise is the same as the subject in the second premise. However, in propositional logic, simple statements are considered as indivisible wholes, and those logical relationships and properties that involve parts of statements such as their subjects and predicates are not taken into consideration.

Propositional logic can be thought of as primarily the study of logical operators. A logical operatoris any word or phrase used either to modify one statement to make a different statement, or join multiple statements together to form a more complicated statement. In English, words such as "and", "or", "not", "if ... then...", "because", and "necessarily", are all operators.

A logical operator is said to be truth-functional if the truth-values (the truth or falsity, etc.) of the statements it is used to construct always depend entirely on the truth or falsity of the statements from which they are constructed. The English words "and", "or" and "not" are (at least arguably) truth-functional, because a compound statement joined together with the word "and" is true if both the statements so joined are true, and false if either or both are false, a compound statement joined together with the word "or" is true if at least one of the joined statements is true, and false if both joined statements are false, and the negation of a statement is true if and only if the statement negated is false.

Some logical operators are not truth-functional. One example of an operator in English that is not truth-functional is the word "necessarily". Whether a statement formed using this operator is true or false does not depend entirely on the truth or falsity of the statement to which the operator is applied. For example, both of the following statements are true:

However, let us now consider the corresponding statements modified with the operator "necessarily":

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