Conjunctive Statements in Propositional Logic

There are 4 kinds of compound statements utilized in symbolic logic, specifically: 

1) conjunctive, 

2) disjunctive, 

3) conditional, and 

4) biconditional 

In these notes, I’ll focus solely on conjunctive statements.

A conjunctive assertion or conjunction is a compound assertion related by the phrase “and.” The part statements in a conjunction are referred to as conjuncts. Allow us to contemplate this instance:

Roses are crimson and jasmines are white.

Clearly, the above assertion is a conjunction as a result of it’s related by the phrase “and.” The primary assertion “Roses are crimson” is the primary conjunct and the assertion “Jasmines are white” is the second conjunct.

In my notes titled “Propositions and Symbols Utilized in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), the image for “and” is • (dot). Now, if we let p stand for “Roses are crimson” and q for “Jasmines are white,” then the assertion “Roses are crimson and jasmines are white” is symbolized as follows:

p • q

In some instances, a conjunctive assertion doesn’t use the phrase “and” as connective. Generally, the next phrases are used as connectives of a conjunctive assertion:

However

Nevertheless

Nonetheless

Although

Whereas

Though

Whereas

Nonetheless

But

Take into account the next examples:

1. Chocolate is scrumptious, however it isn’t a superb meals for folks with diabetes.
2. Lucas is enjoying, whereas Rob is finding out.
3. The instructor was already shouting, but the scholars stay very noisy.

In instances the place there are not any phrases that signify a conjunction, a comma (,) or a semi-colon (;) might point out that the assertion is a conjunction. Take into account the instance beneath:

Though the human individual is mortal, she will be able to dwell lengthy.

Symbolizing Conjunctive Statements

I’ve been symbolizing statements above and in my earlier posts, however it isn’t till now that I’ll particularly speak about symbolizing statements.

Firstly, logicians often put the variables or constants that can symbolize the assertion proper after the assertion per se. Take into account the examples beneath:

Chocolate is scrumptious, however it isn’t a superb meals for folks with diabetes. (p, q)

Please observe that the variables offered after the assertion symbolize the part statements respectively. Thus, within the instance above, the variable p represents the primary part assertion “Chocolate is scrumptious,” whereas q represents the second part assertion “It’s not a superb meals for folks with diabetes.”

Secondly, when symbolizing statements, we have to put correct punctuations and negation if vital. Thus, within the instance above, the assertion “Chocolate is scrumptious” is represented by p, whereas the assertion “It’s not a superb meals for folks with diabetes” is represented by q. If we aren’t cautious, we might symbolize the assertion as follows: p • q. Nevertheless, if we analyze the assertion, we discover that the second part accommodates a negation signal “It’s not the case.” Therefore, the assertion “Chocolate is scrumptious, however it isn’t a superb meals for folks with diabetes” is symbolized as follows:

p • ~q

You will need to observe that typically the phrase “and” just isn’t truth-functional, that’s, it doesn’t join two unbiased propositions. Thus, if this happens, we must always symbolize the proposition merely as a easy proposition. Take into account the next instance:

Bread and butter is a good mixture.

Clearly, the “and” within the instance above just isn’t truth-functional as a result of it doesn’t join two truth-functional propositions or sentences. It’s because we can’t say that “Bread is an ideal mixture” and “Butter is an ideal mixture.” Therefore, the proposition “Bread and butter is a good mixture” is symbolized merely as:

p

Nevertheless, if we now have the instance

“John and Mary are watching TV”

then we now have to represent this as:

p • q

It’s because the “and” right here is truth-functional, that’s, it connects two unbiased propositions or sentences. For positive, it’s attainable for us to say “John is watching TV” and “Mary is watching TV.” In different phrases, each John and Mary are watching TV.

Guidelines in Conjunction

1. A conjunction is true if and provided that each conjuncts are true.
2. If at the very least one of many conjuncts is false, then the conjunction is false.

The reality desk beneath illustrates this level.

The reality desk above says:

1) If p is true and q is true, then p • q is true.

2) If p is true and q is false, then p • q is false.

3) If p is false and q is true, then p • q is false.

4) If p is false and q is false, then p • q is false.

Now, given the rule in conjunction, how can we decide the truth-value of the conjunctive assertion p • ~q?

Allow us to suppose that the truth-value of p is true and q is fake. So, if p is true and q false, then the assertion p • ~q is true. As an example:

The illustration above says that p is true and q is fake. Now, earlier than we apply the rule in conjunction within the assertion p • ~q, we have to simplify ~q first as a result of the truth-value “false” is assigned to q and to not ~q. If we recall our dialogue on the rule in negation, we realized that the negation of false is true. So, if q is false, then ~q is true. Thus, on the finish of all of it, p • ~q is true if p is true and q is false.