7.5.1. Valid Argument Form for Premises

A functionally complete set of logical operators allows every compound proposition to be represented using only those operators. In our case, we want to prove that the set consisting of conjunction (AND), disjunction (OR), and negation (NOT) is functionally complete. Any compound proposition involving implications can be rewritten using these operators.

A functionally complete set of logical operators means that you can create any logical statement using just a small subset of operators. The three fundamental operators - conjunction (AND), disjunction (OR), and negation (NOT) - can achieve this. For instance, if your starting proposition includes an implication (p → q), you can replace it with its equivalent form using negation and disjunction: ¬p ∨ q. By repeatedly applying this substitution, you can express any logical statement solely using AND, OR, and NOT.

Think of this like cooking with a minimal set of ingredients. If you have a few basic ingredients like salt (negation), pepper (disjunction), and oil (conjunction), you can create a variety of dishes, just like how these logical operators can create any logical statement.

When you encounter a bi-implication (p ↔ q), it can be rewritten as the conjunction of two implications (p → q) and (q → p). This means that every instance of bi-implication can also be expressed using only conjunction, disjunction, and negation.

A bi-implication indicates that two propositions are equivalent. You can break this down into two implications that use only the conjunction and disjunction operators along with negation. For example, (p ↔ q) can be rewritten as (p → q) ∧ (q → p). By substituting the implications with their negated forms, you can eventually express this entire statement only with AND, OR, and NOT operators.

Imagine two friends agreeing to meet only if both are free. If you think of their availability as a set of 'conditions', you can express their meeting arrangements using simpler rules, showing that fundamental conditions lead to complex agreements based on those two.

It is possible to represent any statement using just the negation and disjunction operators. Specifically, any conjunction can be expressed as a combination of negations and disjunctions utilizing De Morgan's laws.

Using only disjunction and negation is equally powerful as using conjunction. For any conjunction, like p ∧ q, we can express it as negation of the negation form: ¬(¬p ∨ ¬q). This follows from De Morgan's law, which allows you to transform AND statements into expressions involving ONLY ORs and NOTs. Thus, having just disjunction and negation enables you to express all logical statements.

Think of this like having a complicated gadget. You can either use direct methods or, using creativity, come up with workarounds to achieve the same outcome. Using just disjunction and negation is like finding clever shortcuts that might take longer initially but ultimately lead to the same result.

An argument form is valid if the conclusion logically follows from the premises. For example, given premises p and p implies q, you can apply Modus Ponens to conclude q.

The validity of an argument relies on whether the conclusion can logically be derived from the premises. A classic example is Modus Ponens, where if you have p and also know p implies q, you can confidently state q must be true. This logical step is the basis for many forms of reasoning in mathematics and philosophy.

Consider a simple decision-making process: If it is raining (p) and I have an umbrella (p implies q), then I will stay dry (concluding q). Here, the outcome (staying dry) relies entirely on the initial conditions (it is raining and I have the umbrella).

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