Logic and Logical Connectives

The video focuses on applying the understanding of logic, specifically logical connectives, to evaluate complex sentences.
Four logical connectives:
- Negation: not the case that p ( $\neg p$ )
- Conjunction: both p and q ( $p \land q$ )
- Disjunction: either p or q ( $p \lor q$ )
- Conditional: if p, then q ( $p \rightarrow q$ ), or p only if q
Simple sentences use only one connective, while complex sentences use more than one.
Examples of complex sentences:
- It's not the case that both P and Q.
- If either P or Q, then R.

Negation: It's not the case that p ( $\neg p$ ).
'p' is a variable replaceable by sentences, statements, or claims.
The truth table for negation indicates when a negation is true or false.
Evaluating a complex negation:
- Replace the variable with a sentence.
- Determine if the replaced sentence (the part being negated) is true or false.
- If the negated sentence is true, the whole negation is false.
- If the negated sentence is false, the whole negation is true.
Example:
- It's not the case that both m and n.
- Could be, "It's not the case that both the Earth is flat and the sky is blue."

What matters in evaluating any negation (simple or complex) is the truth value of the sentence being negated.
If the negated sentence is true, the entire negation is false.
- If the negated sentence is false, the entire negation is true.

Conjunction: both P and Q ( $p \land q$ ).
Truth table for conjunction:
- Both P and Q are true: conjunction is true.
- P is true, Q is false: conjunction is false.
- P is false, Q is true: conjunction is false.
- Both P and Q are false: conjunction is false.

When evaluating complex conjunctions, determine the truth values of the individual conjuncts (the parts being conjoined).
If both conjuncts are true, the whole conjunction is true.
If one or both conjuncts are false, the whole conjunction is false.

Disjunction: either p or q ( $p \lor q$ ).
When evaluating a disjunction, determine the truth values of the individual disjuncts.
Truth table for disjunction:
- Both disjuncts are true: disjunction is true.
- One disjunct is true: disjunction is true.
- Both disjuncts are false: disjunction is false.

Conditional: if this, then that, or this only if that ( $p \rightarrow q$ ).
When evaluating a conditional, determine the truth values of the antecedent (the 'if' part) and the consequent (the 'then' part).
Truth table for conditional:
- Antecedent true, consequent true: conditional is true.
- Antecedent true, consequent false: conditional is false.
- Antecedent false, consequent true: conditional is true.
- Antecedent false, consequent false: conditional is true.

Assume: m is true, n and o are false.
Problem 1: Negation of a conditional. It's not the case that if m then n.
- Identify as a complex negation.
- The sentence being negated is a conditional (if m, then n).
- Evaluate the conditional: m is true, n is false, so the conditional is false.
- Since the conditional being negated is false, the whole negation is true.
Problem 2: Conjunction involving a negation. (Not n) and o.
- Identify as a conjunction.
- One conjunct is a negation (not n).
- Evaluate (not n): n is false, so (not n) is true.
- The other conjunct (o) is false.
- Since one conjunct is true and the other is false, the whole conjunction is false.
Problem 3: Conditional involving a disjunction. If m, then (n or o).
- Identify as a conditional.
- The consequent is a disjunction (n or o).
- Evaluate the disjunction: n and o are both false, so the disjunction is false.
- The antecedent (m) is true.
- Since the antecedent is true and the consequent is false, the whole conditional is false.

When evaluating complex sentences:
- Determine the main connective (negation, conjunction, disjunction, conditional).
- Evaluate the parts of the sentence based on the relevant truth table.
- Work from the inside out, evaluating simpler parts before the whole.

Example: A highly complex sentence using all four connectives. If (m and (not(n or o))), then …
The same basic principles apply, breaking down the sentence into its parts and evaluating each part.

These principles can be applied to ordinary English sentences.
Example 1: "The Earth is flat and the sky is blue."
- Identify as a conjunction.
- "The Earth is flat" is false.
- Therefore, the whole conjunction is false.
Example 2: "If all humans are mammals, then two plus two is five."
- Identify as a conditional.
- "All humans are mammals" is true (antecedent).
- "Two plus two is five" is false (consequent).
- Therefore, the whole conditional is false.