# Logical Implication

Definition. A formula $P$ logically implies a formula $Q$ if and only if for all Boolean valuations, if $[P]=T$ then $[Q]=T$.

$P⊨Q$

which is pronounced “P entails Q” or “P logically implies Q”.

$P⊨Q$ iff $⊨P⇒Q$ iff $P⇒Q$ is a tautology. (where the meta-symbols are in black and the formulas are in red and substituting some formulas for P and Q)

So |= P means true |= P.

### Valid Arguments

Definition. An argument is a valid argument if in all Boolean valuations where the premises have the value $T$, the conclusion has the truth value $T$. $P_{1},P_{2},…,P_{n}⊨C$

- $P_{1},P_{2},…,P_{n}$ are
**premises** - $C$ is the
**conclusion**

An argument is valid if and only if the conjunction of the premises logically imply the conclusion.

$⊨P$ means formula $P$ is a tautology. We can also say $P$ is “valid” to mean $P$ is a Tautology.

To prove an valid argument, with:

- Transformational Proof: Assume premises, and them together, and arrive to the conclusion, i.e. $P_{1},P_{2},…∣=Q$
- Natural Deduction: Just whatever the argument is
- Semantic Tableaux: Use the premises and the negation of the conclusion.

### Invalid Arguments

Definition. An argument is invalid if and only if there is at least one Boolean valuation in which the premises are $T$, but the conclusion is $F$.

To show that an argument is invalid, **find a counterexample**: a Boolean valuation where the premises are $T$ and the conclusion is $F$.

- Natural Deduction: Have the premises and conclude False, i.e $P_{1},P_{2},…,P_{n}∣−false$

### What about Logical Equivalence

See page 110 of the notes. (not to be confused), the above is stronger

Remember that |= is not ⇚⇛. The formula C can be “weaker” than the conjunction of the premises. “Weaker” means T in more Boolean valuations.

For example, $a∧b⊨a∨b$ The formula a ∨ b is T in some Boolean valuations where a ∧ b is not T.