Consider P, -P → Q
Valid, because there are no situations in which the premises are true.
-
There is no row in which all premises are true
-
so the conclusion is true in all rows in which all premises are true
there is no row in which all premises are true, yet the conclusion false.
Ex falso sequitur quodlibet
principle of explosion
PL is an explosive system of logic, ie. From inconsistence formulas anything follows.
You can derive anything from something of the form of premises: α, -α ╞ γ
Falsum and Verum
Falsum is a nullary connective, has 0 inputs and makes an output, which is always zero. Thus, falsum is a wff by itself.
Verum is a nullary connective, has 0 inputs and makes an output, which is always one. Thus verum is a wff by itself.
eg. -Pv┴, PvQ, ┬ → Q^┬
→ Valid
If you add falsum to something, you’re not really adding anything. It’s just zeroes.
Some general rules of falsum and verum
┬≈-┴
┴≈-┬
┬≈ αv- α
┴ ╞ α
α ╞ ┬
α ╞ ┴ iff ╞ -α
A set of formulas is inconsistent iff that set ╞ ┴
this is responsive to the conjunction of all formulas in the set also being a contradiction
P → Q (P implies Q, or if P then Q): The Material Conditional
P is the antecedent, and Q is the consequent.
Implication is not commutative. The order in which you mention P and Q matters.
Aristotle’s philosophy is heavily based on hylomorphism, everything is made up of form and matter.
Thus, in Aristotle there is a material and a formal conditional. The formal conditional is ╞, whilst the material conditional is →.
→ is a binary propositional connective
if α and β are wff, I can put the binary propositional connective in between.
→ is a truth functional connective
α β | α → β
1 1 1
1 0 0
0 1 1
0 0 1
Is equivalent to ”-αvβ”
-α → β ≈ αvβ
However, it is a matter of philosophical debate whether the if-then statement can be truth functional.
α → ┴ ≈ -α
cf. reductio ad absurdum
If San Marino wins the World Cup in football, I will eat my shoe
San Marino winn not win the World Cup in fooball (because I never eat my shoe).
With [→, ¬] everything can be properly expressed, they are expressively adequate and thus [→, ┴] are also expressively adequate.
P → Q and Q→ P are not commutative
1 1
0 1
1 0
1 1
Modus Ponens: α → β: α╞ β
Modus Tollens: α → β: -β╞ -α
Hypothetical Syllogism: α→β→γ α╞γ
Reasoning by cases: αvβ, α→γ, β→γ, γ╞γ
Strengthening the antecedent: a → g ╞
Weakening the antecedent: