equivalence is legitimate. The proposition P is equivalent to the proposition ~~P, for example. Proofs Using Logical Equivalences Rosen 1.2 List of Logical Equivalences List of Equivalences Prove: (p q) q p q (p q) q Left-Hand Statement q (p q) Commutative (q p) (q q) Distributive (q p) T Or Tautology q p Identity p q Commutative Prove: (p q) q p q (p q) q Left-Hand Statement q (p q) Commutative (q p) (q q) Distributive Why did we need this step? In other words, equivalent propositions have the same truth value in all possible circumstances: whenever one is true, so is the other; and whenever one is false, so is the other. ~S)] can be replaced with [~(P
(~A ∨ (C ∨ ~D)) is equivalent to ((~A ∨ C) ∨ ~D) by association. Equivalence statements. :: [p
Q) ●
Equivalence can be defined as truth under the same conditions (and, since
Equivalence can be defined as truth under the same conditions (and, since truth … the rule with the WFF's used in the mapping, and holding the constants constant,
(The antecedent of the conditional is replaced with an equivalent formula by DM.). Statements that say the same thing, or are equivalent to one another are very important to a system of logical deduction. (The consequent of the conditional is replaced with an equivalent formula by Com.). correspond to the sentential variables in the side of the rule pattern that is
Q)
→
(P • (Q ⊃ R)) is equivalent to (P • (~R ⊃ ~Q)) by contraposition. (The second conjunct is replaced with an equivalent formula by Contra.). matched to generate a substitution instance of the other side of the rule. :: [(p ● q) ▼
↔
other without any loss of or change in meaning. Our system utilizes 11 different rules of
(P ⊃ (Q • R)) is equivalent to (~P ∨ (Q • R)) by implication. For example, if we wish to apply the rule
If any two well-formed formulas (WFFs) are logically equivalent, they represent the same proposition. Boolean Algebra. (q
truth is bivalent, falsity under the same conditions). the rule. Q) ▼~
instance of the other side of the rule. →
In the language of high
Propositional Logic Equivalence Laws. rule and then using that mapping create a substitution
~S)] and vice versa. (A tilde is “factored out” from the two conjuncts and the ‘•’ is replaced with a ‘∨’. Whenever the truth table columns for the dominant
Since logically equivalent WFFs represent the same proposition, they can be substituted for one another in any context, even when they appear as components of a larger WFF. q) :: (~ p ▼
. (p ● r)], (p
For example, by De Morgan’s law, we can replace ~(A • B) with (~A ∨ ~B) and vice versa: we can replace (~A ∨ ~B) with ~(A • B). →
we get (~P
formulas are equivalent and one can be substituted for the other. Q), we first need to map the
The analogy isn’t perfect, though. legitimate with a truth table. Share ← → In this tutorial we will cover Equivalence Laws. (A pair of tildes is removed from the right side of the biconditional by DN. equivalence. need to map the formula we wish to modify onto one side of of the equivalence
If two formulae are equivalent, one version may be substituted for the
r]
:: (~p
Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); nam… → q)
In fact, it is somewhat misleading to say that P and ~~P are two different propositions. Moreover, the substitution can go in either direction. In other words, equivalent propositions have the same truth value in all possible circumstances: whenever one is true, so is the other; and whenever one is false, so is the other. As you know, for instance, if we have a true conjunction, we can infer that either of its parts is true. obvious: our WFF (P
Here are a few more examples: (A ⊃ (B ∨ C)) is equivalent to (A ⊃ (C ∨ B)) by commutation. Rules of Inference Modus Ponens p =)q Modus Tollens p =)q p ˘q) q )˘p Elimination p_q Transitivity p =)q ˘q q =)r) p ) p =)r Generalization p =)p_q Specialization p^q =)p q =)p_q p^q =)q Conjunction p Contradiction Rule ˘p =)F q ) p) p^q « 2011 B.E.Shapiro forintegral-table.com. material implication to the formula (P
They are: [p
). p = It … q) :: (~p ▼ ~q). By replacing the sentential variables in the right hand side of
to "replacing equals with equals.".
By memorizing a few simple equivalence rules, we can more easily recognize when two sentences mean the same thing—a useful skill in philosophy. (R
Equivalence rules. Familiarity with equivalence rules is also necessary for constructing logical proofs, as we’ll see on the next page. ▼q). Let's try a more complex example and apply a DeMorgan
maps onto the left hand side of the rule (p
This rule is analogous in some ways to the distributive property of addition over multiplication. DeMorgan's transformation: The following truth table establishes that the
In this case, the WFF maps onto the left hand side of the
I. DeMorgan’s Rule . And, if you’re studying the subject, exam tips can come in handy. Rules of Equivalence or Replacement. (~(Q • R) ⊃ ~P) is equivalent to (P ⊃ (Q • R)) by contraposition.