↔ :: [(p ● q) ▼ operators in a pair of formulas are identical, those formulas are said to be equivalent. DeMorgan's transformation: The following truth table establishes that the ● q) → (R  → Consider the equivalence rule known as Boolean Algebra. Here are six inference rules worth memorizing: (x • (y • z)) is equivalent to ((x • y) • z), (x ∨ (y ∨ z)) is equivalent to ((x ∨ y) ∨ z), (x ≡ (y ≡ z)) is equivalent to ((x ≡ y) ≡ z). Q) ● ▼q). formulas are equivalent and one can be substituted for the other. q) :: (~ p ▼ Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); nam… (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. → q) EQUIVALENCE RULES (Rules of Replacement) Whenever the truth table columns for the dominant operators in a pair of formulas are identical, those formulas are said to be equivalent. formula onto the rule. So, now we use the WFF's in the original WFF that we get (~P original WFF a substitution instance. the rule with the WFF's used in the mapping, and holding the constants constant, → Return to Tutorials Index    We can think of the ‘~’ (tilde) outside the parentheses as being “distributed” to each of the components inside the parentheses, in the same way we distribute a multiple: 2×(3+4) = 2×3 + 2×4. maps onto the left hand side of the rule (p If any two well-formed formulas (WFFs) are logically equivalent, they represent the same proposition. (The consequent of the conditional is replaced with an equivalent formula by Com.). ~S)] can be replaced with [~(P In fact, it is somewhat misleading to say that P and ~~P are two different propositions. school algebra, replacing one expression with an equivalent expression amounts equivalence. ↔ → This rule is similar to the commutative property of addition and multiplication in mathematics: 1+2 = 2+1 and 2×3 = 3×2. (~A • ~(B ⊃ C)) is equivalent to ~(A ∨ (B ⊃ C)) by De Morgan’s law. Moreover, the substitution can go in either direction. → Equivalence can be defined as truth under the same conditions (and, since ● instance of the other side of the rule. → transformation to the following WFF: so the first task is to map the original WFF onto q), [(p By memorizing a few simple equivalence rules, we can more easily recognize when two sentences mean the same thing—a useful skill in philosophy. ). Each of our equivalence rules can be verified as Propositional Logic Equivalence Laws. They are: [p 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. Two statements are said to be equivalent if they have the same truth value. (~(Q • R) ⊃ ~P) is equivalent to (P ⊃ (Q • R)) by contraposition. q) ::  (~p ▼ ~q). (A tilde is “factored out” from the two conjuncts and the ‘•’ is replaced with a ‘∨’. Share ← → In this tutorial we will cover Equivalence Laws. Since P is equivalent to ~~P by the “double negation” rule, for example, (Q • P) is likewise equivalent to (Q • ~~P), by that same rule. Q) ▼~ ~S)] and vice versa. (P • (Q ⊃ R)) is equivalent to (P • (~R ⊃ ~Q)) by contraposition. Since the rule states: we need to determine of which side is our correspond to the sentential variables in the side of the rule pattern that is The proposition P is equivalent to the proposition ~~P, for example. Alternatively, we can imagine “factoring out” the tilde from each component inside the parentheses, just as we factor out a multiple: 2×3 + 2×4 = 2×(3+4). other without any loss of or change in meaning. ~p), (p :: (~p Q) p = It … (p ● r)], (p truth is bivalent, falsity under the same conditions). The analogy isn’t perfect, though. material implication to the formula (P  → By replacing the sentential variables in the right hand side of ), ((~A • ~B) ⊃ C) is equivalent to (~(A ∨ B) ⊃ C) by De Morgan’s law. matched to generate a substitution instance of the other side of the rule. This rule is analogous in some ways to the distributive property of addition over multiplication. → Since columns 3 and 9 are identical, the In this case the mapping is r)]. Let's try a more complex example and apply a DeMorgan An equivalence rule is a pair of equivalent proposition forms, with lowercase letters used as variables for which we can substitute any WFF (just as we did previously with inference rules). rule. legitimate with a truth table. Whenever the truth table columns for the dominant ● (q ▼ r)] (~(A ∨ B) ∨ C) is equivalent to ((A ∨ B) ⊃ C) by implication. q) :: (~ q 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. (R  Recall that two propositions are logically equivalent if and only if they entail each other. Recall that two propositions are logically equivalent if and only if they entail each other. When “distributing” or “factoring out” the tilde, we also have to change the ‘•’ to a ‘∨’ or, ((P • Q) ≡ ~~R) is equivalent to ((P • Q) ≡ R) by double-negation. As you know, for instance, if we have a true conjunction, we can infer that either of its parts is true. ▼ Q). They mean exactly the same thing; they are just different ways of representing the same proposition. We get: so we know that ~[(P → :: [p need to map the formula we wish to modify onto one side of of the equivalence Equivalence can be defined as truth under the same conditions (and, since truth …

## logical equivalence rules

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