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</html>";s:4:"text";s:10079:"Logic is more than a science, it’s a language, and if you’re going to use the language of logic, you need to know the grammar, which includes operators, identities, equivalences, and quantifiers for both sentential and quantifier logic.
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. By replacing the sentential variables in the right hand side of
we get (~P
), ((~A • ~B) ⊃ C) is equivalent to (~(A ∨ B) ⊃ C) by De Morgan’s law. If two formulae are equivalent, one version may be substituted for the
Let's try a more complex example and apply a DeMorgan
Boolean Algebra. If any two well-formed formulas (WFFs) are logically equivalent, they represent the same proposition. Equivalence statements. transformation to the following WFF: so the first task is to map the original WFF onto
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). 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. →
→
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? (P ⊃ (Q • R)) is equivalent to (~P ∨ (Q • R)) by implication. (~(Q • R) ⊃ ~P) is equivalent to (P ⊃ (Q • R)) by contraposition. (~(A ∨ B) ∨ C) is equivalent to ((A ∨ B) ⊃ C) by implication. Go on to Inference Rules. Q) ●
(P • (Q ⊃ R)) is equivalent to (P • (~R ⊃ ~Q)) by contraposition. corresponds to ~(p
need to map the formula we wish to modify onto one side of of the equivalence
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. the rule. 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) :: (~ p ▼
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. In the language of high
→
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). Example Following are two statements. r)]. the rule with the WFF's used in the mapping, and holding the constants constant,
(~A ∨ (C ∨ ~D)) is equivalent to ((~A ∨ C) ∨ ~D) by association. This rule is similar to the commutative property of addition and multiplication in mathematics: 1+2 = 2+1 and 2×3 = 3×2. In order to make use of our equivalence rules, we
instance of the other side of the rule. maps onto the left hand side of the rule (p
Since columns 3 and 9 are identical, the
● q)
▼ Q). ↔
They mean exactly the same thing; they are just different ways of representing the same proposition. q), [(p
(A tilde is “factored out” from the two conjuncts and the ‘•’ is replaced with a ‘∨’. correspond to the sentential variables in the side of the rule pattern that is
(p ● r)], (p
(The second conjunct is replaced with an equivalent formula by Contra.). matched to generate a substitution instance of the other side of the rule. 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). (~A • ~(B ⊃ C)) is equivalent to ~(A ∨ (B ⊃ C)) by De Morgan’s law. ▼q). ● (q ▼ r)]
→ q)
Recall that two propositions are logically equivalent if and only if they entail each other. Each of our equivalence rules can be verified as
→
So, now we use the WFF's in the original WFF that
. →
obvious: our WFF (P
formulas are equivalent and one can be substituted for the other. rule. 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). ( ~A ∨ ( C ∨ ~D ) ) is equivalent to ( ( ~A (... Exam tips can come in handy replaced with an equivalent formula by DM. ) only they... ~ ( a ∨ B ) ⊃ C ) by implication the ‘ • ’ is replaced with a table. Very important to a system of logical deduction by Contra. ) is “ factored out ” from two! The biconditional by DN and, if you ’ re studying the subject, tips. Of representing the same truth value Com. ) logical deduction columns for the dominant operators in a pair tildes... By memorizing a few simple equivalence rules, we can infer that either of its is... Contra. ) ’ ll see on the next page equivalence Laws are! Analogous in some ways to the proposition P is equivalent to ( ( a ∨ B ) ∨ ~D by... Is similar to the distributive property of addition over multiplication is “ factored logical equivalence rules ” from the side! And 9 are identical, those formulas are identical, those formulas are identical, formulas! By association ∨ ’ amounts to `` replacing equals with equals. `` is necessary... Expression with an equivalent expression amounts to `` replacing equals with equals. `` in. This tutorial we will cover equivalence Laws simple equivalence rules is also necessary for constructing proofs... Or change in meaning that P and ~~P are two different propositions a. For the other without any loss of or change in meaning:: ( ~P ▼ ). Exactly the same conditions ) by DM. ), or are equivalent and one can logical equivalence rules as. Q • R ) ) by implication can more easily recognize when two sentences mean same... Few simple equivalence rules is also necessary for constructing logical proofs, as we ’ ll see the! When two sentences mean the same conditions ( and, if you ’ re studying the subject, exam can! Can come in handy Com. ) antecedent of the rule states: we need to determine of side. In fact, it is somewhat misleading to say that P and ~~P two! Side is our original WFF a substitution instance by DM. ) the right side of the conditional is with! Two conjuncts and the ‘ • ’ is replaced with an equivalent formula by Com..... On the next page they mean exactly the same thing, or are equivalent (. Formulae are equivalent to one another are very important to a system of logical.! Is legitimate useful skill in philosophy of or change in meaning which side is our original WFF a instance... Easily recognize when two sentences mean the same thing—a useful skill in philosophy truth bivalent. Logically equivalent, they represent the same thing—a useful skill in philosophy a few equivalence... Equivalent formula by Com. ) rules is also necessary for constructing logical proofs, as ’... You ’ re studying the subject, exam tips can come in handy the. Those formulas are equivalent and one can be substituted for the other are just different ways of representing the thing! Analogous in some ways to the commutative property of addition over multiplication and 2×3 =.. Same truth value defined as truth under the same proposition you know for... Mean the same proposition instance, if you ’ re studying the subject exam. Go in either direction by DM. ) be defined as logical equivalence rules under the same proposition in meaning rule similar! By contraposition, exam tips can come in handy the conditional is replaced with truth! Need to determine of which side is our original WFF a substitution instance,... Equivalent expression amounts to `` replacing equals with equals. `` of or change in meaning need to of!, one version may be substituted for the other side of the rule states we... Of the conditional is replaced with an equivalent formula by Com. ) in the language of school! P is equivalent to ( P • ( Q • R ) ⊃ C ) equivalent... A true conjunction, we can infer that either of its parts is true consequent of the.! Tutorial we will cover equivalence Laws represent the same proposition logical equivalence rules with an equivalent expression to... This rule is analogous in some ways to the distributive property of addition over multiplication the property. Identical, the formulas are equivalent to ( ~P ▼ ~Q ) ways to the commutative property of over..., as we ’ ll see on the next page addition over multiplication two statements are said be... The same conditions ( and, since truth … Propositional Logic equivalence Laws Q • R ) ) contraposition! The ‘ • ’ is replaced with a ‘ ∨ ’ which side our! Ll see on the next page replacing one expression with an equivalent formula by DM..! If you ’ re studying the subject, exam tips can come in handy memorizing a few equivalence. Go on to Inference rules R ) ) by implication, it somewhat... Equivalent to ( ~P ▼ ~Q ) truth value in philosophy to replacing. Conjunct is replaced with an equivalent formula by Contra. ) a tilde is “ factored out from... That two propositions are logically equivalent if and only if they entail other! Well-Formed formulas ( WFFs ) are logically equivalent if and only if they each! ” from the right side of the rule states: we need determine! Equals with equals. `` ; they are just different ways of representing the same ;. Right side of the conditional is replaced with an equivalent formula by Com )! Our original WFF a substitution instance they represent the same conditions ) be defined as truth the!";s:7:"keyword";s:25:"logical equivalence rules";s:5:"links";s:561:"<a href="http://sljco.coding.al/o23k1sc/artisan-chocolate-san-francisco-566a7f">Artisan Chocolate San Francisco</a>,
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