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Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Math 10130 Lecture 32 9 April 2008 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Last time 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Last time Last time: 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Last time Last time: We considered quantiﬁed formulas in several diﬀerent contexts. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Last time Last time: We considered quantiﬁed formulas in several diﬀerent contexts. Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Last time Last time: We considered quantiﬁed formulas in several diﬀerent contexts. Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ” - true when the domain of discourse is the set of natural numbers N 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Last time Last time: We considered quantiﬁed formulas in several diﬀerent contexts. Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ” - true when the domain of discourse is the set of natural numbers N - false when the domain of discourse is the set of integers Z 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Last time Last time: We considered quantiﬁed formulas in several diﬀerent contexts. Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ” - true when the domain of discourse is the set of natural numbers N - false when the domain of discourse is the set of integers Z Principle: Truth or falsity of a statement depends on the context. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Alternation of Quantiﬁers 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Alternation of Quantiﬁers Consider the following structure: 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Alternation of Quantiﬁers Consider the following structure: Domain of discourse: people 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Alternation of Quantiﬁers Consider the following structure: Domain of discourse: people Mxy stands for “x loves y ” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Alternation of Quantiﬁers Consider the following structure: Domain of discourse: people Mxy stands for “x loves y ” Q: Do ∀x ∃y Mxy and ∃y ∀x Mxy express the same proposition? 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Comparing the two formulas 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Comparing the two formulas ∀x ∃y Mxy says, “For each person x, there is a y such that x loves y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Comparing the two formulas ∀x ∃y Mxy says, “For each person x, there is a y such that x loves y .” In other words, “Everybody loves somebody.” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Comparing the two formulas ∀x ∃y Mxy says, “For each person x, there is a y such that x loves y .” In other words, “Everybody loves somebody.” ∃y ∀x Mxy says, “There is a person y such that for all persons x, x loves y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Comparing the two formulas ∀x ∃y Mxy says, “For each person x, there is a y such that x loves y .” In other words, “Everybody loves somebody.” ∃y ∀x Mxy says, “There is a person y such that for all persons x, x loves y .” In other words, “Somebody is loved by everybody.” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A tough example 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A tough example Domain of discourse: the set of natural numbers, N 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A tough example Domain of discourse: the set of natural numbers, N Mxy stands for “x ≤ y ” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A tough example Domain of discourse: the set of natural numbers, N Mxy stands for “x ≤ y ” Let’s discuss this beast of a formula: 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A tough example Domain of discourse: the set of natural numbers, N Mxy stands for “x ≤ y ” Let’s discuss this beast of a formula: ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) Mxy & ¬x = y ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) Mxy & ¬x = y ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) (Mxz & ¬x = z) &(Mzy & ¬z = y ) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) Mxy & ¬x = y ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) (Mxz & ¬x = z) &(Mzy & ¬z = y ) Mxz & ¬x = z Mzy & ¬z = y 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A huge syntactic tree ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) 6 ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) Mxy & ¬x = y 1 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) 5 (Mxz & ¬x = z) &(Mzy & ¬z = y ) 4 Mxz & ¬x = z 2 Mzy & ¬z = y 3 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula 1 Mxy & ¬x = y says “x < y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula 1 Mxy & ¬x = y says “x < y .” 2 Mxz & ¬x = z says “x < z.” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula 1 Mxy & ¬x = y says “x < y .” 2 Mxz & ¬x = z says “x < z.” 3 Mzy & ¬z = y says “z < y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula 1 Mxy & ¬x = y says “x < y .” 2 Mxz & ¬x = z says “x < z.” 3 Mzy & ¬z = y says “z < y .” 4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or “x < z < y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula 1 Mxy & ¬x = y says “x < y .” 2 Mxz & ¬x = z says “x < z.” 3 Mzy & ¬z = y says “z < y .” 4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or “x < z < y .” 5 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) says “There is a z such that x < z < y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula 1 Mxy & ¬x = y says “x < y .” 2 Mxz & ¬x = z says “x < z.” 3 Mzy & ¬z = y says “z < y .” 4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or “x < z < y .” 5 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) says “There is a z such that x < z < y .” 6 (the big formula) says “For all x and y , if x < y , then there is a z such that x < z < y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula, continued 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula, continued “For all x and y , if x < y , then there is a z such that x < z < y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula, continued “For all x and y , if x < y , then there is a z such that x < z < y .” Is this true of all pairs of natural numbers? 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula, continued “For all x and y , if x < y , then there is a z such that x < z < y .” Is this true of all pairs of natural numbers? Consider the pair (0, 1). 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula, continued “For all x and y , if x < y , then there is a z such that x < z < y .” Is this true of all pairs of natural numbers? Consider the pair (0, 1). 0 < 1, but there is no z such that 0 < z < 1. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Breaking down the formula, continued “For all x and y , if x < y , then there is a z such that x < z < y .” Is this true of all pairs of natural numbers? Consider the pair (0, 1). 0 < 1, but there is no z such that 0 < z < 1. Thus, the formula is false. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context Let’s change the domain of discourse to the set of rational numbers Q, i.e. the set of postive and negative fractions. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context Let’s change the domain of discourse to the set of rational numbers Q, i.e. the set of postive and negative fractions. Is the formula ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) true or false in this new context? 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context Let’s change the domain of discourse to the set of rational numbers Q, i.e. the set of postive and negative fractions. Is the formula ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) true or false in this new context? Remember, the formula is translated “For all x and y , if x < y , then there is a z such that x < z < y .” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context Let’s change the domain of discourse to the set of rational numbers Q, i.e. the set of postive and negative fractions. Is the formula ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) true or false in this new context? Remember, the formula is translated “For all x and y , if x < y , then there is a z such that x < z < y .” Suppose we have rational numbers x and y such that x < y . 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context Let’s change the domain of discourse to the set of rational numbers Q, i.e. the set of postive and negative fractions. Is the formula ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) true or false in this new context? Remember, the formula is translated “For all x and y , if x < y , then there is a z such that x < z < y .” Suppose we have rational numbers x and y such that x < y . Q: Is there a z such that x < z < y ? 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context Let’s change the domain of discourse to the set of rational numbers Q, i.e. the set of postive and negative fractions. Is the formula ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) true or false in this new context? Remember, the formula is translated “For all x and y , if x < y , then there is a z such that x < z < y .” Suppose we have rational numbers x and y such that x < y . Q: Is there a z such that x < z < y ? x+y A: Yes. Let z = 2 . 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Changing the context Let’s change the domain of discourse to the set of rational numbers Q, i.e. the set of postive and negative fractions. Is the formula ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) true or false in this new context? Remember, the formula is translated “For all x and y , if x < y , then there is a z such that x < z < y .” Suppose we have rational numbers x and y such that x < y . Q: Is there a z such that x < z < y ? x+y A: Yes. Let z = 2 . Thus, the formula is true. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation “consistent” means true in some situation 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation “consistent” means true in some situation “inconsistent” means true in no situation 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation “consistent” means true in some situation “inconsistent” means true in no situation Now, we have the following new deﬁnitions: 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation “consistent” means true in some situation “inconsistent” means true in no situation Now, we have the following new deﬁnitions: A closed predicate formula is 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation “consistent” means true in some situation “inconsistent” means true in no situation Now, we have the following new deﬁnitions: A closed predicate formula is logically true if it is true in every structure, 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation “consistent” means true in some situation “inconsistent” means true in no situation Now, we have the following new deﬁnitions: A closed predicate formula is logically true if it is true in every structure, consistent if it is true in some structure, and 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic Recall. . . “logically true” means true in every situation “consistent” means true in some situation “inconsistent” means true in no situation Now, we have the following new deﬁnitions: A closed predicate formula is logically true if it is true in every structure, consistent if it is true in some structure, and inconsistent if it is true in no structure. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula We will work with a vocabulary consisting of a 1-place predicate Gx and a 2-place predicate Mxy . 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula We will work with a vocabulary consisting of a 1-place predicate Gx and a 2-place predicate Mxy . To show: ∀x (¬Gx ⇒ ∃y Mxy ) is consistent. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula We will work with a vocabulary consisting of a 1-place predicate Gx and a 2-place predicate Mxy . To show: ∀x (¬Gx ⇒ ∃y Mxy ) is consistent. That is, we need to ﬁnd a structure in which ∀x (¬Gx ⇒ ∃y Mxy ) is true. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures? 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures? No. It is false in the following structure: 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures? No. It is false in the following structure: Domain: people 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures? No. It is false in the following structure: Domain: people Gx: “x is male” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures? No. It is false in the following structure: Domain: people Gx: “x is male” Mxy : “x is married to y ” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures? No. It is false in the following structure: Domain: people Gx: “x is male” Mxy : “x is married to y ” Then the formula says, “For every person x, if x is not male, then x is married.” 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A consistent formula, continued Here’s the desired structure: Domain: people Gx: “x is single” Mxy : “x is married to y ” In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if x is not single, then x is married.” True. Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures? No. It is false in the following structure: Domain: people Gx: “x is male” Mxy : “x is married to y ” Then the formula says, “For every person x, if x is not male, then x is married.” False. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. Assume that ∀x Fx ⇒ ∃x Fx is false in S. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. Assume that ∀x Fx ⇒ ∃x Fx is false in S. Thus, ∀x Fx is true in S, but ∃x Fx is false in S. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. Assume that ∀x Fx ⇒ ∃x Fx is false in S. Thus, ∀x Fx is true in S, but ∃x Fx is false in S. ∀x Fx is true in S: Every object in the domain of S has the property F . 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. Assume that ∀x Fx ⇒ ∃x Fx is false in S. Thus, ∀x Fx is true in S, but ∃x Fx is false in S. ∀x Fx is true in S: Every object in the domain of S has the property F . ∃x Fx is false in S: There is no object in the domain of S has the property F . 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. Assume that ∀x Fx ⇒ ∃x Fx is false in S. Thus, ∀x Fx is true in S, but ∃x Fx is false in S. ∀x Fx is true in S: Every object in the domain of S has the property F . ∃x Fx is false in S: There is no object in the domain of S has the property F . Impossible! 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. Assume that ∀x Fx ⇒ ∃x Fx is false in S. Thus, ∀x Fx is true in S, but ∃x Fx is false in S. ∀x Fx is true in S: Every object in the domain of S has the property F . ∃x Fx is false in S: There is no object in the domain of S has the property F . Impossible! Thus, ∀x Fx ⇒ ∃x Fx is true in S. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula Vocabulary: 1-place predicate Fx. To show: ∀x Fx ⇒ ∃x Fx is logically true. Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by abstract reasoning about structures. Let S be any structure. Assume that ∀x Fx ⇒ ∃x Fx is false in S. Thus, ∀x Fx is true in S, but ∃x Fx is false in S. ∀x Fx is true in S: Every object in the domain of S has the property F . ∃x Fx is false in S: There is no object in the domain of S has the property F . Impossible! Thus, ∀x Fx ⇒ ∃x Fx is true in S. But S is an arbitrary structure, so this holds for all structures. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 1. ∀x Fx (assume) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 1. ∀x Fx (assume) 2. ??? (∀E 1) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 1. ∀x Fx (assume) 2. ??? (∀E 1) . . . . . . . . . . . . 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 1. ∀x Fx (assume) 2. ??? (∀E 1) . . . . . . . . . . . . ? ∃x Fx (∃I, ??) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 1. ∀x Fx (assume) 2. ??? (∀E 1) . . . . . . . . . . . . ? ∃x Fx (∃I, ??) ? ∀x Fx ⇒ ∃x Fx (⇒I, 1-??) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 1. ∀x Fx (assume) 2. ??? (∀E 1) . . . . . . . . . . . . ? ∃x Fx (∃I, ??) ? ∀x Fx ⇒ ∃x Fx (⇒I, 1-??) 10130 Lecture 32 Review A bit more on the alternation of quantiﬁers One last example Logical Theory for Predicate Logic A logically true formula, continued Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx. 1. ∀x Fx (assume) 2. ??? (∀E 1) . . . . . . . . . . . . ? ∃x Fx (∃I, ??) ? ∀x Fx ⇒ ∃x Fx (⇒I, 1-??) We’ll do this later. . . 10130 Lecture 32