What is a Quantifier? - Definition from Techopedia
Establishing a relationship between exclusive-or (XOR) and the unique existential decoder, unique existential quantifier, universal quantifier. The existential quantifier is used to assert that some individual or individuals have A traditional way of illustrating the relationship between the quantifiers is . among them. Propositional logic: Statements about similar objects and relations needs to be stand for the existential and the universal quantifier.) (ψ. ϕ ∧.).
Predicates and Quantifiers
Only B's are A's. You should be able to figure this out when you need it so that you don't have to memorize it.
If "all" sentences and "only" statements are conditionals that reverse each other Tips 182734then "all and only" statements conjoin such conditionals; they are biconditionals.
For example, "She picked out all and only the bad grapes": We can paraphrase "None but ripe bananas are edible" in many equivalent ways. No bananas except ripe ones are edible.
Only ripe bananas are edible. A banana is edible only if it is ripe. All edible bananas are ripe. Don't let these many forms confuse you. The articles "a" and "an" sometimes take existential, sometimes universal, quantifiers. Because there is no hard and fast rule, paraphrase the English before translating.
We usually don't reach section 5. As in propositional logic, a common translation mistake is to omit necessary parentheses. But in predicate logic, there are two reasons to insert parentheses, not just one: To resolve ambiguities of operator precedence. See Tip 15above.1.3 Quantifiers
To mark the scopes of quantifiers. Make sure that each quantifier has the scope it needs. Is every variable inside the scope of some quantifier? Is every variable inside the scope of the right quantifier? If you understand the role of parentheses, then you should understand that these two expressions are not equivalent: If our universe contains some things that are A's but not B's, and other things that are B's but not A's, and nothing that is neither, then the first of these expressions will be false and the second true.
Moreover, the first is truth-functional compound, while the second is not.
Peter Suber, "Translation Tips"
Multiply General Monadic When do you need more than one quantifier? The answer is not simple. You will not err if, in translating compound statements, you add a new quantifier for every component which would need a quantifier if you were to translate it as an entire statement unto itself. But on the other hand, sometimes you can get away with fewer. For example, "All cats are mammals, but no cats are birds".
Each component of this compound would require a quantifier if translated separately. Hence, you will not err if you use two quantifiers: But on the other hand, the second quantifier is unnecessary in this case, provided the second component uses the variables of the first and is put within the scope of the first quantifier: We can prove that these two translations are equivalent by deriving each from the other.
The two translations in In this sentence, the two components no longer share a common subject, but they still require the same kind of quantifier and are still joined by conjunction. As before, you will not err if you use two quantifiers: But we can still dispense with the second: These two translations are also provably equivalent. When do you need a new variable letter? The answer is simple: If you are using prenex normal form introduced in Tip 43belowthen you must use a new variable letter for each new quantifier.
But the "logically never" part of the rule in Tip 42 still stands, because you need not ever use prenex normal form.
The two don't interfere with each other. Hence they may use the same variable letter without ambiguity. But it is confusing to readers to see the same letter when it may not refer to the same objects. We might mistakenly infer that the things that are A's are the same things that are B's; but that would not be warranted.
To prevent such mistakes, make it a habit to use new variable letters with each new quantifier in the same statement: Even when quantifier scopes nest inside each other, the quantifiers do not interfere with each other and may unambiguously use the same letter: We can and should parse such expressions under the rule that inner quantifiers should be read have their scopes settled, variables bound before outer quantifiers.
But for the sake of poor humans who must read these, use new letters with new quantifiers: A predicate logic expression is in prenex normal form if 1 all its quantifiers are clustered at the left, 2 no quantifier is negated, 3 the scope of each quantifier extends to the right end of the expression, 4 no two quantifiers use the same variable, 5 every letter used by a quantifier is used later in the expression as a bound variable.
Every predicate logic expression can be cast in prenex normal form.
- Derived Rules and Applications
- Existential quantification
- Universal quantification
Translating English sentences into prenex normal form is easier and more natural for most people than using any other format. Repeating a reference, with constants. To refer to the same object more than once in an expression, with constants, you must use the same constant in each reference. The only way to enforce univocity of reference with constants is to use the same letter.
Repeating a reference, with variables. To refer to the same object more than once in an expression, without constants, two conditions must be met: Here the "they" refers back to the dancing humans; we use the same variable "x" and put it in the scope of the same quantifier as the component about dancing humans.
In addition both the latter two expressions fail by leaving the final occurrence of the final variable free, creating a propositional function instead of a proposition; see Tip 31above. There is another way to repeat a reference with variables that violates both the conditions stated in the previous rule.
Requantifying on dancers in the second clause doesn't mean we are talking about a different group of people. You see, there are only seven different ways to construct non-atomic formulae—with a single formula, we can negate it, universally quantify over it, or existentially quantify over it, and with two formulae, we can form a conjunction, disjunction, conditional, or biconditional from them.
Now, here's the trick: We only have to consider cases where the immediate subformulae are themselves in prenex normal form—that is, we only have to demonstrate that we can convert any formula whose immediate subformulae are all in prenex normal form into a formula in prenex normal form.
Universal quantification - Wikipedia
After all, if the immediate subformulae are not in prenex normal form, we can just use the same conversion techniques to the immediate subformulae themselves. Okay, so let's start with negation. We need to be able to convert a formula of the following form into prenex normal form: Qn F This, however, is a trivial matter, as the derived rules for quantifiers and negations make evident.
We just need to move the negation in past all the quantifiers, changing the type of each quantifier from universal to existential or vice versa along the way. So much for negations. Next, we have universally and existentially quantified formulae to consider, but if our immediate subformula is already in prenex normal form, then adding another quantifier won't change anything there, so we're already done.
The only cases we have left to worry about, then, are those where the main connective of our formula is one of the binary connectives. Here's how we'll do it: First, we'll change any bound variables that the two immediate subformulae have in common, so that no two quantifiers in the formula as a whole have the same variable of quantification. Once we've taken care of that, it's actually an easy matter to bring all the quantifiers out to the ''front'' of the formula, one at a time.
We won't get into a detailed demonstration of this point, here—we'll let you take care of it in the exercises by completing a number of derivations. Now that we have dealt with the binary connectives, we've done what we set out to do—demonstrate that for any formula whose immediate subformulae are all in prenex normal form, we can find a formula in prenex normal form that is equivalent to the original formula.
As we mentioned, this also suffices to demonstrate that we can find a prenex equivalent for any formula—if its immediate subformulae are not in prenex normal form, we just have to find prenex equivalents to the subformulae first.
At this point, you might be a little suspicious—after all, what if the subformulae of the subformulae are not in prenex normal form?