"Statement Logic Extended with Predicates & Quantifiers for Complex Arguments", Study notes of Reasoning

Download "Statement Logic Extended with Predicates & Quantifiers for Complex Arguments" and more Study notes Reasoning in PDF only on Docsity! 9.1: Predicates and Quantifiers Expressive Limitations of Statement Logic Many intuitively valid arguments in ordinary language cannot be represented as valid in statement logic. For example: • This house is red. Therefore, something is red. • All logicians are exceptional. Saul is a logician. Therefore, Saul is excep- tional. • No politicians are honest. Some politicians are administrators. Therefore, some administrators are not honest. • Every horse is an animal. Therefore, every head of a horse is a head of an animal. Represented in statement logic (under appropriate schemes of abbreviation), these arguments look like this: • H ∴ R • M. S ∴ T • P. A ∴∼ H • H ∴ A Obviously, all three of these symbolized arguments are invalid in statement logic. (You should know how to prove that!) The problem: The validity of the above arguments rests in large measure on “subsentential” components of the constituent statements — e.g., the name Saul , the plural common noun logicians, the verb phrase is exceptional , and the “quantifier” All . Because the basic unit of statement logic is the atomic statement, statement logic is incapable of representing these features of the arguments that are crucial to their validity. In this chapter, we will extend statement logic with the apparatus and methods necessary for representing the above arguments adequately and proving their validity. Predicates and Quantifiers Recall the four Standard Forms of categorical statements: Categorical Statement Form Universal affirmative All S are P. Universal negative No S are P. Particular affirmative Some S are P. Particular negative Some S are not P. To represent these statements, we need to supplement the language of state- ment logic with new elements corresponding to those noted above. The first kinds of elements we need are those corresponding to names and verb phrases. Individual constants: a, ..., u Predicate letters: A, ..., Z Individual constants will represent names of individual things like the logician Saul Kripke, the city of Austin, and the number 17. Predicate letters standing alone are just statement letters. However, when combined with individual con- stants and names they stand for verb phrases like ‘is a man’ and ‘is mortal’, and relational predicates like ‘is the head of’ (which we will not study in this section). 2 Translating Categorical Statements Universal Affirmatives Let ‘L’ stand for ‘is a logician’ again and let ‘E’ stand for ‘is exceptional’. • Ordinary English: “All logicians are exceptional.” • Can be paraphrased as: “Everything is such that, if it is logician, then it is exceptional.” • In logical English (“Logicese”): “For all individuals x, if x is a logician, then x is exceptional.” • Fully translated: (x)(Lx → Ex) In general, “All S are P ” is translated into predicate logic as (x)(Sx → Px), where x is any variable. Note that the arrow usually goes with the universal quantifier . Universal Negatives Let ‘P’ stand for ‘is a politician’ and let ‘H’ stand for ‘is honest’. • Ordinary English: ‘No politicians are honest’ . • Can be paraphrased as: ◦ Everything is such that, if it is politician, then it is not honest. ◦ It is false that something is both a politician and honest. 5 • In logical English: ◦ For all individuals x, if x is a politician, then x is not honest. ◦ It is not the case that there is an individual x such that x is a politician and x is a honest. • Fully translated: ◦ (x)(Px →∼Hx) ◦ ∼(∃x)(Px • Hx) In general, “No S are P ” is translated into predicate logic as either (x)(Sx →∼Px) or ∼(∃x)(Sx • Px), where x is any variable. Particular Affirmatives Let ‘P’ stand for ‘is a politician’ again and let ‘A’ stand for ‘is an administrator’. • Ordinary English: “Some politicians are administrators.” • Can be paraphrased as: “Something is such that it is both a politician and an administrator.” • In logical English: “For some individual x, x is a politician and x is an admin- istrator.” • Fully translated: (∃x)(Px • Ax) In general, “Some S are P ” is translated into predicate logic as (∃x)(Sx •Px), where x is any variable. Note that the dot usually goes with the existential quantifier . 6 Particular Negatives • Ordinary English: “Some administrators are not honest.” • Can be paraphrased as: “Something is such that it is both an administrator and not honest.” • In logical English: “For some individual x, x is an administrator and x is not honest.” • Fully translated: (∃x)(Ax • ∼Hx) In general, “Some S are not P ” is translated into predicate logic as (∃x)(Sx • ∼Px), where x is any variable. Stylistic Variants for Categorical Statements Recall that each type of categorical statement has both a standard form and multiple stylistic variants that, essentially, mean the same thing. Universal affirmative: (x)(Sx → Px) Universal negative: (x)(Sx →∼Px) ∼(∃x)(Sx • Px) All men are animals. No politicians are logicians. Every man is an animal. No politician is a logician. Each man is an animal. All politicians are nonlogicians. Men are animals. No one is a politician if a logician. Any man is an animal. There are no politicians who are logicians. Anything that is a man is an animal. All politicians fail to be logicians. Only animals are men. Only nonlogicians are politicians. 7