SYMBOLIZING ORDINARY LANGUAGE IN
THE SINGLY QUANTIFIED PREDICATE CALCULUS
When symbolizing in the predicate calculus first you have to look at the subject term(s) and ask yourself: is the subject referring to a particular individual? If it is a proper noun, a demonstrative phrase (a phrase beginning with "this," that," "these," or "those"), or an indexical ("here" = "this place," "there" = "that place," "now" = "this time," or "then" = "that time"), then it is referring to an individual. In this case you have singular proposition. If the subject term(s) are referring to a class, whether in whole or in part, then you have a general proposition. Classes are generally referred to by common nouns, sometimes with no quantifier, or sometimes with a quantifier (e.g. "all," "some," "no," "none," "many," "few," "plenty," etc.).
Singular propositions: These are propositions in which the named individual in the subject is indicated by a proper noun or demonstrative phrase; e.g.,
"Socrates is a man."
With M symbolizing the class "man" ans "s" is the individual constant symbolizing "Socrates," this is symbolized simply as:
Singular propositions with demonstrative phrases in the subject are handled the same way:
"These books are not logic books."
With L symbolizing the class of "logic books" and "b" is the individual constant symbolizing "these books," it is symbolized as:
Proper nouns are easy to identify; demonstrative phrases generally have "this," "these," "that," or "those" in them. ("Now" = "this time"; "then" = "that time"; "here" = "this place"; "there" = that place). In each case the term referring to the subject is treated as a name of an individual object (what "individuals" or "objects" are, is a question left for the metaphysician), and is symbolized with a lower case letter from "a" through "t", known as an "individual constant." The term in the predicate is treated as identifying a class or "property" (what "properties" or "class defining characteristics" are is a question the logician gladly hands over to the metaphysician), and is symbolized by an upper case letter from "A" through "Z".
General Propositions: The general propositions relate two classes or what Aristotle first called "categories," designated as the "subject term" S, and the "predicate term" P. They differ from singular propositions in the subject does not name a single individual but refers to any individual in general which is a member of the subject class. For this reason the subject is symbolized by an individual "variable", x.
The variable by itself is "unquantified" or "free".The subject may have either "universal quantity" if it refers to all members of the subject class, or "particular quantity" if it refers to some subset, formally defined as "at least one," of the subject class. "Universal quantity" is symbolized by the "universal quantifier, "(x)" (which should be regarded as a single symbol, even though it is three key strokes on the keyboard), expressed in English by the "standard" words "All" and "No"; and "particular quantity" is symbolized by the "existential quantifier," "($x)" always expressed in English by the word "Some," which by convention is given the meaning of "at least one." General propositions also have "quality" by which we mean that they either include, ("affirmative") or exclude ("negative") members of the subject class in or from the predicate class. This yields the pattern of the four logical forms of what were traditionally called "standard form categorical propositions" abbreviated by the first four vowels: A: universal affirmative; E: universal negative; I: particular affirmative; and O: particular negative. They are represented in words and symbols in the following array known as "The Square of Opposition":
In order to make a string of symbols stand for a proposition, the "free variable" must be "bound" by a "quantifier."
Affirmative Negative Universal A: All S are P
(x)(Sx => Px)
E: No S are P
(x)(Sx => ~Px)
Particular I: Some S are P.
($x)(Sx · Px)
O: Some S are not P.
($x)(Sx · ~Px)
As an approach to learning how to translate English sentences into the predicate calculus, our strategy is to consider first "translating" any given proposition to the familiar A,E,I, and O standard form categorical propositions. Once in standard form, all propositions will translate according to the patterns given in the Square of Opposition above. In ordinary language relatively few sentences will appear already as in standard form A, E, I, or O propositions. However, it is possible to "translate" very many, but not all, ordinary language statements into one or more of the four standard form categorical propositions.The important thing to remember here is that the major relation in the universals (A and E) is always an implication, but the major relation in the particulars (I and O) is always a conjunction.
The following guidelines should be helpful:
Non standard quantifiers: Many synonyms (or synonyms from the point of view of logic) of the standard quantifiers can be replaced directly by "all," "no," or "some." Remember that "some" means simply "at least one," so even though words like "many," "several," or "few" are not synonyms in the ordinary sense, since they all mean "at least one," from the point of view of the predicate calculus, they can all be translated as "some." All finite numbers as they appear are translated in the predicate calculus can be treated as "at least one." (Arguments which depend on specific relations between numbers cannot be expressed directly in singly quantified propositions.) Synonyms of the universals include the following: every, any, everyone, everything, everywhere, anyone, anything, one who, he who, she who, whoever, whosoever, nobody, no one, always, and never, etc. The articles "A," "an," and "the" can be either universal or particular quantifiers depending on context. Thus "A bat is a mammal." is translated "All bats are mammals." and symbolized (x)(Bx => Mx), but "A bat flew in the window." clearly does not mean "all bats" but "at least one" and is symbolized ($ x)(Bx · Fx), where "Fx" = "x flew in the window". "The bear is a carnivore." becomes "All bears are carnivores." and is symbolized (x)(Bx => Cx), but "The bear is fed every afternoon." becomes "Some bear is a creature who is fed every afternoon." or ($x)(Bx · Fx) where "Fx" = "x is fed every afternoon".
No explicitly stated quantifier: In many contexts a quantifier is "understood" from the context, even though it is not explicitly stated. Here common sense and context must be your guide. Thus "Bears are carnivores." obviously is intended to mean "All bears are carnivores." or (x)(Bx => Cx) but "Bears are hunted in Alaska." means "Some bears are creatures hunted in Alaska." or ($x)(Bx · Hx)
Verbs other than "to be": Standard form categorical propositions always have a form of the verb "to be" connecting the subject and predicate classes. Most intransitive verbs present no problem and can be translated either directly or with an "-er" form and a form of the verb "to be": "Some men drink." becomes "Some men are drinkers." and is symbolized ($ x)(Mx · Dx) where "Dx" = "x is a drinker." But transitive verbs with a direct object can be analyzed as defining a predicate class by transforming them into predicate nouns with a form of the verb to be: "All men breath air." becomes "All men are air-breathers." symbolized by (x)(Mx => Ax) where "Ax" = "x is an air breather." "Some students buy books." becomes "Some students are book-buyers." or ( $x)(Sx · Bx) with "Bx" = "x is a book-buyer.'
Adjectives which must be treated as defining classes: Sometimes an adjective will be easily rendered directly as a noun, without having to specify e.g. "All S are beautiful." can be translated as "All S are beauties." Sometimes in order to define a class for the predicate term, it may be necessary to provide a noun or substantive phrase acting as a noun for the adjective to modify, traditionally called a "parameter." Thus for example "valuable" might be treated as referring to the class of "valuable objects" while "exciting" may be taken to refer to the class of "exciting events." Such terms are generally taken from the "universe of discourse" of the argument, which is, roughly, what sort of things the argument is about, e.g. "persons," "places," "objects," "beings," "creatures," "events," etc. (The "universe of discourse" chosen is not a function simply of the given proposition, but must reflect the universe of the whole argument in the which the given proposition is either premise or conclusion. Thus for example, the above suggestion of "valuable objects" might be appropriate for an argument in which other propositions could be about "useful objects" or "beautiful objects" while in a different argument where other premises were about, say "rich persons" or "famous persons," "valuable persons" might be a better parameter.)
Compounds in the predicate: When the proposition to be symbolized predicates two or more predicates to the subject, then a simple conjunction is used. These may be expressed as either two (or more) nouns, two (or more) adjectives or an adjective and noun combination: All pigs are herbivores and quadrupeds" symbolized (x)[Px (Hx ù Qx)]; "All pigs are fat and lazy." is symbolized (x)[Px => (Fx · Lx)], and "All pigs are fat herbivores." is symbolized (x)[Px =>(Fx · Hx)]. Many propositions will appear with one or more adjectives in the subject or predicate which will need to be symbolized as a separate class, the conjunction of which will become the "predicate class". All of the adjectives or nouns used to identify the predicate class must be conjoined as a separate unit forming the predicate. Thus "All bears are omnivorous quadrupeds." become (x)[Bx => (Ox · Qx)], where Ox = x is an omnivorous creature,(with "creature" used as a "parameter" from the presumed "universe of discourse"). A disjunction in the predicate is usually signalled by the "either...or" construction: "All vertebrates are either bipeds or quadrupeds." and is symbolized by a simple disjunction in the predicate: (x)[Vx => (Bx v Qx)]. Note: Be careful when denying the predicate in an E proposition, that you deny the whole predicate, not its separate conjuncts. Thus the possibly true statement "No man is a fat athlete." becomes (x)[Mx => ~(Fx · Ax)], and is not symbolized (x)[Mx =>( ~Fx · ~Ax)] which would be the correct symbolization of the probably false conjunction of two statements: "No man is fat" and "no man is an athlete." If this last compound were to be rendered as a single E statement, it would have to be with a disjunction in the predicate instead of a conjunction: "All men are either not-fat or not-an-athlete."
Compounds in the subject: Simple disjunctions in the subject are symbolized directly by a disjunction and present no problems: "Any student or faculty member may be invited." is symbolized by the obvious (x)[(Sx v Fx) => Ix]. But the same statement might just as well have been expressed by the English conjunction "All students and faculty may be invited." Since the first is symbolized by a disjunction, the latter, which is equivalent, must also be symbolized by a disjunction, not a conjunction, even though it is "translating" the English word "and." It would be wrong to symbolize the latter as (x)[(Sx · Fx) => Ix] for this would say that all who were invited were both students and faculty, which is a much different class from the class of those who are either students or faculty. Many familiar phrases using "and" are not in fact conjunctions of two subject classes but in fact disjunctions in spite of the fact that we use "and". Thus "Ladies and gentlemen are invited." clearly does not refer to the class of objects which are both "ladies and gentlemen" (a presumably empty class) but clearly is intended to refer to the class of individuals who are ladies or gentlemen.
Terms in non-standard order: The quantifiers "All," "No," and "Some," or their synonyms, may appear in a proposition but not in the appropriate order. Generally the components can be rearranged into standard order, but be sure that the correct term stays the subject. "Students are all intelligent." becomes "All students are intelligent." (x)(Sx => Ix) and is not translated "All intelligent people are students." or symbolized (x)(Ix Sx). "All is fair in love and war." becomes "All actions in love or war are fair actions." symbolized (x)[(Lx v Wx) => Fx] with Lx = x is an action in love, and Wx = x is an action in war.
Exclusive propositions: Propositions in which one class is introduced by "only" or "none but" can be translated into standard form universal propositions with "All" or "No" but the term introduced by "only" or "none but" must be in the predicate and therefore is symbolized as the consequent of the main implication. Thus "Only members are invited." becomes "All invited persons are members." symbolized as (x)(Ix => Mx); "None but students may join." becomes "All persons who may join are students." or (x)(Jx => Sx). The order of terms in the English sentence is not significant. Thus, the former example could just have well been said, "Those invited are none but members." "Only if" has the same function as a simple "only". Thus "Only if you study, will you pass." or "You will pass, only if you study." means "All those who will pass are those who study." and is symbolized (x)(Px => Sx). However, the order of the terms in the symbolic translation for universal propositions is crucial because the symbol for implication, =>, does not commute. "Only citizens can vote." or "Citizens only are allowed to vote." means the true statement "All voters are citizens." (x)(Vx => Cx), not the false statement "all citizens are voters" (x)(Cx =>Vx) (because persons under 18 years of age are citizens but can't vote, and of course there are citizens who just don't bother to vote).
Existential claims: Claims which assert or deny the existence of members of classes can be expressed as categorical propositions. Assertions of existence are handled as I propositions. Thus "There are man-eating tigers." becomes "Some tigers are man-eaters." ($x)(Tx · Mx) with Mx = "x is a man-eater". But negative existential claims (claims that assert something doesn't exist) are contradictories of the I and hence appear as E propositions (note that contradictories are across the diagonals of the Square of Opposition). Thus "There are no tame sharks." can be rendered "No sharks are tame." (x)(Sx => ~Tx) and "Nothing is both a reptile and warm-blooded." becomes "No reptile is a warm-blooded creature." (x)(Rx => ~Wx).
Denials of universal affirmatives: The denial, i.e. "contradiction," of a universal affirmative A proposition is always an O proposition (across the diagonal of the Square): thus "Not all students are brilliant." becomes "Some students are not brilliant." or ($x)(Sx · Bx). (The similar sentence, "All students are not brilliant." is poorly constructed and ambiguous, possibly meaning the E, "No students are brilliant." or the O, "Some students are not brilliant.") Unfortunately quantifiers which may be treated as synonymous for affirmations, need not be for denials. Thus although "every" and "any" are synonyms for affirmative A propositions, the same is not true of their denials, E propositions. "Not every" has the familiar meaning of denying the A to form an O proposition: "Not every student passed." becomes "Some students are not those who passed." ($ x)(Sx · ~Px), but "not any" is synonymous with "none" and hence requires the E proposition: "Not any students passed." is equivalent to "No students passed." symbolized as (x)(Sx => ~ Px).
Exceptive propositions: Propositions such as "All except faculty must pay tuition." must be translated as a conjunction of two separate categorical propositions:
"All nonfaculty are tuition- payers." (x)(~Fx => Tx)
and "No faculty members are tuition- payers." (x)(Fx => ~Tx).
Remember that all of the above remarks about quantifiers refer to general propositions or propositions in which no individual is "named" as subject, or in other words in which the "name" of an individual does not appear. Propositions in which a named individual appears as the subject are not categorical propositions at all, and so do not have a variable in the subject and do take quantifiers at all (see above: "Singular Propositions").
There are of course many possibilities for propsitions not touched on here, and becoming adept at translating from ordinary English to standard form propositions is a considerable "art" acquired only with practice. Always allow common sense and context to dictate the best possible translation.