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Material Type: Notes; Class: LOGIC AND SCIENTIFIC REASONING; Subject: Philosophy; University: University of Texas - Austin; Term: Fall 2001;
Typology: Study notes
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First-order Predicate Logic
Odysseusis hurting me.
Nobody is hurting me.
Predicates:
is hurting me.
is hurting.
gives to.
Predicates represented by upper—case letters. If followed by no names: complete statement. If followed by one name: quality or property. If followed by two names: binary relation. etc.
Nonvariable (constant) name: an expression used to name a particular, individual thing (person, place,time). Names are represented by lower-case letters from a through s.
The symbol for identity: =
Socrates is wise. Socrates is Plato's teacher.
Ws s = p
Socrates is Plato's teacher. You are not Socrates. ˇ You are not Plato's teacher.
No swans are white. Œx(Sx á —Wx) —´x(Sx & Wx)
The language of predicate logic introduces two new logical symbols: the universal quantifier (Œ) and the existential quantifier (´). These can be used to translate the words something and everything: Something is coming. ´xCx Everything is coming. ŒxCx Every sentence in English contains a main verb. These verbs come in several kinds: Intransitive verbs: sleeps, walks, thirsts. These take a subject only. Transitive verbs: loves, moves,
irritates. These take a subject and a direct object. Ditransitive verbs: gives (as in A gives B C). These take a subject, a direct object and an indirect object. Verbs shall be translated by means of predicate letters. We shall follow these letters by one, two or three underscores, indicating whether the verb is intransitive, transitive or ditransitive. walks ó W_ likes ó L_ _ gives ó G_ _ _ As I described earlier, proper names shall be symbolized by lower-case letters from a through o (the individual constants). We will precede these constants by a question mark, to indicate that the name must be combined with a predicate in order to produce a complete sentence. An underscore shall stand for a missing
Proper names are only one kind of term. Another kind is a noun phrase like 'a boy' or 'all circles', in which a determiner ('a', 'all') is combined with a common noun ('boy', 'circle') or common noun phrase. A common noun phrase can be created by starting with a common noun and adding a relative clause ( a clause headed by 'that', 'who', 'whom' or 'which'), and adjective or a prepositional phrase. The class of common noun phrases can be defined as follows:
boy boy in Austin boy who lives in Austin dark-haired boy dark-haired boy who lives in Austin boy who lives in a capital city A relative clause results from replacing the subject, direct object or indirect object of a complete sentence by 'that', 'who', 'whom' or 'which' (where 'who' must always replace the subject and 'whom' the direct object). A prepositional phrase is the result of combining a preposition (like 'in', 'on', 'through', etc.) with a noun phrase. First, it is necessary to specify the translation of the determiners of English. The following table gives the translation of the most common determiners.
translating 'only' you should put the first predicate in the consequent instead. Although the variable x is used throughout this table, in fact one should introduce a new variable every time you translate a new determiner in an English sentence. So, if you've already used x, you should translate the next determiner using y instead. Some swans are white. [´x( ?ö x & ?ï x)] [S_] [W_]
[´x(Sx & ?ï x)] [W_]
´x(Sx & Wx) All swans are white. [Œx( ?ö x á ?ï x)] [S_] [W_]
[Œx(Sx á ?ï x)] [W_]
Œx(Sx á Wx) No swans are white. [—´x(?ö x & ?ï x)] [S_] [W_]
[—´x(Sx á ?ï x)] [W_]
—´x(Sx & Wx)
Relative clauses consist of relative pronouns combined with predicates. The relative pronouns 'that', 'who', 'whom' and 'which' are all translated in the same way: that' = who' = whom' = which' = [(?ö _ˆ & ?ï _ˆ )]
In this translation, the shadow question marks stand for two different predicates, but the underscores, which share the same index, stand for one individual.
Prepositional phrases consist of prepositions plus noun phrases. A preposition is translated by means of a two-place predicate, a predicate letter followed by two underscores. This predicate represents the spatial or temporal relationship encoded by the preposition. For example, the translation of the predicate 'in' includes the two- place predicate I_ˆ_Ô, which represents the relationship of the thing corresponding to _ˆ's being inside the thing represented by _Ô. The full translation of a preposition is as follows: in' = [(?ö ˆ & Iˆ _Ô )]
The following translations illustrate the use of relative clauses and prepositional phrases:
All students who attend will do well.
[Œx(?ö x á ?ï x)][S?][[(?ð _ˆ & ?å _ˆ )]
[A?] [W?] [Œx( ?ö x á ?ï x)] [(S ˆ & ?å _ˆ )]
[A?] [W?] [Œx( ?ö x á ?ï x)] [(S _ˆ & A ˆ)] [W]
[Œx((Sx & Ax) á ?ï x)] [W_]
Œx((Sx & Ax) á Wx)
Some students who attend will do well. [´x(?ö x & ?ï x)] [S?] [[(?ð _ˆ & ?å _ˆ )]
[A_] [W_]
[´x( ?ö x & ?ï x)] [(S _ˆ & ?å _ˆ )]
[A_] [W_]
[´x( ?ö x & ?ï x)] [(S _ˆ & A ˆ)] [W]
[´x((Sx & Ax) & ?ï x)] [W_]
´x((Sx & Cxc) & Wx)
very material difference in what order we combine the elements. In general, when translating written text, we should start on the right and work to the left. Consider the following simple example:
Every boy kisses a girl. [Œx( ?ö x á ?ï x)] [B_] [K_Ô _ ]
[´y(?ð y& ?å y )] [G_Â]
[Œx( ?ö x á ?ï x)] [B?] [K_Ô _ ]
[´y(Gy & ?å y )]
[Œx( ?ö x á ?ï x)][B_][´y(Gy & K_Ô y)]
[Œx(Bx á ?ï x)] [´y(Gy & K_Ô y)]
Œx(Bx á ´y(Gy & Kxy))
Pronouns (he, she, it, him, her) should be translated by means of constants or variables. In translating a pronoun in an English sentence, one should try to discern which preceding noun phrase in the sentence is the antecedent of the pronoun. Once you have identified this,
you can use the corresponding constant or variable as your translation of the pronoun. For example:
Jane came, and Mike greeted her. (Cj & Gmj)
A visitor arrived, and he asked for Sally. ´x((Vx & Ax) & A'xs)
Every student joined a club that accepted him. Œx(Sx á ´y(Cy & Ayx)) There are several special cases where the rule given above for the translation of 'a', 'an' and 'some' has to be broken. In these exceptional cases, we should translate these words as having the same meaning as 'every'.
2', (´y(Py & Lay) á Gay) 4'. Œx((Sx & ´y(Cy & Gxy)) á Pxy)
2". Œy((Py & Lay) á Gay) 4". ŒxŒy((Sx & (Cy & Gxy)) á Pxy)
"Any" tries to take a wider scope than its position in the sentence would indicate and always moves into a positive context, but it cannot leapfrog over more than one connective.
If you love everyone, you are fortunate. (Œx(Px á Lax) á Fa) If you love anyone, you are fortunate. Œx(Px á (Lax á Fa)) I didn't catch every fish. —Œx(Fx á Cbx) I didn't catch any fish. Œx(Fx á —Cbx)
