The subject of propositional logic, which we dealt with in the first part, are those logical relations between sentences that depend on the logical connectives (negation, conjunction, disjunction, conditional, and biconditional). The sentences that are not formed of simpler sentences through logical connectives are considered structurally simple in propositional logic and, accordingly, are symbolized by simple symbols – propositional letters (p, q, r ...). However, there are inference whose logical validity or invalidity depends on the internal structure of these “simple” (for propositional logic) sentences. Consider, for example, the following logically valid inference:
|(1)||All humans are mortal.|
|Socrates is a human.|
|Socrates is mortal.|
The premises and the conclusion are sentences that are not formed of simpler sentences through logical connectives, so they are simple (atomic) from the point of view of propositional logic and should be symbolized by propositional letters. As a result, we get the following inference scheme:
Important for the logical validity of (1) is the word “all”, which is part of the sentence “All humans are mortal”. Unlike logical connectives, it does not connect sentences into new sentences. To be able to properly evaluate such inferences as logically valid, logic needs to take into account such logical words. We saw that traditional logic does that but it does not take into account the logical connectives. The part of modern logic that does both is the predicate logic. Predicate logic contains in itself propositional logic but it has additional resources and much greater capabilities. In fact, it is what is primary meant by “modern logic” – it is the classical part of modern logic.
In traditional logic, we simply referred to terms without differentiating between them. In contrast, in predicate logic we differentiate between general terms and singular terms.
A singular term is a word or expression that denotes (refers to) a particular thing. Paradigmatic among them are proper names (“Socrates”, “John Lennon”, etc.). Another typical example are the so-called definite description – expressions such as “Earth’s highest mountain” or “the President of France”, which denote certain things without being their names. Other examples are pronouns or expressions containing pronouns such as “she”, “that door”, etc. The name “Socrates” refers to the man Socrates; the expression “Earth’s highest mountain” refers to Mount Everest and so on. What makes a singular term such is the function it is generally intended to perform in language (to denote a thing) – it may not be able to perform it. For example, the proper name “Pegasus” or the definite description “the largest number” are singular terms although they do not succeed in denoting anything (there are no winged horses or a number greater than any number).
General terms, for their part, are words or expressions that do not denote things but are true or false of things. Such are the common nouns (for example, “human”), the adjectives (“wise”), the verbs (“sleeps”). As with singular terms, they may be individual words (such as the examples just given) as well as compound expressions, such as “desperately bowed his head”, which is a verb phrase. General terms are also called predicates (hence “predicate logic”). As we have said, it is characteristic of a predicate that it does not denote a certain thing but is true or false of each single thing. Thus, the predicate “philosopher” is true of each philosopher, and it is false of each human that is not philosopher as well as of each thing that is not human. The predicate “sleep” is true of everything that is sleeping now and false of everything else, etc.
In addition to general terms that are true or false of things regardless of other things like “human”, “sleeps”, etc., there are general terms that are true or false of two things taken together, i.e. of a thing relatively to a thing. For example, the predicate “larger” is true of Europe with respect to Australia or, to put it another way, it is true of the pair made up of Europe and Australia, and it is false of Europe with respect to Asia (false of the pair consisting of Europe and Asia). Since the order of the pairs in question is important, they are called ordered pairs. The predicate “teacher of” is true of the ordered pair (Plato, Aristotle) because Plato is a teacher of Aristotle but false of the ordered pair (Aristotle, Plato) because Aristotle is not a teacher of Plato. There are general terms that are true or false of ordered triples (3-tuples), ordered 4-tuples, … etc. For example, the predicate “between” is true of the ordered triple (Mediterranean Sea, Europe, Africa) because Mediterranean Sea is between Europe and Africa. The predicate „…gave…to…in exchange for…“ would be true of the ordered 4-tuple (Alice, Alice’s bicycle, Bob, Bob’s car) if Alice has given her bicycle to Bob in exchange for his car. The terms that are true or false of things regardless of other things are sometimes called absolute general terms, and those that are true or false of ordered pairs, triples,… etc. are called relative general terms. We will also refer to the former as one-place predicates and to the latter as two-place, three-place… predicates (collectively, many-place predicates).
The difference between singular and general terms is often indicated by the use of a definite or indefinite article. For example, “the friend of Alice” is a singular term while “a friend of Alice” is a general term. The function of the first expression is to refer to a certain thing while that of the second expression is be true or false of things. “A friend of Alice” will continue to be a general term even if Alice has a single friend or no friends. Similarly, “a mother of Alice”, unlike “the mother of Alice”, is a general term although it is not possible for Alice to have more or less than one mother. It does not matter of how many things a general term is true – whether of zero, one or more – and it does not matter whether a singular term succeeds in denoting something or not. What matters is the general function or purpose of the expression. The fact of the matter in the world is irrelevant to logic as it takes into account, so to say, all possible worlds, not just ours.
Predicate logic uses different types of symbols for the singular and the general terms. We will represent symbolically singular terms by lowercase Latin letters (“a”, “b”, “c”...), which we will call constants, and general terms by uppercase Latin letters usually starting with “F” (“F”, “G”, “H”...), which we will call predicate letters.
Simple sentences such as “Socrates is human”, “Sunset is red”, “This dog sleeps”, etc. are represented symbolically by a predicate letter followed by a constant. For example, if we symbolize the predicate “human” with “F” and the singular term “Socrates” with “a”, the sentence “Socrates is human” will be represented symbolically by “Fa”.
We will use the same predicate letters (F, G, H...) to symbolize many-placed predicates, too. For example, if we represent symbolically the two-place predicate “killed” with “F” and the singular terms “Brutus” and “Caesar” with “b” and “c” respectively, the sentence “Brutus killed Caesar” will be symbolized by “Fbc”. Similarly, if we use “G” for the three-place predicate “… is jealous of … and …”, “a” for the singular term “Othello”, “b” for “Desdemona”, and “c” for “Cassio”, the sentence “Othello is jealous of Desdemona and Cassio” will be symbolized with “Gabc”.
Sentences such as the above, which are formed by connecting a (one- or many-place) predicate with singular terms (one or more), are called atomic. Accordingly, their symbolic representations are called atomic formulas.
When we specify what predicate a predicate letter will symbolize, it has to be clear whether the predicate is one-place, two-place ... etc. and where exactly those places are – where singular terms can be put in the predicate to form an atomic sentence. To this end, we will mark the places in question by ellipses (“…”), writing, for example, “…is human” instead of “human”, “...killed…” instead of “killed”, “…is jealous of…and…” instead of simply “jealous”. Thus, the number of ellipses will determine the number of places. “Jealous”, for example, may be used as a two-place, three-place, and even one-place predicate. By explicitly indicating the places for the singular terms in it, we specify how we intend to use it. The ellipses will also serve another purpose. Let us see what it is by an example. Suppose we want to represent symbolically the geometry sentence “Point A is between points B and C”. We symbolize the three-place predicate “between” with, say, “F” and denote the points A, B and C with the constants “a”, “b” and “c” respectively. Then there are at least two different ways to represent the sentence: with “Fabc” or with “Fbac”. In the first case, we are guided by the order in which the singular terms “point A”, “point B” and “point C” occur in the sentence “Point A is between points B and C”. In the same way, in the formula “a” comes first, then comes “b” and then “c”. In the second case, we are guided by the meaning of the sentence. Since it tells us that point A is between points B and C, in the symbolic representation “a” stands between “b” and “c”. So far, both ways would be admissible, as the only rule for symbolizing atomic sentences we have adopted is that the predicate letter stands before the constants, and both “Fabc” and “Fbac” satisfy it. However, we have a problem here. What if we are presented with the formula “Fabc” together with the interpretation of its symbols, but we are not given a sentence that the formula is supposed to represent. Then we cannot be sure what it means. Does it mean that the point A is between points B and C, or that B is between A and C, or something else? The formula is ambiguous. To remove the possibility for such ambiguities, we will adopt the rule that the order of the constants in an atomic formula has to be the same as the order of the places in the predicate to which the formula corresponds. As ellipses show where these places are, the importance of their usage becomes obvious. Because of the new rule, under the above interpretation of its symbols, the formula “Fabc” means that A is between B and C, not that B is between A and C. Similarly, if “F” represents the predicate “…killed…”, and “b” and “c” refer to Brutus and Caesar respectively, “Fbc” will mean that Brutus killed Caesar (not that Caesar killed Brutus). The reason is that the interpretation of “F” is the predicate “…killed…”, in which the place for the name of who killed is before the place for the name of who has been killed, and “b” is before “c” in the formula.
Since we can now symbolize atomic sentences, we can also symbolize sentences that are formed from atomic statements through logical connectives. Here are two examples:
|Either Alice is angry with Bob, or Bob is angry with Alice.|
|Fab ∨ Fba|
|F – …is angry with…|
|а – Alice|
|b – Bob|
|If Bob has not said hello to Alice, she is angry with him.|
|¬Gba → Fab|
|F – …is angry with…|
|G – …has said hello to…|
|а – Alice|
|b – Bob|
The examples clearly show the deepening of the logical analysis compared to propositional logic. In propositional logic, we would symbolize the two sentences with, say, “p∨q” and “¬r→p” (where “p” corresponds to “Alice is angry with Bob”, “q” to “Bob is angry with Alice” and “r” to “Bob has said hello to Alice”). Now the formals have the same general form (the first is a disjunction and the second a conditional whose antecedent is negated) but the analysis has also entered the internal structure of the atomic sentences, which propositional logic does not analyze (“p”, “q”, and “r”).
We should try to maximally explicate logical form of sentences. If in the second sentence we symbolized with “G” the predicate “…has not said hello to…” instead of “…has said hello to…”, the antecedent of the conditional (“Bob has not said hello to Alice”) would have been symbolized with “Gba” instead of “¬Gba”. Then the logical form of the whole sentence would not be maximally explicated because the antecedent in it has the form of a negation – something that would not be seen in the symbolic representation. This could have a negative effect on the evaluation of the validity of inferences of which this sentence is part.
Here are some more examples of symbolizing sentences obtained by connecting atomic sentences with logical connectives:
|If the Pope is а human, he is not sinless.|
|Fa → ¬Ga|
|F – …is a human|
|G – …is sinless|
|а – the Pope|
|If Bob turned red, he has heard Alice’s remark.|
|Fb → Gba|
|F – …turned red|
|G – …has heard…|
|а – Alice’s remark|
|b – Bob|
|Bob blames and hates himself.|
|Fbb ∧ Gbb|
|F – …blames…|
|G – …hates…|
|b – Bob|
|(1) Is each of the following terms general or singular?|
|1)||older than John|
|2)||the brother of Alice|
|3)||a brother of Alice|
|5)||a prime minister|
|6)||the prime minister|
|7)||a friend of the prime minister|
|8)||the friend of a prime minister|
|9)||the fat man at the door|
|10)||the assassin of a president of the United States of America|
|(2) Symbolize the following sentences. In each case, explicitly state the interpretation of the predicate letters and constants used.|
|1)||The abbot of the monastery is old and wise.|
|2)||Alcibiades respects Socrates.|
|3)||Peter and John hate each other.|
|4)||Mary does not respect Peter but she does respect John.|
|5)||What he said now is trustworthy but what he said earlier is not.|
|6)||Romeo and Juliet are in love with each other.|
|7)||Peter and John are brothers of Alice and Mary respectively.|
|8)||Mary is pretty but Alice is prettier.|
|9)||The rescuer found the child frozen and dehydrated.|
|10)||Although Alice and Bob love each other, they hurt each other.|
|11)||Oedipus is both Jocasta’s son and husband, and she is both his mother and wife.|
|12)||Mary is the young and beautiful, but not happy, wife of Peter.|
|13)||If Peter hasn’t bought the car from John, he has bought it from Paul.|
|14)||K2 is higher than Broad Peak but both are lower than Everest.|
|15)||If the lake is to the west of the mountain and the village is to the west of the lake, then the mountain is to the east of the village.|
|16)||Bob is neither Alice’s son or brother but she loves him more than herself.|
|17)||If John didn’t recognize this man, he either haven’t met him before or he has a poor memory.|
|18)||If John respects himself, he will deceive neither his family nor his friends.|
|19)||If Peter is not with his wife, he feels lonely, and if he is with her, she annoys him.|
|20)||Grandpa exchanged the cow for two hens, and them he exchanged for a bag of plums, but Grandma was not angry with him either for the cow or for the hens.|