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Introduction to Philosophical Logic

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Prolegomena

What is logic?

Logic is the study of the consistency of beliefs. For beliefs to be consistent it must be possible for them to obtain at the same time. For example, it is illogical to believe that the sky is completely blue and that the sky is completely red because the sky being entirely blue is inconsistent with its being entirely red, i.e. it is not possible for the sky to be entirely red at the same time as its being entirely blue.

Logic is also a study of "logical consequence", i.e. what follows by necessity from something else. By studying inconsistency of beliefs, philosophers are able to study the validity of arguments, as will be shown later. Methods of finding whether certain arguments are valid is described later.

The symbolisation of these sentences, known as formalisation, simplifies and quickens this process. It also enables the philosopher to clarify ideas using an unambiguous language in which to represent thoughts. The sophistication of the language used enables greater insights into the significance of these thoughts (and a cursory analysis of more logical languages is described in Other Logics.

Aims of this book

This book aims to give the reader a basic understanding of logic and its relationship with philosophy, rather than a more mathematical approach to advanced logic. It is designed to provide the reader with a grasp of terms such as "valid", "consistency", "entailment", which often arise in philosophy, to help the reader comprehend the philosophical issues that use them. Philosophical debates of certain issues are not developed here, and are at most briefly mentioned. Certain assumptions are made; the reader is advised to consider them.

The final part of this book (Other Logics) is a development and extension of the principles described herein. It is designed to interest rather than fully inform the reader of these matters.

Consistency and Inconsistency

Logic is a study of the consistency of beliefs. A belief is part of a psychological state in which a person thinks, is under an impression or believes that the universe has some property. They are readily represented by sentences, a linguistic interpretation. Such sentences can then be evaluated for truth.

Truth

What is truth? What sort of thing could be considered true?

1. Shut that door!
2. Are you going to the party?
3. Would that I were not hungry.
4. A panda eats shoots and leaves.

Of the above sentences, only one, it seems, is capable of being assigned a truth value (considered true or false). The last sentence is true, but what of the others? The first is a command. It can be obeyed or disobeyed, but cannot be true (or false). The second is a question and again cannot be true or false. The third expresses a wish; it may be true that the person uttering it wishes he or she were not hungry, but it itself cannot be evaluated for truth. It shall be said that only declarative statements are evaluable for truth.

A test, perhaps, for determining if a sentence is declarative or not is to ask if the following question is a meaningful, grammatical question in English:

• Is it true that P?

where P is the sentence being analysed.

1. Is it true that shut that door?
2. Is it true that are you going to the party?
3. Is it true that would that I were hungry?

None of the above is meaningful English. These therefore fail the test.

So what is it for a declarative sentence to be true. There are many truth theories, each defining truth slightly differently. The definition of truth adopted in this book comes from the correspondence theory, which states that a sentence P is true if and only if the state of affairs designated by P obtains; "it is true that P" is true if P. Falsehood is similarly defined for a sentence: a sentence is false if the state of affairs it designates do not exist. Although this seems straightforward and platitudinous, there are complications, explanation of which roams outside the scope of this book. The reader is urged to investigate these!

For the sake of simplicity it will be assumed that all declarative sentences are true or false and so, if a sentence is true, its negation is false. The negation of a sentence P is any sentence that shares its meaning precisely with the sentence

it is not the case that P.

Definition of Consistency

Logic is concerned with the consistency of sets of sentences (used to represent beliefs) and with consistent outcomes to processing logical formulae containing operands and operatives. A set of sentences is said to be consistent if and only if there is at least one possible situation in which they are all true. So, the following set of sentences is consistent:

1. Grass is green.
2. It is not the case that the moon is green.
3. Most humans have ten fingers.
4. Grasshoppers have six legs and dogs have four.
5. Earth is a cube.

Note that not all of the sentences are true (sentence 5). However, if the Earth were a cube, it need not be that any of the other sentences are false. There is a possible situation in which they are all true. In this way, logicians are not interested as such in the actual truth value of a sentence (whether it is in fact true or false), but more in a sentence's possible truth values.

Consider the following English sentences:

1. The moon is entirely constituted from cheese.
2. The moon is only partly constituted from cheese.

There is no possible situation in which 1 and 2 are both true, that is to say, it is impossible for them both to be true at the same moment. Such a pair of sentences are said to be inconsistent. A set of sentences is said to be inconsistent if there is no possible situation in which they are all true.

A set of sentences may consist of only one sentence. In which case, if there is a possible situation in which that sentence is true, the sentence is said to be consistent. If there is no such situation, the sentence is said to be inconsistent.

For example, "grass is green" and "snow is blue" are examples of consistent sentences (and the set containing them and only them is consistent). The sentences "grass is green" and "grass is blue" (when taken individually) are consistent sentences, but any set containing them both is inconsistent.

The sentence "2+2=5" is inconsistent. There is no situation in which it is true.

If two sentences cannot be both true and cannot be both false, they are said to be contradictory. For example, "Socrates was a philosopher" and "Socrates was not a philosopher" are contradictory statements.

If two sentences cannot be both true at the same time (form an inconsistent set), they are said to be contrary. For example, "I have exactly 10 fingers" and "I have exactly 9 fingers" are contrary (both statements cannot be true, but both could be false). All contradictory statements are contrary.

Logical Possibility and Necessity

What exactly is meant by possible where it is used above? The type of possibility being considered is logical. If something is considered logically possible, it could have been the case that it was true, regardless of whether it is actually true or not. So, it could have been true that Charles I was not beheaded, for example if he had evaded the executioner, or that the moon is made of cheese, if, for example, the universe was slightly different to how it actually is.

Necessity can be defined in terms of possibility. It is, for example, impossible that Charles I was both beheaded and not beheaded, i.e. it is not possible that he was beheaded and not beheaded. A statement about impossibility is a statement about necessity, i.e. the way things must be. To say something is impossible is to say that no possible situation exists where it is true. If no situation exists where it is true, it must be false - it is necessarily false and so its negation is necessarily true.

1. It is not the case that P is possible.
2. It must be the case that it is not the case that P is true.

The reader should strive to understand that 2 follows from 1 (and indeed that 1 follows from 2) before continuing. Some examples will clarify this (and extend the idea):

1. *It is not possible that two plus two makes five.

 *It must be that two plus two does not make five.

1. *It is not possible that four is not the square of two.

 *It must be that four is the square of two.

1. *According to the definition above, it must be that a sentence is either true or false.

 *According to the definition above, it could not be that a sentence were neither true nor false.

A sentence that describes a state of affairs that is not possible is said to be necessarily false. A sentence the negation of which describes a state of affairs that is not possible is said to be necessarily true.

Arguments

Definition of an argument

An argument (in the context of logic) is defined as a set of premises and a conclusion where the conclusion and premises are separated by some trigger word, phrase or mark known as a turnstile.

For example:

1 I think; therefore I am.

There is only one premise in this argument, I think. The conclusion is I am and the turnstile is therefore (although the semi-colon may be thought of as part of the turnstile).

2 All men are mortal. Socrates was a man. So, Socrates was mortal.

In this example there are two premises and the turnstile is so.

In English (and all other natural languages), the conclusion need not come at the end of an argument:

3 Pigs can fly. For pigs have wings and all winged animals can fly.

4 I am a safe driver: I have never had an accident.

Here the turnstiles (for and :) seem to indicate where the premises come as opposed to where the conclusion comes. Other examples of turnstiles (indicating either conclusion or premises) are: so, thus, hence, since, because, it follows that, for the reason that, from this it can be seen that.

Sound arguments

A sound argument is an argument that satisfies three conditions. -True premises -Unambiguous premises -Valid logic

If one of these conditions is unsatisfied then the argument is unsound, though in the case of ambiguous premises, not necessarily so.

The first, second and fourth arguments (depending of course, on who "I" is - it is being assumed it is the author!) are all sound. Here are some more:

5 Grass is green. The sky is blue. Snow is white. Therefore coal is black.

6 Grass is green. The sky is blue. Snow is white. Therefore pigs can fly.

7 2+2=4; hence 2+2=4

8 2+2=4; hence 2+2=5

Note that it is not necessary that the truth value of the conclusion play a role in determining whether an argument is sound. This can be determined by considering the truth of the premises and the validity of the argument. However, if the conclusion is not true, the argument is not sound.

A sound argument always has consistent premises. This must be the case, since there is a possible situation (namely reality) in which they are all true.

Valid arguments

Of greater interest to the logician are valid arguments. A valid argument is an argument for which there is no possible situation in which the premises are all true and the conclusion is false.

Of the above arguments 2, 3 and 7 are valid. The reader should consider whether argument 1 is valid (read Meditations on First Philosophy by Descartes, chapters 1, 2).

It does not matter whether the premises or conclusion are actually true for an argument to be valid. All that matters is that the premises could not all be true and the conclusion false. Indeed, this means that an argument with inconsistent premises is always valid. There is no situation under which such an argument has all premises true and so there is no situation under which such an argument has all premises true and conclusion false. Hence it is valid. Similarly, an argument with a necessary conclusion can in no situation have all true premises and a false conclusion, since there is no situation in which the conclusion is false.

An argument with the single premise 'The conclusion is true.' is valid (regardless of the conclusion). An argument with the conclusion 'The premises are all true.' is also valid.

According to the definition of truth given previously, if the conclusion is false, its negation is true. Hence a valid argument can also be defined as an argument for which there is no possible situation under which the premises and the negation of the conclusion are all true. Hence, a valid argument is an argument such that the set of its premises and the negation of the conclusion is inconsistent. Such a set (the union of the set of premises and the set of the negation of the conclusion) is known as the counter-example set. It is called the counter-example set for the following reason: if a possible situation is found in which the members of this set are all true (and so the set is found to be consistent), this situation provides a counter-example to the arguments being valid, i.e. the existence of such a situation proves that the argument is not valid.

Counter-examples do not exist only for arguments, but also for statements:

Prime numbers are always odd 2 provides a counter-example (a number) to this statement. All animals have four legs Human beings provide a counter-example (a type of animal). Years are 365 days long Leap years provide a counter-example (a type of year). Years designated by a number divisible by four are leap years The year 1900 provides a counter-example (a particular year). It always rains in England A singularly sunny day in September (today, when written - a particular interval of time) provides a counter-example.

Counter-examples to declarative sentences refute their truth and are classes of things (thing being understood very broadly here) or particular things. Counter-examples to arguments refute their validity and are possible situations designated by sets of sentences (the counter-example set). Some clarification of the situation is often needed.

For example, take argument 4. It will be modified slightly as follows:

John has never had an accident; therefore, John is a safe driver.

It will be assumed here that accident means car accident and driver means motorist and safe means not liable to cause an accident. The counter-example set is:

John has never had an accident. John is not a safe driver.

Clarification by example: it may be that John has never driven in his life (and so never had an accident) because he is blind (and so cannot be considered a safe driver).

As mentioned, an inconsistent counter-example set implies that a conclusion is valid because it means that there is no situation under which the premises are all true and the conclusion is false (the negation of the conclusion is true).

Take argument 2; the counter-example set is:

All men are mortal. Socrates was a man. Socrates was not mortal.

These sentences cannot all be true at once. If Socrates was a man and he was not mortal, it could not be that all men are mortal. If all men are indeed mortal and Socrates was not mortal, he could not have been a man. If all men are mortal and Socrates was a man, he must have been mortal. Hence the counter-example set is inconsistent and the argument is valid.

Use a similar approach to show that arguments 3 and 7 are valid (and use it to consider argument 1). This method is known as reductio ad absurdum (which translates literally from Latin as "reduction to absurdity"). The negation of the conclusion is absurd given the truth of the premises and so the conclusion must be true.

Propositional Calculus

Natural languages, such as English, are flawed. One such flaw is ambiguity, another is that they are tedious to write out if they are long. However, sentences can be formalised into a symbolic, logical language that contains neither of these flaws (although, as the discerning reader will discover, natural languages have many advantages over these languages).

The first such language to be considered is the propositional calculus.

Complex sentences and sentence functors

The reductio ad absurdum method can be extended to complex sentences. A complex sentence is a sentence made up of smaller sentences, for example "Sarah can swim but she cannot dive" is made up of the declarative sentences "Sarah can swim" and "She cannot dive", linked by the conjunction "but". Such words and phrases that link other declarative sentences are sentence functors. A functor is the part of a language that stands for a function. For instance "y=sin(x)" is a statement (in a particular mathematical notation) about the function sine and "sin()" is the functor that stands for this function in that statement. Function is not defined here. If you are unsure of its meaning, please refer to Wikipedia.

A 'sentence function' takes declarative sentences as input and yields declarative sentences as output. Sentence functors stand for such functions. A sentence functor is then a string of words and sentence variables that becomes a declarative sentence if each sentence variable is replaced by a declarative sentence (here "string" just means a series of symbols). This is true for English, French, Spanish, German, Greek, Latin and so on, but all words and declarative sentences must be in the same language.

The composition of sentences

Sentences comprise parts known as constituents. A constituent will be defined as a string of symbols that is meaningful by itself. This definition may be regarded as slightly empty and the meaning of constituent is probably best understood intuitively and by example. The following are constituents of the sentence "The cat sat on the mat":

the

cat

the cat

sat on the mat

on the mat

the cat sat on the mat

All of the above are meaningful in themselves. Words are meaningful by themselves and so all individual words are constituents of any sentence that contains them (words can be considered atomic parts of language, that is they cannot be broken down further into other constituents - inflections, prefixes and suffixes are ignored here). The meaning of each constituent shown is the same as its meaning as a part of the sentence. The meaning of the constituent must be the same as it meaning in the sentence of which it is part, otherwise (although a string identical to the constituent may appear in the sentence) it is not a constituent of the sentence. For example, consider the sentence "the man who wrote on the blackboard was old".

The string "the blackboard was old" is meaningful, however its meaning is no part of the sentence above, which is asserting that the man, not the blackboard, was old. Hence, "the blackboard was old" is not a constituent of that sentence. Similarly, and perhaps less clearly, "the man" is not a constituent of the sentence. The meaning of "the man" is not the meaning conveyed in the sentence. Were "the man" to be a constituent, it would suggest that a particular man had already been identified. However, this is not the case since the restriction "who wrote on the blackboard" is needed. So, "the man" is not a constituent, but "the man who wrote on the blackboard" is.

Ambiguity

An ambiguous sentence is one that has more than one meaning. Crudely speaking, there are two types of ambiguity: structural and lexical. A lexical ambiguity arises where one word (or perhaps a phrase) has more than one meaning; for example "fast" in the sentences "He drove very fast" and "He was fast asleep". A structural ambiguity arises in a constituent because it is unclear what that constituent's constituents are. To clarify this, the notion of scope is introduced.

The scope of a constituent is defined as the smallest constituent containing that constituent and something else besides. So, in the above example, the scope of "blackboard" is "the blackboard"; the scope of the first "the" is "the man who wrote on the blackboard".

Consider the sentence "He told me to be careful this evening". Was this warning discussed in this sentence issued this evening or was it about this evening? It is unclear exactly what the constituents of this sentence are: it is unclear what the scope of "this evening" is. Is the scope of "this evening" "to be careful this evening", or is it the whole sentence "He told me to be careful this evening"? It can be seen that a new language might be devised to clear up such ambiguities: "He told me [to be careful this evening]"; "[He told me to be careful this evening]". Propositional calculus is an entirely unambiguous language, using a bracketing system as shown here. This will be seen in the next part of the book.

Sentence functors

To repeat the definition given earlier, sentence functors are strings of words and sentence variables such that if all the sentence variables are replaced by any declarative sentences, the whole becomes a declarative sentence. To fully understand this definition, it is necessary to know what a sentence variable is. A sentence variable is something (usually represented by the Greek letters psi, phi or chi) that can be assigned as its value any declarative sentence.

One example of a sentence functor was discussed in the section on consistency: it is not the case that phi

Other examples of a sentence functors are:

phi and psi I know that phi It is obvious that phi Either phi, or psi and chi

The following are not English sentence functors (consider whether the string obtained by replacing the sentence variable with a declarative sentence is in itself a declarative sentence):

Mary and phi Is it true that phi? phi is true (but "phi" is true is a sentence functor) Whomever phi should stand up for themselves

The last example forms a declarative sentence when phi is replaced by some declarative sentences (for example, if replaced with "Jack is bullying"), but not all (e.g. "the sky is blue"), so it is not a sentence functor.

The number of places of a sentence functor is the number of different sentence variables it contains.

Either phi, or psi and chi is a 3-place sentence functor. Either phi or psi is a 2-place sentence functor. Either it is the case that psi or it is not the case that psi is a 1-place sentence functor.

An n-place sentence functor is satisfied by certain ordered n-tuplets of declarative sentences. An ordered n-tuplet of declarative sentences is a list of n different declarative sentences in a particular order. In particular, those ordered n-tuplets of declarative sentences that satisfy an n-place sentence functor yield a true declarative sentence when they, in order, replace the sentence variables of a sentence functor.

The ordered pair (grass is green, snow is black) satisfies it is the case that phi, but not that psi; whereas the ordered pair (snow is black, grass is green) does not.

A sentence is a sentence functor with no sentence variables, i.e. a sentence is a 0-place sentence functor.

Truth tables

When determining what sentences satisfy what sentence functors, the logician is interested in their truth value (rather than the actual meaning or sense). This information can be summarised in the form of a truth table.

A truth table stipulates all combinations of truth for a given set of sentences and what the truth value of a sentence functor is for each combination.

Consider the sentence functor phi and psi. The truth table for this sentence functor is drawn up as follows:

P	Q	P and Q
T	T	T
T	F	F
F	T	F
F	F	F

Notice that the letters P and Q are used rather than phi and psi. Sentence variables cannot bear truth values. P and Q instantiate actual sentences, the truth values of which are considered below these letters in the table: 'T' stands for when that sentence is true and 'F' stands for when it is false. Note also that "P and Q" is a sentence, not a 2-place sentence functor (sentence functors with at least one sentence variable cannot bear truth values). To know what the value of the declarative sentence yielded by replacing the sentence variables of the sentence functor with sentences of various truth values, the row (known as a structure) containing the desired truth values is selected and the letter in that row below the complex sentence taken.

In the above example, the sentence "P and Q" is true when P is true and Q is true but false for any other values of P and Q (so when P is true but Q is false, "P and Q" is false).

Consider the truth table of the sentence functor Hume knew that phi.

P	Hume knew that P
T	-
F	F

The structure where P is false is false for "Hume knew that P", for it is not possible to know something that is false. However, the structure where P is true has the symbol "-" in it (this will be referred to as a blank). This symbol does not mean that the sentence is neither true nor false in this structure. It means that there are true sentences that satisfy this functor and there are sentences that do not satisfy this functor. For example, Hume knew that his (Hume's) first name was David. However, he did not know that Russell was (or rather would be from Hume's perspective) a 20th century philosopher.

Predicate Calculus

Predicate calculus, also called Logic Of Quantifiers, that part of modern formal or symbolic logic which systematically exhibits the logical relations between sentences that hold purely in virtue of the manner in which predicates or noun expressions are distributed through ranges of subjects by means of quantifiers such as “all” and “some” without regard to the meanings or conceptual contents of any predicates in particular. Such predicates can include both qualities and relations; and, in a higher-order form called the functional calculus, it also includes functions, which are “framework” expressions with one or with several variables that acquire definite truth-values only when the variables are replaced by specific terms. The predicate calculus is to be distinguished from the propositional calculus, which deals with unanalyzed whole propositions related by connectives (such as “and,” “if . . . then,” and “or”).

The traditional syllogism is the most well-known sample of predicate logic, though it does not exhaust the subject. In such arguments as “All C are B and no B are A, so no C are A,” the truth of the two premises requires the truth of the conclusion in virtue of the manner in which the predicates B and A are distributed with reference to the classes specified by C and B, respectively. If, for example, the predicate A belonged to only one of the B’s, the conclusion then could possibly be false—some C could be an A.

Modern symbolic logic, of which the predicate calculus is a part, does not restrict itself, however, to the traditional syllogistic forms or to their symbolisms, a very large number of which have been devised. The predicate calculus usually builds upon some form of the propositional calculus. It then proceeds to give a classification of the sentence types that it contains or deals with, by reference to the different manners in which predicates may be distributed within sentences. It distinguishes, for example, the following two types of sentences: “All F’s are either G’s or H’s,” and “Some F’s are both G’s and H’s.” The conditions of truth and falsity in the basic sentence types are determined, and then a cross-classification is made that groups the sentences formulable within the calculus into three mutually exclusive classes—(1) those sentences that are true on every possible specification of the meaning of their predicate signs, as with “Everything is F or is not F”; (2) those false on every such specification, as with “Something is F and not F”; and (3) those true on some specifications and false on others, as with “Something is F and is G.” These are, respectively, the tautologous, inconsistent, and contingent sentences of the predicate calculus. Certain tautologous sentence types may be selected as axioms or as the basis for rules for transforming the symbols of the various sentence types; and rather routine and mechanical procedures may then be laid down for deciding whether given sentences are tautologous, inconsistent, or contingent—or whether and how given sentences are logically related to each other. Such procedures can be devised to decide the logical properties and relations of every sentence in any predicate calculus that does not contain predicates (functions) that range over predicates themselves—i.e., in any first-order, or lower, predicate calculus.

Calculi that do contain predicates ranging freely over predicates, on the other hand—called higher-order calculi—do not permit the classification of all their sentences by such routine procedures. As was proved by Kurt Gödel, a 20th-century Moravian-born American mathematical logician, these calculi, if consistent, always contain well-formed formulas such that neither they nor their negations can be derived (shown tautologous) by the rules of the calculus. Such calculi are, in the precise sense, incomplete. Various restricted forms of the higher-order calculi have been shown, however, to be susceptible to routine decision procedures for all of their formulae.