As an introduction to philosophical logic, and a foundation from which to approach deeper philosophical subjects, I will be reading and summarizing Mark Sainsbury’s “Logical Forms: An Introduction to Philosophical Logic” (2nd ed), and occasionally adding my own thoughts or questions relating to the content as I go.
Chapter 1: Validity
The study of logic is concerned with whether we have good reasons for our beliefs.
We can mean different things when we talk about “reasons for belief”. For example:
1. Reason in the sense of an explanation as to how someone came to believe something. “He believes x because he read it in a book.”
2. Reason in the sense of justification; a proposition or set of propositions that aim to provide evidence for a belief. Logic is concerned with this second kind of reason.
Justification is a relational term. The question is, how does one set of propositions (call them premises) come to justify another proposition (the conclusion)?
We will be looking at deductive logic, which is concerned with the formal validity or invalidity of arguments. A valid argument is one in which the premises, if accepted as true, guarantee the truth of the conclusion. The premises cannot be true, while the conclusion is false.
Validity is therefore a relation between the premises and the conclusion. It is irrelevant whether or not the premises actually are true. Validity only ensures that, if the premises were true, then conclusion would also necessarily be true. Validity is therefore a formal property of arguments. It is something arguments have or fail to have, by virtue of the form of the propositions and the relations between them, not with the actual content of the propositions.
“Inductive logic”, on the other hand, is concerned with arguments that aim to show that their conclusions are probably true. Evaluating inductive arguments requires more than just examining their formal characteristics. It requires evaluating arguments by the contents of their propositions, and in the light of certain unstated, background assumptions or theories.
How can we test whether a deductive argument is valid? That is, how do we know that, if the premises were true, the conclusion must also be true?
Consider the argument: “Fido is a dog [premise]. Therefore he cannot sing ‘Happy Birthday' [conclusion]”. You might think that it is impossible for the conclusion to be false, given the truth of the premise, and therefore this is a valid argument. But to a logician, this argument is not valid.
To see why, we could appeal to a distinction between logical possibility and physical possibility. It is not physically possible for a dog to sing, but suppose you had never heard of a "dog" and were unaware of this fact. Then you would have no way of judging the validity of the argument.
While it is true as a matter of physical fact that dogs cannot sing, it nevertheless might be logically possible that dogs could sing, if the facts about the world were different. We could then say that an argument is only valid if it were logically impossible for the premises to be true and the conclusion false. But now we will need to provide a definition of logical possibility.
We could define logical impossibility in terms of consistency and inconsistency: two propositions are inconsistent if they cannot both be true at the same time. For example, the propositions “today is Monday” and “tomorrow is Friday”. These two propositions are inconsistent because they cannot both be true simultaneously. (Note, however, that they could both be false.)
We can also define a subset of inconsistency, known as contradiction. If two propositions cannot both be true nor both be false, they are contradictory. And a subset of contradiction is negation, which is simply contradiction that uses the word “not” or an equivalent word or phrase. Eg. “Today is Monday” is the negation of “today is not Monday.”
Note:
These definitions sound simple, but they can trip you up if you’re not careful. Consider the propositions “I hope she will arrive soon” and “I hope she will not arrive soon”. These are inconsistent (they cannot both be true), but they are not contradictory (they could both be false; I might have no feeling one way or the other).
Or consider the propositions “you must walk on the grass” and “you must not walk on the grass”. These propositions are not negations of each other, despite the use of the word “not”. Once again, both propositions could be false; walking on the grass may be permitted but not mandatory. The negation of “you must walk on the grass” could be stated as “it is not the case that you must walk on the grass”.
We can now use these notions to define validity. We said that a valid argument is one in which it is logically impossible for its premises to be true and its conclusion false. But if the conclusion is true, then its contradictory must be false. So we can rephrase our definition as: it is logically impossible for the premises to be true and the contradictory of the conclusion to be true as well. And we have said that if two sentences cannot both be true then they are inconsistent. Therefore, an argument is valid if its premises are inconsistent with the contradictory of the conclusion.
Digression:
So now we have defined validity in terms of logical possibility, and logical possibility in terms of consistency. But how do we determine whether two propositions are consistent or inconsistent? Isn’t it simply by weighing them up in our minds, and declaring it logically possible or impossible for them both to be true? So what is logically impossible is inconsistent, and what is inconsistent is logically impossible. Is this a circular definition? And does this represent a problem for the foundations of logic?
The answer (I think) is that this is not so much a problem for logic, as it is a feature of our language. After all, every word is defined in terms of other words, in an ultimately circular manner. Languages work nevertheless because the speakers of that language share a broadly similar worldview, and agree on the conventional meanings of their words. The circularity in the definition of “inconsistent” is nothing special in this regard. We simply have to agree on whether any two propositions are inconsistent, based on our shared understanding of the words of the sentences concerned.
This does not undermine the foundations of logic, however, because language, while it is conventional, is not arbitrary. We use language to refer to things in the world. So to the extent that words are conventional, we can disagree over the meaning of a proposition. But to the extent that words refer to the world, there is a fact of the matter about whether the proposition is true or not.
Suppose we ask if the following two propositions are inconsistent: “Socrates is a man. Socrates is not a man.” They certainly seem to be inconsistent, but can we be sure? Do we need some kind of independent test? How could we argue against a person who held that they were not inconsistent? To do so, we would not need to appeal to some mysterious “logical laws” which ensure that something cannot both be and not be. We could simply appeal to the conventions of language, and point out that by asserting both the first and the second propositions, our imaginary speaker has not actually asserted anything at all. He has not made a claim that can be evaluated, because he has denied the claim in the very act of making it. And if he were to maintain that, actually, it is possible for Socrates to both be and to not be a man, we would just have to conclude that he is not using the same definitions for his words as we are.
Therefore we can rely on a consensus understanding of the meaning of words to determine the consistency of a set of propositions, without this undermining the usefulness of the concept of validity.
Some properties of Validity:
Validity logically guarantees the preservation of truth. If you start with true premises, and you argue validly, you will always end up with true conclusions.
However, valid arguments can have false premises and/or false conclusions.
Furthermore, invalid arguments can have true premises and/or true conclusions.
Monotonicity: There are no degrees of validity; an argument is either valid or it is not. And because validity is a product of the relation between premises and conclusion, once an argument has “attained” validity, it cannot be made invalid by adding new premises, no matter what those premises are.
Transitivity: chaining arguments together will preserve validity. This means that, if “A, therefore B” is valid, and “B, therefore C” is valid, then “A, therefore C” is also valid.
“B" does not need to be stated as a premise because it is already established by “A”.
Reflexivity: if the conclusion of an argument is also a premise of the same argument, the argument is automatically valid (regardless of what other premises there are). For instance, if your premises are “A”, “B”, and “C”, the conclusion “therefore B” follows validly.
This shows that circular arguments are valid (though they may not be very useful).
An argument with inconsistent premises is valid, whatever its conclusion (this is called the “principle of explosion” - if you start with inconsistent premises you can validly deduce anything). But since inconsistent premises cannot all be true, we cannot really infer anything from these arguments.
For example, given the premises “A” and “not A”, you can validly derive the conclusion “Therefore B”. This argument is valid, but it cannot tell us anything about the truth of B, because one of the premises must be false.
If it is logically impossible for a proposition to be false, then any argument with that proposition as conclusion will be valid.
eg. If pigs can fly, then the moon is made of cheese.
Pigs can fly.
Therefore either Socrates is mortal or Socrates is immortal.
The conclusion is necessarily true (Socrates must be either mortal or immortal). Therefore the argument satisfies the technical criterion of validity: it is impossible for the premises to be true while the conclusion is false.
Relevance and Persuasiveness:
An argument can be sound (meaning that its premises are true) and valid, but still useless. eg. “Canberra is the capital of Australia. Therefore all dogs are dogs.” The premise of this argument is true, so the argument is sound. And the conclusion cannot be false, so the argument is valid. Nevertheless, it is hard to see how this argument could be relevant to anyone. Nor would it be persuasive to anyone who did not already accept the conclusion.
Propositions and Sentences:
Declarative sentences are sentences that can be affirmed as true or false (“the car is red”), as opposed to interrogative sentences which ask how things are (“is the car red?”), or imperative sentences which order things to be a certain way (“close the door”).
Subjunctive sentences express a hypothetical (“were the car red”), and can be used in if-then sentences (so-called "subjunctive conditionals").
A proposition is what is expressed by a meaningful, declarative sentence.
It can be useful to think in terms of propositions, rather than sentences, because the meaning of a sentence can change with context, eg. the phrase "I am hungry" means one thing if I say it, but something different when you say it, because the word “I” does not refer to the same entity in both cases. The proposition is what is affirmed or expressed by the sentence, so in this case we can talk about two different propositions being expressed by the same sentence.
Truth Conditions:
Truth conditions are defined as the circumstances in which a sentence is true. The sentence “snow is white” is true given the circumstance that snow is white. This seems like a simple concept, but it allows us to make a new definition of validity:
An argument is valid if the truth conditions of the conclusion contains the truth conditions of all of the premises. That is, any circumstances in which the premises are true will also be circumstances in which the conclusion is true. The truth conditions of the latter set of circumstances "contains" the truth conditions of the former.
This has the consequence that you can replace a sentence in an argument without affecting its validity, as long as the new sentence has the same truth conditions as the previous sentence.
Formal Validity and Argument Forms:
An argument form is a sentence or set of sentences containing a number of variables. The variables represent any words that can be substituted into the sentence. When the variables in an argument form are substituted with appropriate words, the result is an argument. We can say that the argument is an instance of that particular argument form.
An example of an argument form might be: “All F are G. x is F. Therefore x is G.” where F, G, and x are the variables (in general, lower-case letters stand for proper names, while upper-case letters stand for predicates - expressions that can be used to describe or qualify names). An instance of such an argument form might be: “All men are mortal. Socrates is a man. Therefore Socrates is mortal.”
An argument form is valid if and only if, necessarily, each of its instances is valid.
An argument is formally valid if it is an instance of a valid argument form.
Are all valid arguments formally valid? That is, do all valid arguments conform to a valid argument-form? No. Consider the argument:
Tom is a bachelor. Therefore Tom is unmarried.
This is valid, but it is not formally valid (its argument form is "x is F. Therefore x is G". This is not a valid argument form, because not all of its instances would be valid).
In fact, all valid arguments instantiate multiple argument forms, some valid and some invalid. For example, we could have said that the previous example is an instance of the argument form “x is a bachelor. Therefore x is unmarried.” This argument form is valid, because every instance of it would be valid (under a conventional definition of “bachelor”). To be formally valid, an argument only has to instantiate one valid argument form, regardless of what other invalid forms it also instantiates.
In order to make useful generalizations about valid and invalid argument forms, we can restrict the definition to say that an argument form consists only of variables and logical constants.
Logical constants include words and phrases such as:
It is not the case that
And
Or
If… then…
If and only if
Some
A
Everything
All
Is
Are
Is the same as
These words and phrases have constant interpretations - they operate the same in any proposition in which they appear. In principle, we could assign a constant interpretation to any expression we choose, but in logic we are interested in words with meanings applicable to any subject area. Logical constants are therefore topic-neutral. (So, for example, we could assign a constant, unambiguous interpretation to the word “bachelor”, but this would not be considered valuable as a logical constant, because its use is limited to very narrow contexts.)
So now we can say that an argument is valid if and only if it can be written as a valid argument form containing nothing but variables and logical constants.
Formalisation:
Formalisation is the translation of natural language sentences into argument forms. Natural language sentences often obscure the logical form of the propositions they express, because natural language is subject to various kinds of ambiguity, and missing information that must be supplied by the audience.
To unambiguously test the validity of a proposition, all the logically relevant features of the proposition would have to correlate with distinct features of the sentence expressing it. The aim of logic is to create artificial languages to facilitate this. Formalisation is the translation of natural language sentences into such an artificial language, so they can be written with only logical constants and variables.
Natural languages can be unsuitable for this purpose in different ways, such as:
Lexical ambiguity - the same word having different meanings.
Structural ambiguity - eg. in the sentence “Tom is a dirty window cleaner” (none of the words on their own are ambiguous, but the sentence as a whole is ambiguous - it can yield two different interpretations). Structural ambiguity even affects the logical constants, eg. "Tom has written a book about everything" - the constants "a" and "everything" are ambiguous (is that a single book that covers all topics, or a different book for every topic?).
Syntactic ambiguity - let us say that two expressions belong to the same syntactic category if they can always be substituted for each other without turning sense into nonsense, either in the premise or in the conclusion of the argument. eg. in the argument “John is a teacher. Therefore someone is a teacher”, replacing "John" with "no one" yields "no one is a teacher. Therefore someone is a teacher". The substitution invalidates the argument, so “John” and “no one” can be said to belong to different syntactic categories. The words themselves are not ambiguous, but they function differently in the syntax of a sentence. We need to be clear about the differences between syntactic categories so we can specify what counts as a valid substitute for the variables in a sentence of propositional logic.
Again, consider two examples of the argument form: F are G. a is an F. So a is G.
Human beings are sensitive to pain. Harry is a human being. So Harry is sensitive to pain.
Human beings have an average height of 5 feet. Harry is a human being. So Harry has an average height of 5 feet.
The former would appear to be valid, but the invalidity of the latter establishes that the argument form is in fact not valid. The rules governing predicates will need more spelling out if we are to generalize about valid argument forms.
There are other ways in which natural language sentences can exhibit syntactic irregularities. For example, take the sentence "I will marry you if you change your religion". This is unambiguous as it stands, but if we preface this sentence with "it is not the case that", it is now open to different interpretations (either, "I could only marry if you did not change your religion" or "I will not marry you even if you change your religion"). The resulting sentence becomes ambiguous, rather than clearly producing a negation of the original proposition.
Formalisation is intended to remove problems of interpretation like those described above. The task is then to be able to pair every natural language sentence with a sentence in a formal language which is, or reveals, the logical form of the original.
We can then discover fully generalizable rules about validity using the formalism, and translate these artificial sentences back into natural language.
It is helpful to keep in mind this intended function of a formal language. There are times when sentences written in this language will behave very differently to a normal English sentence. For example, we saw above how we can use inconsistent premises to prove absolutely anything, or derive a true conclusion from premises we know to be false. Such arguments, while technically valid given our definitions, are not conducive to conveying useful information about the real world. It is natural to ask what is the point of such a language, then, if it can diverge so widely from how we actually communicate?
But a formal language is not designed for everyday communication. It is designed so that its sentences can be unambiguously and unchangingly interpreted. This means that we can be sure when the conclusion of our argument follows validly from its premises, and when it does not. We can always translate the formal language sentence into some natural language equivalent if we wish (thereby risking the introduction of some ambiguity), but the reverse is not the case; we cannot always translate a natural language sentence into an argument form, as we saw above, because of ambiguities of interpretation. The formal language (and, we might say, the study of logic itself) is a tool we use to derive conclusions, and to test their validity, in an unambiguous way. Such, at least, is the goal. We have yet to discuss how we might attempt to construct such a language.
No comments:
Post a Comment