In the excursus on formal languages in the last blog post, we already got to know the signs and more general expressions of propositional logic. We have seen that there are special character strings, so-called tautologies, which are always true, i.e. independent of the truth values of the individual characters. We had also already introduced such a tautology, namely (A ∧ (A → B)) → B.
Here we will show how logically correct reasoning rules can be formulated with the help of tautologies. But let’s first ask which simple tautologies still exist and how to generate tautologies in general. Interesting in this context are also character strings, which are always false, because from their negation one can also gain a tautology.
Tautologies
Let us first introduce prominent tautologies:
A ∨¬A is in any case true,
because either A is true or ¬A. There is no such thing as a third according to our requirements. This statement is called the statement on the excluded third.
On the other hand, A ∧ ¬A is wrong in any case, because the statement A and the statement ¬A cannot be true at the same time. For example, it cannot be that it is raining and not raining at the same time. A and ¬A contradict each other. One generally calls a compound expression, which is false regardless of the truth values of the individual statements, a contradiction. Then, the following applies to the negation:
¬(A ∧ ¬A) is true in any case.
This statement is called the principle of contradiction.
Here we should now also list the tautology from the last blog post again.
(A ∧ (A → B)) → B is in any case true.
How do we find further tautologies in order to be able to form further rules of inference?
It can be shown that the propositional logic can be regarded as an axiomatic-deductive system. Axioms can be all tautologies of the form (by Kutschera & Breitkopf, Alfred, 2007, p. 69).
A → (B → A),
(A → (B → C)) → ((A → B) → (A → C)),
(¬A → ¬B) → (B → A)
and as a rule of inference or concluding rule the modus ponens, which we already mentioned in an earlier blog post. In a moment, we will really introduce this concluding rule by deriving it from a tautology.
All logical expressions, that can now be derived from these axioms using the modus ponens, are tautologies again. So you can set up as many concluding rules as you want. Only a few will be needed.
Now we understand why Wittgenstein says: The propositions of logic are tautologies. So the propositions of logic say nothing (Wittgenstein, 2006, pp. No. 6.1, 6.11).
From Tautologies to Concluding Rules
Let’s have a look at the truth table for the tautology (A ∧ (A → B)) → B again:
A | B | A → B | A ∧ ( A → B) | (A ∧ (A → B)) → B |
1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
0 | 1 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 1 |
Let us first consider the first line, in which both premises A and A → B are true. From the second column of this first row we then infer that B is true. The statement B must therefore necessarily be true if both A and A → B is true. Only in this way can the character of tautology be respected.
This is now a conclusion that results from inspection of the truth table of a tautology. No reasoning can be more elementary. This is also the conclusion with which all other concluding rules can be won. So here we have the origin of logical reasoning before us, the “mother” of all reasoning rules.
You write, with the sign “⊨” for a logical conclusion:
A ∧ (A → B) ⊨ B,
but frequently also in a form in which the individual premises are separated only by a comma:
A, (A → B) ⊨ B.
The symbol “⊨” is not a sign of propositional logic, but an abbreviation for the phrase “follows logically from this” in colloquial language. Otherwise we should be able to count with this symbol like with “∧” or “∨”. It only expresses the relationship between the statements A ∧ (A → B) and B in the meta language, our colloquial language: In the case that the statement A is true, and in the case that the statement B follows from A, then B is true.
You may be confused at first and ask why all the effort. They knew that a long time ago. It is indeed trivial, in the truest sense of the word, because in the Middle Ages the word “trivial” was coined from insights gained in trivium, the lowest level of education in a monastic school. This trivium, in turn, was named in ancient times for a place to which three paths lead, and where many who share the same opinion can gather.
Here, however, in a formal language, what Aristotle has already defined is very concretely realized: “A conclusion is thus a speech in which, with certain assumptions, something other than the presupposed follows with necessity on the basis of the presupposed”. (after Schupp, I, p.267). The emphasis is on “necessity”.
This is the modus ponens. It is the most prominent logical conclusion, was already known in ancient times by the philosophers of the Stoa and subject of many discussions in the Middle Ages (see “the Logic of the Stoics”).
We still want to investigate what this conclusion tells us if one of the premises or both are wrong. In any case, the entire premise is then wrong, because the individual premises are linked by a “∧”. We extract the two relevant columns from the table above and arrange them somewhat differently:
A ∧ (A → B) | B |
1 | 1 |
0 | 0 |
0 | 1 |
0 | 0 |
So, if the total premise is false (2nd to 4th line), B can be true, but also false, i.e. nothing can be said about the truth value of B. Everything can be deduced from a false premise. At first this is surprising, and in the history of logic it has been discussed for a long time. But if one can derive both B and ¬B, then the conclusion is meaningless.
From the derivation of the modus ponens we can learn how to create a general final rule: From every tautology that can be found in the form
M → B, you can use the final rule
M ⊨ B
because then it follows immediately from the truth table for “M → B” that under the condition that M is true, also B must be true, because this implication M → B is a tautology after assumption and therefore true.
This is a statement on the meta level, not in the calculation of the syntactic level. However, we know that because M → B is a tautology, we can go from the character string M to the character (string) B on the syntactic level without leaving the realm of true statements on the semantic level. We write this in the form of
M ⊢ B,
and call this operation a derivation at the syntactic level. That what is a conclusion on the semantic level is called a derivation on the syntactic level.
At the syntactic level, we can therefore pass from the character string M to the character or character string B without leaving the realm of true statements at the semantic level.
But now one knows how to “calculate” on the syntactic level, namely according to the rules for the formation of character strings and according to the concluding rules, by which one may transform certain character strings into others, which are also usually shorter. Such a system of calculation rules is called a “calculus”.
The calculus of propositional logic therefore has nothing to do with any meaning of the statements in terms of content. To a certain extent, it only provides the tracks on which truth of statements can be safely transported from premises to conclusion. Wrong statements on such tracks lead to arbitrariness. Without the truth of the premises “everything is nothing”. That’ll still occupy us.
Two remarks are in place here:
Let us consider the premises:
A: = „2 + 2 = 4“,
B: ≔ “Freiburg is located in the south of Germany”.
The statements A and B are true, thus also the implication A → B, i.e. if 2 + 2 = 4, then Freiburg lies in the south of Germany”. Thus, the expression A → B is well formed, but futile. Then the logical conclusion
A, A → B ⊨ B
is also meaningless. That doesn’t have to be irritating. Even in our colloquial language we can form grammatically correct sentences that are meaningless: “The moon babbles red suit.” The tracks are not responsible for what goes on them. Incorrectly formed sentences or not well formulated expressions are senseless anyway.
You often hear people say: “It’s logical, isn’t it?” and the speaker means that the conclusion immediately makes sense to him. This feeling, however, probably does not refer to the concluding rule, but to the implication, which is of course true for the speaker. The speaker thus confuses the logical conclusion with his assumption that his premise is true. He should say, “It’s a plausible assumption for the modus ponens, isn’t it?” He would be met with venerable understanding.
Important Concluding Rules in Applications
The modus ponens is probably the most prominent concluding rule, even in generalized form, which reads:
(A_{1} ∨ A_{2} ∨ … ∨ A_{n} ∨ A) ∧ (¬A ∨ B_{1} ∨ B_{2} ∨ … ∨ B_{m})
⊨ A_{1} ∨ A_{2} ∨ … ∨ A_{n} ∨ B_{1} ∨ B_{2} ∨ … ∨ B_{m}.
The underlined portion of the premise represents exactly the simple modus ponens. This term then also appears accordingly as B_{1} in the conclusion.
The premise here is based on a character string in normal or standard form, into which each character string can be brought by a systematic transformation process. In general, this is the normal form:
(A_{1} ∨ A_{2} ∨ … ∨ A_{n}) ∧ (B_{1} ∨ B_{2} ∨ … ∨ B_{m}).
Such procedures and applications of inference rules as the generalized modus ponens are implemented in so-called resolution algorithms. The length of the character string can be successively reduced. Within the “Prolog” program, for example, such an algorithm can be processed on a computer.
A particularly popular inference rule in mathematics, already known in ancient times, is proof by contradiction:
If one wants to prove that a statement A implies another statement B, one first assumes for the proof that besides a premise A also ¬B is true premise. If then one can deduce a contradiction from it, ¬B cannot be true. So B must be true, because there’s no such thing as a third. Here it is shown that the negation of an assumption can be reduced to a contradiction. In the Middle Ages this form of proof was therefore also called “reductio ad adsurdum”.
The contradiction can show itself e.g. in the fact that one can deduce from A ∧ ¬B a statement C, and in addition, the statement ¬C. In order to show that this strategy can also be represented as a concluding rule, one only needs to find the corresponding form M → B. This is
((A ∧ ¬B) → C) ∧ ((A ∧ ¬B) → ¬C) → (A → B).
This is indeed a tautology and therefore the concluding rule:
((A ∧ ¬B) → C) ∧ ((A ∧ ¬B) → ¬C) ⊨ (A → B).
In a somewhat different form one uses the proof by contradiction if one wants to know whether a statement B is contained in a knowledge base W and thus also true. So, you’re asking whether
W ⊨ B
applies. This is the case if W → B, i.e. ¬W ∨ B is a tautology. Since ¬W ∨ B can be transformed into ¬(W ∧ ¬B), we must therefore ask whether ¬(W ∧ ¬B) is a tautology and thus W ∧ ¬B is a contradiction. So, we can see that a statement B can be deduced from a knowledge base W if
W ∧ ¬B
is a contradiction. That is also plausible: If the information of B is contained in W, the contradiction B ∧¬B must show up somehow with the evaluation of the expression W ∧ ¬B.
In order to now show that W ∧ ¬B leads to a contradiction, the expression W ∧ ¬B is transformed into the disjunctive normal form within the framework of the calculus and then the receiving expression is successively reduced with the help of the generalized modus ponens until an expression is shown that represents a contradiction – or not, depending on whether the statement B is contained in the knowledge base W or not.
The proof of the Pythagoreans that there are infinitely many prime numbers is e.g. of this form. The statement B is then: There are infinitely many prime numbers. The statement ¬B is: There are only finitely many prime numbers. The knowledge base consists of the rules of arithmetic for integers.
On this basis of ¬B and with the knowledge of W one shows then that one can always find a new prime number to every set of finitely many prime numbers, thus ¬B is false, in contradiction to the assumption that W ∧ ¬B is true.
This article was very helpful just now. Thank you for the thorough analysis and clear explanations.