52. Primative Propositions

Harold R. (Hal) Foster’s Prince Valiant

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PRIMITIVE PROPOSITIONS. *1^.1. Anything implied by a true elementary proposition is true. Pp*¹.

The above principle will be extended in _*9 to propositions which are not elementary. It is not the same as "if p is true, then if p implies q, q is true.”

This is a true proposition, but it holds equally when p is not true and when p does not imply q. It does not, like the principle we are concerned with, enable us to assert q simply, without any hypothesis. We cannot express the principle symbolically, partly because any symbolism in which p is variable only gives the hypothesis that p is true, not the fact that it is true*². The above principle is used whenever we have to deduce a proposition from a proposition. But the immense majority of the assertions in the present work are assertions of propositional functions, i.e. they contain an undetermined variable. Since the assertion of a propositional function is a different primitive idea from the assertion of a proposition, we require a primitive proposition different from *1^.1, though allied to it, to enable us to deduce the assertion of a propositional function "ψx" from the assertions of the two propositional functions "φx" and " φx ⊃ ψx." This primitive proposition is as follows:

*1^.11. When φx can be asserted, where x is a real variable, and φx ⊃ ψx can be asserted, where x is a real variable, then ψx can be asserted, where x is a real variable.  Pp.

This principle is also to be assumed for functions of several variables.

Part of the importance of the above primitive proposition is due to the fact that it expresses in the symbolism a result following from the theory of types, which requires symbolic recognition. Suppose we have the two assertions of propositional functions "┣ . φx " and "┣ . φx ⊃ ψx "; then the "x" in φx is not absolutely anything, but anything for which as argument the function "φx" is significant; similarly in "φx ⊃ ψx" the x is anything for which "φx ⊃ ψx" is significant. Apart from some axiom, we do not know that the x's for which "φx ⊃ ψx " is significant are the same as those for which "φx" is significant. The primitive proposition *1^.11, by securing that, as the result of the assertions of the propositional functions "φx" and " φx ⊃ ψx," the propositional function “ψx” can also be asserted, secures partial symbolic recognition, in the form most useful in actual deductions, of an important principle which follows from the theory of types, namely that, if there is any one argument ⍺ for which both “φx and “ψx” are significant, then the range of arguments for which "φx" is significant is the same as the range of arguments for which "ψx" is significant. It is obvious that, if the propositional function "φx ⊃ ψx" can be asserted, there must be arguments a for which "φ⍺ ⊃ ψ⍺ " is significant, and for which, therefore, "φ⍺" and "ψ⍺" must be significant. Hence, by our principle, the values of x for which “φx” is significant are the same as those for which "ψx" is significant, i.e. the type of possible arguments for φx (cf. post 7) is the same as that of possible arguments for ψx. The primitive proposition *1^.11, since it states a practically important consequence of this fact, is called the "axiom of identification of type."

Another consequence of the principle that, if there is an argument ⍺ for which both φ⍺ and ψ⍺ are significant, then φx is significant whenever ψx is significant, and vice versa, will be given in the "axiom of identification of real variables," introduced in _*1^.72. These two propositions, _*1^.11 and _*1^.72, give what is symbolically essential to the conduct of demonstrations in accordance with the theory of types.

The above proposition _*1^.11 is used in every inference from one asserted propositional function to another. We will illustrate the use of this proposition by setting forth at length the way in which it is first used, in the proof of _*2^.06. That proposition is

"┣ :. p  ⊃ q . ⊃ : q ⊃ r . ⊃ . p ⊃ r."

We have already proved, in _*2^.05, the proposition

┣ :. q ⊃ r . ⊃ : p ⊃ q . ⊃ . p ⊃ r.

It is obvious that _*2^.06 results from _*2^.05 by means of _*2^.04, which is

┣ :. p . ⊃ . q ⊃ r : ⊃ : q . p ⊃ r.

For if, in this proposition, we replace p by q ⊃ r, q by p ⊃ q, and r by p ⊃ r, we obtain, as an instance of _*2^.04, the proposition

┣ :: q ⊃ r . ⊃ : p ⊃ q . ⊃ . p ⊃ r :. ⊃ :. p ⊃ q . ⊃ : q ⊃ r . ⊃. p ⊃ r     (1),

and here the hypothesis is asserted by _*2^.05. Thus our primitive proposition _*1^.11 enables us to assert the conclusion.

_*1^.2.  ┣ : p v p . ⊃ . p  Pp.

This proposition states: "If either p is true or p is true, then p is true." It is called the "principle of tautology," and will be quoted by the abbreviated title of "Taut." It is convenient, for purposes of reference, to give names to a few of the more important propositions; in general, propositions will be referred to by their numbers.

_*1^.3. ┣ : q . ⊃ . p v q   Pp.

This principle states: "If q is true, then 'p or q' is true." Thus e.g. if q is "today is Wednesday" and p is "today is Tuesday," the principle states: "If today is Wednesday, then today is either Tuesday or Wednesday." It is called the "principle of addition," because it states that if a proposition is true, any alternative may be added without making it false. The principle will be referred to as "Add."

_*1^.4. ┣ : p v q . ⊃ . q v p  Pp.

This principle states that "p or q" implies "q or p." It states the permutative law for logical addition of propositions, and will be called the "principle of permutation." It will be referred to as "Perm."

_*1^.5. ┣ : p v (q v r) . ⊃ q v (p v r)  Pp.

This principle states: "If either p is true, or 'q or r' is true, then either q is true, or 'p or r' is true." It is a form of the associative law for logical addition, and will be called the "associative principle." It will be referred to as "Assoc." The proposition

p v (q v r) . ⊃ . (p v q) v r,

which would be the natural form for the associative law, has less deductive power, and is therefore not taken as a primitive proposition.

_*1^.6. ┣ :. q ⊃ r . ⊃ : p v q . ⊃ . p v r  Pp.

This principle states: "If q implies r, then 'p or q' implies 'p or r.'" In other words, in an implication, an alternative may be added to both premises and conclusion without impairing the truth of the implication. The principle will be called the "principle of summation," and will be referred to as "Sum."

_*1^.7. If p is an elementary proposition, ~p is an elementary proposition.  Pp.

_*1^.71. If p and q are elementary propositions, p v q is an elementary proposition.  Pp.

_*1^.72. If φp and ψp are elementary propositional functions which take elementary propositions as arguments, φp v ψp is an elementary propositional function.  Pp.

This axiom is to apply also to functions of two or more variables. It is called the "axiom of identification of real variables." It will be observed that if φ and ψ are functions which take arguments of different types, there is no such function as "φx v ψx," because φ and ψ cannot significantly have the same argument. A more general form of the above axiom will be given in _*9.

The use of the above axioms will generally be tacit. It is only through them and the axioms of _*9 that the theory of types explained in the Introduction becomes relevant, and any view of logic which justifies these axioms justifies such subsequent reasoning as employs the theory of types.

This completes the list of primitive propositions required for the theory of deduction as applied to elementary propositions.

*¹The letters "Pp" stand for " primitive proposition," as with Peano.

*² For further remarks on this principle, cf. Principles of Mathematics, ~ 38.

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Here primitive propositions are examined closely. It is helpful to learn their constitution and why no formal logic can go forward without them. Also the benefit to our system is that they are few in number; and this limit helps us to avoid errors in using it.

The numbered designations explain the mechanical process of each primitive proposition; and how they may be used as blocks to build our equations and logical expressions. In our use of logic, either philosophical or mathematical, I feel it is most useful in fact to not-use it. As we find any situation where logic presents an unusual result, in a written work, a technical discussion, or even a social one; rather than thinking and speaking in technical terms it is always more effective and pleasant to internalize our logic in more human terms.

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Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.