Frege (§12) Begriffsschrift, Chapter 1 (Geach transl.), [on combinations of structures], summary

by Corry Shores
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[The following is summary. Bracketed commentary is my own. Please forgive my typos, as proofreading is incomplete.]

Summary of

 

Gottlob Frege

 

Begriffsschrift, Chapter 1

(Geach transl.)

 

§12

[on combinations of structures]

 

 

 

Brief Summary: 

In Frege’s notation system, we write:

 

¬(∀z)Xz

“Not everything has the property X”

 

 

(∀z)¬Xz

“there is not something with the property X”

 

 

¬(∀z)¬Λz

“There are Λs”

 

 

(∀z)(Xz→Pz)

“If something has the property X, then it has also the property P”

“every X is a P”

“all Xs are Ps”

 

 

(∀z)(Xz→¬Pz)

“what has the property Ψ has not the property P”

“no Ψ is a P”

 

 

¬(∀z)(Λz→Pz)

“some Λs are not Ps”

 

 

¬(∀z)(Mz→¬Pz)

“some Ms are Ps”

“it is possible for an M to be a P”

 

 

 

 

 

Summary

 

 

Frege will now explain his notation system for certain combinations of symbols. [Suppose we wanted to write

¬(∀x)Mx

Or, “not everything is movable.” See Agler, Symbolic Logic, section 6.5. This means that the predicate M does not hold for all things.] If we want to notate “find something, say Δ such that X(Δ) is denied. We may thus render it as: “there are some things that have not the property of X” (Frege 19). We would write that as

(Frege 19)

 

[Now suppose instead that we wanted to write

(∀x)¬Mx

Or, “Nothing is movable.” Again see Agler, Symbolic Logic, section 6.5. This means that the predicate M holds for absolutely none of the objects in the domain.] Suppose instead we wanted to notate “ ‘Whatever a may be, X(a) must always be denied,’ or ‘there is not something with the property X,’ or (calling something that has the property X, a X) ‘there is no X’ ” (Frege 19). We would write it as:

 

[The next idea might be something like the following. Suppose we have ‘for all x, x is not P.’ This means that none of the x’s are P. Now suppose we have ‘it is not the case that for all x, x is not P.” This would mean that we cannot say that none of the x’s are not P. It also would be making no claim whether or not all of them are either. In other words, it would seem to mean that some x’s are P. That might be something like what Frege does next.]

is denied by

This may be rendered as ‘there are Λ’s.

(Frege 19)

 

[For the next part, recall first how Agler (Symbolic Logic section 6.5.2) made translations for a formation that might now with respect to this new context be something like ‘all S’s are P’s.’ He gave this sort of a bridging translation:

(3) (∀x)(Zx→Hx)

(3_B) For every x, if x is a zombie, then x is happy.

(3_B*) Choose any object you please in the domain of discourse; if that object is a zombie, then it will be also be happy.

(3_E) Every zombie is happy.

(Agler 271, 272)

So suppose we want to say that all X’s are P’s. We would then say that if  z is an X then z is a P. Given the conditional structure, this means that it cannot be that z is an X and z is not a P. Here are the other possibilities, however:

z is an X and z is a P

z is not an X and z is a P

z is not an X and z is not a P.

All of these situations conform to the notion that all X’s are P’s. As we will see, Frege will render this structure using the conditional notation.]

means: ‘whatever may be substituted for a, the case in which P(a) would have to be denied and X(a) affirmed does not occur.’ It is thus possible that, for some possible meanings of a,

P(a) must be affirmed and X(a) affirmed; for others,

P(a) must be affirmed and X(a) denied; for others again,

P(a) must be denied and X(a) denied.

| We can thus give the rendering: ‘If something has the property X, then it has also the property P,’ or ‘every X is a P,’ or ‘all X’s are P’s.’

(Frege 19-20)

 

[The next structure Frege says is for causal connexions. I do not know the terminology well enough to understand why this has something to do with causality, as I cannot discern any such meaning. For this formation, recall (again from Agler Symbolic Logic section 6.5.2):

(4) (∀x)(Zx→¬Hx)

(4_B) For every x, if x is a zombie, then x is not happy.

(4_B*) Choose any object you please in the domain of discourse consisting of human beings (living or dead); if that object is a zombie, then it will not be happy.

(4_E) No zombies are happy.

(Agler 271, 272)

Frege will show now how to make this structure.]

This is the way causal connexions are expressed.

means: ‘no meaning can be given to a such that P(a) and Ψ(a) p. 24] could both be affirmed.’ We may thus render it as ‘what has the property Ψ has not the property P’ or ‘no Ψ is a P.’

(Frege 20)

[Now, if we wanted to say that all S’s are P’s, then we would say that for all x, if x is S, then x is P. And if we wanted to say that no S’s are P’s, then we would say, for all x, if x is S then x is not P. But if we want to say that some S’s are not P’s, then we would say that it is not the case that for all x, if x is S then x is P. For, if not all are something (and not none are something), then some are not that something. I think.]

denies

and may be therefore rendered as ‘some Λs are not Ps’

(Frege 20)

 

[Suppose we said, ‘all M’s are not P’s’. This means there is no M which is a P. Now suppose we said, “it is not the case that all M’s are not P’s’. That would mean that some M’s are P’s.]

denies that no M is a P and thus means ‘some Ms are Ps’ or ‘it is possible for an M to be a P.’

(Frege 20)

 

 

 

 

From:

Frege, Gottlob. “Begriffsschrift (Chapter 1)”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).

 

 

Or if otherwise noted:

Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.

 

 

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Frege (§11) Begriffsschrift, Chapter 1 (Geach transl.), “Generality", summary

by Corry Shores
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Summary of

 

Gottlob Frege

 

Begriffsschrift, Chapter 1

(Geach transl.)

 

§11 Generality

 

 

 

Brief Summary: 

In Frege’s notation system, we denote universal quantification by making an indent into the judgment stroke, above which goes the universally quantified variable, and that variable is also placed in parentheses next to the predicate symbol.

The scope is determined by which stroke (and which part of the stroke) the indent is located on. For example:

 

 

Summary

 

 

[Recall from section 10 that we can formulate a function using a judgment stroke. For example, we can express an indeterminate function of argument A, that is, Φ(A) with the judgment stroke as

which is read, “A has the property Φ”. An indeterminate function for two arguments may be written Ψ(A,B) and with the judgment stroke as

I am not entirely certain, but Frege’s point now seems to be that in these prior cases, A and B were determinate arguments. In other words, they seem to be like constants that are being predicated. We can instead think of them as being like algebraic variables. Since he says that the notation goes back to what we see above when we substitute something for the letter, that makes me think it is something like a variable. He will give a separate notation for this other situation. The Gothic letter seems like it is the predicate’s variable, and the concave indent into the judgment stroke above which the variable is written again seems to indicate that it is universal quantification. Many details are unclear to me, so let me quote for now:]

In the expression for a judgment, the complex symbol to the right of ⊢ may always be regarded as a function of one of the symbols that occur in it. Let us replace this argument with a Gothic letter, and insert a concavity in the content-stroke, and make this same Gothic letter stand over the concavity: e.g.:

This signifies the judgment that the function is a fact whatever we take its argument to be. A letter used as a functional symbol, like Φ in Φ(A), may itself be regarded as the argument of a function; accordingly, it may be replaced by a Gothic letter, used in the sense I have just specified. The only restrictions imposed on the meaning of a Gothic letter are the obvious ones: (i) that the complex of symbols following a content-stroke must still remain a possible content of judgment (§2); (ii) that if the Gothic letter occurs as a functional symbol, account must be taken of this circumstance. All further conditions imposed upon the allowable substitutions for a Gothic letter must be made part of the judgment. From such a judgment, therefore, we can always deduce any number we like of judgments with less general content, by substituting something different each time for the Gothic letter; when this is done, the concavity in the content-stroke vanishes again. The horizontal stroke that occurs to the left of the concavity in

is the content-stroke for [the proposition] that Φ(ɑ) holds good whatever is substituted for ɑ; the stroke occurring to the right of p. 2o] the concavity is the content-stroke of Φ(ɑ) – we must here imagine something definite substituted for ɑ.

(Frege 16)

[Frege’s next point will be about how we can understand the meaning of these formulations when we remove the judgment stroke. See section 2. We would just have the content of the judgment, but not the claim that it is the case, I think.]

By what was said before about the meaning of the judgment-stroke, it is easy to see what an expression like

| means.

(16-17)

[For the next point, let us recall from section 5 how the conditionals are structured. When we have

This would be read, “If B, then A.” Let me quote the following, then discuss it.]

This expression may occur as part of a judgment, as in

(17)

[Here I think he is simply saying that the universally quantified expressions can be found in more complex structures.]

[I do not quite get the next point. He might be saying the following. Suppose we say that for all x, x is P. Now let us negate that. It is not the case that for all x, x is P. This can still mean that there some substitution, like Pa, that is true. The point is that not all the substitutions are true. So begin again with the simple case of: for all x, x is P. Here, we can infer that any possible substitution will be true. However, from ‘it is not the case that for all x, x is P’, we cannot infer anything, because we do not know which if any of the constants is P. This idea also holds in the conditional structure. Suppose we have ‘if for all x, x is P, then A.’ From this we likewise cannot infer that ‘if Pa, then A’. The reasoning for this instance I am less certain about. One possibility is that the whole conditional can be true if the antecedent is false. So supposing that the antecedent is false in our example, we cannot infer that ‘if Pa, then A’ because perhaps Pa makes the antecedent true while A is false, and thus the whole conditional is false.

It is obvious that from these judgments we cannot infer less general judgments by substituting something definite for ɑ, as we can from

[seems to end the sentence here.]

serves to deny that Χ(ɑ) is always a fact whatever we substitute for ɑ. But this does not in any way deny the possibility of giving ɑ some meaning Δ such that X(Δ) is a fact.

means the case in which

is affirmed and A denied does not occur. But this does not in any way deny the occurrence of the case in which X(Δ) is affirmed and A denied; for, as we have just seen, X(Δ) may be affirmed and nevertheless

denied. Thus here likewise, we cannot make an arbitrary substitution for a without prejudice to the truth of the judgment.

(17)

[Frege will say that this is why we need to indicate the scope of the quantifier. I am not sure how that would change the prior cases. Before we gave the example, ‘if for all x, x is P, then A.’ Perhaps he is saying that the situation would be different if we instead said, ‘for all x, if x is P, then A.’ I am not sure. But maybe in this case we can derive X(Δ). I do not see how, but I also do not yet understand how the example we just saw demonstrates the need for indicating scope. At any rate, he further shows how quantifier scope works in his system.]

This explains why we need the concavity with the Gothic letter written on it; it delimits the scope of the generality signified by the letter. A Gothic letter retains a fixed meaning only within its scope; the same Gothic letter may occur within various scopes in the same judgment, and the meaning we may ascribe to it in one scope does not extend to any other scope. The scope of one Gothic letter may include that of another, as is shown in p. 21]

In this case different letters must be chosen; we could not replace e by a. It is naturally legitimate to replace a Gothic letter everywhere in its scope by some other definite letter, provided that there are still different letters standing where different letters | stood before. This has no effect on the content. Other substitutions are permissible only if the concavity directly follows the judgment stroke, so that the scope of the Gothic letter is constituted by the content of the whole judgment. Since this is a specially important case, I shall introduce the following abbreviation: an italic letter is always to have as its scope the content of the whole judgment, and thus scope is not marked out by a concavity in the content stroke. If an italic letter occurs in an expression not preceded by a judgment stroke, the expression is senseless.

(Frege 18)

 

[I may not be following the ideas of italic and Gothic letters very well. In the German version the differences might be a little more apparent. Here are some examples.

(Begriffsschrift und andere Aufsätze p.21)

So we have lower case italic (Roman) letters, lowercase Gothic, and uppercase Greek. There is also the possibility that there are uppercase Roman letters, like A, B, and P, but I am inclined to think they are uppercase Greek, as I do not know why predicates would be distinguished. My guess is that predicates take uppercase Greek letters; predicate constants take lowercase Roman letters; and predicate variables take Gothic letters. It seems that what he says next is something like universal introduction. See Agler Symbolic Logic section 8.1.3.]

An italic letter may always be replaced by a Gothic letter that does not yet occur in the judgment; in this case the concavity must be inserted immediately after the judgment-stroke. E.g. for

we may put

since a occurs only in the argument-position within X(a).

Likewise it is obvious that from

 we may deduce

 if A is an expression in which a does not occur, and a occupies only argument-positions in Φ(a). If

is denied, we must be able to specify a meaning for a such that Φ(a) is denied. Thus if

were denied and A affirmed, we should have to be able to specify a meaning for a such that A was affirmed and Φ(a) denied.
(18)

[I may be missing his next point. It might be that since we have a conditional, our substitution cannot make it that the antecedent is affirmed and the consequent is denied. Let me quote.]

But since we have

we cannot do so; for this formula means that whatever a may be the case in which Φ(a) would be denied and A affirmed does not | occur. Hence we likewise cannot both deny

and affirm A: i.e.

... Similarly when we have several conditional strokes.

(Frege 18-19)

 

 

 

Frege, Gottlob. “Begriffsschrift (Chapter 1)”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).

 

An image comes from:

Frege, Gottlob. Begriffsschrift und andere Aufsätze. 2nd edn.  Hildesheim/Zürich: Geor Olms, 1993.

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Nolt (11.1) Logics, ‘Modal Operators’, summary

 

by Corry Shores

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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]

 

 

 

Summary of

 

John Nolt

 

Logics

 

Part 4: Extensions of Classical Logic

 

Chapter 11: Leibnizian Modal Logic

 

11.1 Modal Operators

 

 

Brief summary:

There are certain operators called alethic modifiers. They include deontic (or ethical) modalities (e.g. ‘it ought to be the case that’, ‘it is forbidden that’); propositional attitudes (‘believes that’, ‘knows that’, ‘hopes that’, ‘wonders whether’); and tenses (‘was’, ‘is’ , and ‘will be’). Propositional attitude modifiers are binary, but all the rest are monadic. ‘It is necessary that Φ’ is written  □Φ and ‘it is possible that Φ’ is written ◊Φ. Alethic modifiers are often duals meaning that one can be converted into another by adding negations around the symbols, as for example:

□Φ↔~◊~Φ

◊Φ↔~□~Φ

 

 

 

Summary

 

There are operators that are inexpressible in predicate logic but nonetheless are still very important called alethic modifiers, which includes ‘must,’ ‘might,’ ‘could,’ ‘can,’ ‘have to’, ‘possibly,’ ‘contingently,’ ‘necessarily’ (Nolt 307). These words express modes of truth or alethic modalities. (The Greek word for truth is alethea, hence their name alethic modifiers). Modal logic studies the syntax and the semantics of alethic modalities (Nolt 307).

 

Modal logic also studies other kinds of propositional modalities, namely, {1} deontic (or ethical) modalities, which are “expressed by such constructions as ‘it ought to be the case that’, ‘it is forbidden that’, etc.” (Nolt 307); {2} propositional attitudes, which are “relations between sentient beings and propositions, expressed by such terms as ‘believes that’, ‘knows that’, ‘hopes that’, ‘wonders whether’, and so on”; and {3} tenses, which include past, present, and future tenses as expressed by the various modifications of the verb ‘to be’: ‘was’, ‘is’ , and ‘will be’ ” (Nolt 307).

 

All these alethic modalities can be understood as operators on propositions (Nolt 307). [We will see how the alethic modalities can be placed upon a proposition just like other logical operators can. The following is quotation.]

 

Consider, for | example, these sentences, all of which involve the application of modal operators (in the broad sense) to the single proposition ‘People communicate’:

 

Alethic Operators

It is possible that people communicate.

It must be the case that people communicate.

It is contingently the case that people communicate.

It could be the case that people communicate.

It is necessarily the case that people communicate.

 

Deontic Operators

It is obligatory that people communicate.

It is permissible that people communicate.

It is not allowed that people communicate.

It should be the case that people communicate.

 

Operators Expressing Propositional Attitudes

Ann knows that people communicate.

Bill believes that people communicate.

Cynthia fears that people communicate.

Don supposes that people communicate.

Everyone understands that people communicate.

Fred doubts that people communicate.

 

Operators Expressing Tenses

It was (at some time) the case that people communicated.

It was always the case that people communicated.

It will (at some time) be the case that people communicate.

(Nolt 307-308)

 

This is not even an exhaustive list of the operators in each category.  Nolt also notes that except for the propositional attitude operators, the rest here are monadic.

With the exception of the operators expressing propositional attitudes, all of those listed here are monadic; they function syntactically just like the negation operator ‘it is not the case that’, prefixing a sentence to produce a new sentence. Thus, for example, the operators ‘it is necessary that’, usually symbolized by the box ‘□’ and ‘it is possible that’, usually | symbolized by the diamond sign ‘◊’, are introduced by adding this clause to the formation rules:

If Φ is a formula, then so are □Φ and ◊Φ.

(Nolt 308-309)

 

The operators for propositional attitudes are binary operators. However, he continues, “unlike such binary operators as conjunction or disjunction, which unite a pair of sentences into a compound sentence, propositional attitude operators take a name and a sentence to make a sentence. The place for this name may be quantified, as in ‘Everyone understands that people communicate’ ” (Nolt 309).

 

Many operators can be converted into one another through negations that flank the signs.

Many modal operators have duals – operators which, when flanked by negation signs, form their equivalents. The operators ‘□’ and ‘◊’, for example, are duals, as the following sentences assert:

□Φ↔~◊~Φ

◊Φ↔~□~Φ

That is, it is necessary that Φ if and only if it is not possible that not-Φ, and it is possible that Φ if and only if it is not necessary that not-Φ.

There are other duals among these operators as well. Consider the deontic operator ‘it is obligatory that’, which we shall symbolize as ‘O’, and the operator ‘it is permissible that’, which we shall write as ‘P’. These are similarly related:

OΦ↔~P~Φ

PΦ↔~O~Φ

That ‘O’ and ‘P’ should thus mimic ‘□’ and ‘◊’ is understandable, since obligation is a kind of moral necessity and permission a kind of moral possibility.

There are also epistemic (knowledge-related) duals. The operator ‘knows that’ is dual with the operator ‘it is epistemically possible, for ... that’ – the former representing epistemic necessity (knowledge) and the latter epistemic possibility. (Something is epistemically possible for a person if so far as that person knows it might be the case.) Symbolizing ‘knows that’ by ‘K’ and ‘it is epistemically possible for ... that’ by ‘E’, we have:

pKΦ↔~pE~Φ

pEΦ↔~pK~Φ

In English: p knows that Φ if and only if it is not epistemically possible for p that not-Φ; and it is epistemically possible for p that Φ if and only if p does not know that not-Φ (‘p’, of course, stands for a person).

There are temporal duals as well. Let ‘P’ mean ‘it was (at some time) the case that’ and ‘H’ mean ‘it has always been the case that’. Then:

HΦ↔~P~Φ

PΦ↔~H~Φ

| Here ‘H’ represents a kind of past tense temporal necessity and ‘P’ a kind of past tense temporal possibility. A similar relationship holds between ‘it always will be the case that’ and ‘it sometimes will be the case that  and between other pairs of temporal operators.

(Nolt 309-310)

 

Nolt then notices how these duals remind us of two laws in predicate logic

∀Φ↔~∃~Φ

∃Φ↔~∀~Φ

(Nolt 310)

And he wonders if the duals are analogous to quantifiers (Nolt 310).

 

 

  

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

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Nolt, Logics, entry directory

by Corry Shores
 

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[Central Entry Directory]
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[Note, starting with section 16.2, the paragraphs are enumerated, and the larger sections are divided by the numbered paragraphs that are summarized in that post.]

An entry directory like this but with the brief summaries included can be found at:
http://piratesandrevolutionaries.blogspot.com/2018/03/nolt-cbs-logics-collected-brief.html

Entry Directory for

John Nolt

Logics

Part 3: Classical Predicate Logic

Chapter 6: Classical Predicate Logic: Syntax

6.3 Identity

Chapter 8: Classical Predicate Logic: Inference

8.5 Identity

Part 4: Extensions of Classical Logic

Chapter 11: Leibnizian Modal Logic

11.1 Modal Operators

 

11.2.1 [basic set-up and evaluation of Leibnizian possible worlds]

11.2.2 [Possible worlds, actual worlds, redefining logical concepts for possible worlds, and some important theorems in possible world semantics]

11.3 A Natural Model?

11.4 Inference in Leibnizian Logic

Chapter 12: Kripkean Modal Logic

12.1 Kripkean Semantics

Chapter 13: Deontic and Tense Logics

13.2 A Modal Tense Logic 

13.2.1 [Basic set-up and evaluation of modal tense logic]

Chapter 14: Higher-Order Logics

 

14.1 Higher-Order Logics: Syntax

  
  

 

Part 5: Nonclassical Logics

Chapter 15: Mildly Nonclassical Logics

15.1 Free Logics

15.2 Multivalued Logics

 

15.3 Supervaluations

15.3.1 [Basic technique of supervaluation]

Chapter 16: Radically Nonclassical Logics

16.1 Infinite Valued and Fuzzy Logics

16.2 Intuitionistic Logics

16.2.1–16.2.29 [Basic set-up of Kripkean intuitionistic semantics]

16.3 Relevance Logics

16.3.1-16.3.21 [basic set-up of dialethic, relevant logic semantics]

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

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John Nolt, entry directory

by Corry Shores
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Entry Directory for

John Nolt

 
[Thanks works.bepress.com]

 

 

Logics

 

Nolt, Logics, entry directory

 

 

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Agler (8.3) Symbolic Logic: Syntax, Semantics, and Proof, 'Sample Proofs', summary

 

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[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. Please forgive my typos, as proofreading is incomplete. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]

 

Summary of

 

David W. Agler

 

Symbolic Logic: Syntax, Semantics, and Proof

 

Ch.8: Predicate Logic Derivations

 

8.3 Sample Proofs

 

Brief summary:

Agler illustrates the rules of the derivation system for making proofs in the language of predicate logic with a set of examples.

 

 

Summary

 

Agler will shows us how to use the quantifier rules for making proofs by working through a series of examples.

 

The first one is for:

(∀x)(Ax→Bx), (∃x)¬Bx ⊢ (∃x)¬Ax

(Agler 350)

[We can see that we will probably want to use modus tollens to derive the negation of the antecedent in the first premise. We cannot do this directly. So we will use an existential elimination assumption for premise 2. Then we can use universal elimination to give us the conditional in a form that we can work with. Then we can get a negated antecedent with a constant. This will allow us to use existential introduction to get the goal proposition, which we will move into the main proof using existential elimination.]

 

1	(∀x)(Ax→Bx)	P
2	(∃x)￢Bx	P
3	| ￢Ba	A/∃E
4	| Aa→Ba	1∀E
5	| ￢Aa	3,4MT
6	| (∃x)￢Ax	5∃I
7	(∃x)￢Ax	2,3–6∃E

(Agler 350)

 

Agler then shows another proof for

(∀x)(∀y)(Pxy∧Qxy), (∀z)Pzz→Sb ⊢ (∃x)Sx

(Agler 350)

[For this, we  see that we could use existential introduction on the consequent in premise 2 to get the goal proposition. For that we should obtain the antecedent. We could get that by using universal introduction on a formula of the form Paa. We could obtain such a formula but using universal elimination on the conjunction in the first premise and then using conjunction elimination to extract the first conjunct.]

 

1	(∀x)(∀y)(Pxy∧Qxy)	P
2	(∀z)Pzz→Sb	P/(∃x)Sx
3	(∀y)(Pay∧Qay)	1∀E
4	Paa∧Qaa	3∀E
5	Paa	4∧E
6	(∀z)Pzz	5∀I
7	Sb	2,6→E
8	(∃x)Sx	7∃I

(Agler 350)

 

Agler then has us consider one final example, a theorem.

⊢ (∀x)[¬(Qx→Rx)→¬Px]→(∀x)[¬(Px→Rx)→¬(Px→Qx)]

(Agler 350)

 

[Since we have a conditional, the strategy will be to assume the antecedent and derive the consequent. So first we make that assumption. Next what we should do is build the main consequent of the conclusion first with the same constants and then use universal introduction to get the form we need. So we will want ¬(Pa→Ra)→¬(Pa→Qa). Again we will assume this antecedent and derive the consequent. At this point, in order to derive ¬(Pa→Qa), we will assume the unnegated form and seek a contradiction, and then use negation introduction. Agler will do this through a series of steps that do not come intuitively to me so I will leave the reasoning to be seen in the proof.]

1	| (∀x)[￢(Qx→Rx)→￢Px]    |    |    |	A/(∀x)[￢(Px→Rx)→￢(Px→Qx)]
2	|   | ￢(Pa→Ra)    |   |	A/￢(Pa→Qa)
3	|   |   | Pa→Qa	A/P∧￢P
4	|   |   | ￢(Qa→Ra)→￢Pa	1∀E
5	|   |   | ￢(￢Pa∨Ra)	2IMP
6	|   |   | Pa∧￢Ra	5DEM+DN
7	|   |   | Pa	6∧E
8	|   |   | Qa	3,7→E
9	|   |   | ￢￢Pa	7DN
10	|   |   | Qa→Ra	4,9MT+DN
11	|   |   | Ra	8,10→E
12	|   |   | ￢Ra	6∧E
13	|   | ￢(Pa→Qa)	3–12￢I
14	| ￢(Pa→Ra)→￢(Pa→Qa)	2–13→I
15	| (∀x)[￢(Px→Rx)→￢(Px→Qx)]	14∀I
16	(∀x)[￢(Qx→Rx)→￢Px]→(∀x)[￢(Px→Rx)→￢(Px→Qx)]	1–15→I

(Agler 351)

 

 

 

Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.

 

 

Some changes to the book quotations may have been made, as based on: http://markdfisher.com/wp-content/uploads/2014/02/PHIL_012_ONLINE_SYLLABUS_SP14-3-1.pdf .

 

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Agler (8.2) Symbolic Logic: Syntax, Semantics, and Proof, 'Quantifier Negation (QN)', summary

 

by Corry Shores

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[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. Please forgive my typos, as proofreading is incomplete. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]

 

Summary of

 

David W. Agler

 

Symbolic Logic: Syntax, Semantics, and Proof

 

Ch.8: Predicate Logic Derivations

 

8.2 Quantifier Negation (QN)

 

Brief summary:

The four underived quantifier rules for making proofs in the language of predicate logic (RL) made a derivation system called RD. To this we add the derived equivalence rule quantifier negation (QN) to make the deduction system RD+. And since it is an equivalence rule, QN can apply to quantifiers that are not main operators.

Quantifier Negation (QN) From a negated universally quantified expression ‘¬(∀x)P,’ an existentially quantified expression ‘(∃x)¬P’ can be derived, and vice versa. Also, from a negated existentially quantified expression ‘¬(∃x)P,’ a universally quantified expression ‘(∀x)¬P’ can be inferred, and vice versa.

¬(∀x)P ⊣ ⊢ (∃x)¬P ¬(∃x)P ⊣ ⊢ (∀x)¬P

QN QN

  (Agler 348)

 

 

 

 

Summary

 

In the prior section 8.1, we devised a system of deduction for making proofs in the language of predicate logic (RL). That system is called RD. We will now add an equivalence rule, namely, quantifier negation (QN) so that we can complete our proofs more efficiently, and all the rules together with this new addition make RD+. The new rule will help us deal with negated quantifiers.

Quantifier Negation (QN) From a negated universally quantified expression ‘¬(∀x)P,’ an existentially quantified expression ‘(∃x)¬P’ can be derived, and vice versa. Also, from a negated existentially quantified expression ‘¬(∃x)P,’ a universally quantified expression ‘(∀x)¬P’ can be inferred, and vice versa.

¬(∀x)P ⊣ ⊢ (∃x)¬P ¬(∃x)P ⊣ ⊢ (∀x)¬P

QN QN

  (Agler 348)

 

 

Agler gives this simple illustration.

 

1	¬(∀x)Px	P
2	(∃x)¬Px	1QN
3	¬(∀x)Px	2QN

(Agler 348)

 

He gives another example that works the same way, but now with a conjunction.

 

1	¬(∃z)(Wzz∧Mz)	P
2	(∀z)¬(Wzz∧Mz)	1QN
3	¬(∃z)(Wzz∧Mz)	2QN

(Agler 349)

 

Agler then shows how this basic procedure works when we have double negations.

 

1	￢(∀z)￢(Wzz→￢Mz)	P
2	(∃x)(Px∧Rx)	P
3	(∃z)￢￢(Wzz→￢Mz)	1QN
4	(∃z)(Wzz→￢Mz)	3DN
5	￢￢(∃x)(Px∧Rx)	2DN
6	￢(∀z)￢(Wzz→￢Mz)	3QN

(Agler 349)

 

Something very important to note is that “Since quantifier negation is an equivalence rule, it can be applied to subformulas within wffs and not merely to whole propositions whose main operator is the negation” (Agler 349). Agler shows this in the following proof.

 

1	￢(∀z)￢(∃y)[Wzy→￢(∀x)￢My]	P
2	￢(∀z)￢(∃y)[Wzy→(∃x)￢￢My]	1QN
3	(∃z)￢￢(∃y)[Wzy→(∃x)￢￢My]	2QN
4	(∃z)(∃y)[Wzy→(∃x)￢￢My]	3DN
5	(∃z)(∃y)[Wzy→(∃x)My]	4DN

(Agler 349)

 

As we can see, in line 2 we used quantifier negation on a quantifier that is not the main operator.

 

Agler notes that QN is a derived rule, which means it can be proved using the other underived quantifier quantifiers rules. This means that we need to make proofs that show:

¬(∀x)P⊣ ⊢ (∃x)¬P
¬(∃x)P⊣ ⊢ (∀x)¬P

Agler will make a proof for just ¬(∀x)P⊢(∃x)¬P, and he leaves the remaining three cases for exercises.

 

1	￢(∀x)Px	P/(∃x)￢Px
2	| ￢(∃x)￢Px	A/contra
3	|    | ￢Pa	A/contra
4	|    | (∃x)￢Px	3∃I
5	|    | ￢(∃x)￢Px	2R
6	| Pa	3–5￢E
7	| (∀x)Px	6∀I
8	| ￢(∀x)Px	1R
9	(∃x)￢Px	2–8￢E

(Agler 349)

 

 

 

 

Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.

 

 

Some changes to the book quotations may have been made, as based on: http://markdfisher.com/wp-content/uploads/2014/02/PHIL_012_ONLINE_SYLLABUS_SP14-3-1.pdf .

 

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