28 Mar 2017

Kaufmann. Introduction to the Theory of Fuzzy Subsets, entry directory

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, Entry Directory]

 

 

 

Entry Directory for

 

Arnold Kaufmann

 

Introduction à la théorie des sous-ensembles flous

à l’usage des ingénieurs

(Fuzzy sets theory)

1. Eléments théoriques de base

/

Introduction to the Theory of Fuzzy Subsets.

Vol.1 Fundamental Theoretical Elements

Ch.1

Notions de base

Fundamental Notions

 

1.1

Introduction

 

1.2

Rappel sur la notion d’appartenance

Review of the notion of Membership

 

1.3

Le concept de sous-ensemble flou

The Concept of a Fuzzy Subset

 

 

 

 

 

 

 

 

 

 

Kaufmann, Arnold. 1975 [1973]. Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs (Fuzzy sets theory). 1: Eléments théoriques de base. Foreword by L.A. Zadeh. 2nd Edn. Paris: Masson.

 

Kaufmann, Arnold. 1975. Introduction to the Theory of Fuzzy Subsets. Vol.1: Fundamental Theoretical Elements. Foreword by L.A. Zadeh. English translation by D.L. Swamson. New York / San Francisco / London: Academic Press.

 

 

.

Arnold Kaufmann, entry directory

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, Entry Directory]

 

 

 

Entry Directory for

 

Arnold Kaufmann

 

kaufmann crop

(Image source)

 

 

Introduction à la théorie des sous-ensembles flous

à l’usage des ingénieurs

(Fuzzy sets theory)

1. Eléments théoriques de base

/

Introduction to the Theory of Fuzzy Subsets.

Vol.1 Fundamental Theoretical Elements

 

Kaufmann’s Introduction to the Theory of Fuzzy Subsets, entry directory

 

 

 

 

 

Image source:

https://www.researchgate.net/figure/271332770_fig1_Fig-2-Elie-Sanchez-and-Arnold-Kaufmann-in-France-1980

 

 

.

Nolt (16.1) Logics, 'Infinite Valued and Fuzzy Logics,’ summary

 

by Corry Shores

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic & Semantics, Entry Directory]

[John Nolt, entry directory]

[Nolt, Logics, entry directory]

 

[The following is summary. All boldface in quotations are mine unless otherwise noted. Bracketed commentary is my own. As proofreading is incomplete, you will find typos and other districting errors. I apologize in advance.]


[Graham Priest’s similar discussion of infinite valued semantics  in his Logic: A Very Short Introduction]

 

 

 

Summary of

 

John Nolt

 

Logics

 

Part 5: Nonclassical Logics

 

Chapter 16: Radically Nonclassical Logics

 

16.1 Infinite Valued and Fuzzy Logics

 

 

 

 

 

Brief summary:

There are vague predicates or concepts which can potentially lead to counter-intuitive inferences. Consider for example the vague predicate in this inference:

A global population of 1,000,000,000 is sustainable.

If a global population of 1,000,000,000 is sustainable, so is a global population of 1,000,000,001.

∴ A global population of 1,000,000,001 is sustainable.

(Nolt 420)

Suppose we reiterate the premises, each time building from the prior conclusion’s numerical value, and adding one more in the process. After a while, the population number will get very large, and the conclusion will no longer be true (it will not satisfy the predicate any more). An infinite-valued semantics can deal with these situations. It allows us to assign partial values to propositions, such that instead of true and false we have 0, 1, and all the decimal values between. In our example, each iteration would receive slightly less of a truth value, and so eventually the reiterations would arrive at 0, corresponding to our intuition that their truth value should decrease as the population number increases. A common valuation scheme is:

1. V(¬Φ) = 1 – V(Φ)

2. V(Φ & Ψ) = min(V(Φ), V(Ψ))

3. V(Φ ∨ Ψ) = max(V(Φ), V(Ψ))

4. V(Φ → Ψ) = 1 + min(V(Φ), V(Ψ)) - V(Φ)

5. V(Φ ↔ Ψ) = 1 + min(V(Φ), V(Ψ)) - max((V(Φ), V(Ψ))

Consider a predicate like “is red”. Suppose we add the following argument to get “fresh blood is red”. This is clearly true. But what about “a sunset is red”? This is partly true. That means the set of things that belong to the predicate “is red” has items whose membership is not entire. They have a certain degree of membership, and that degree matches the truth value of the proposition predicating them. In other words, if “a sunset is red” has the truth value 0.2, that means sunsets only have a membership degree of 0.2 in the set of red things. These are fuzzy sets (that is, sets whose membership is a matter of degree), and they were invented by Lofti Zadeh. He also applied fuzzy sets to the logical values that can be assigned, such that a range of values would be assigned and certain values in that range are themselves assigned a partial value for their degree of membership in that range of truth values. Such a semantics based on fuzzy truth values is called a fuzzy logic. While such fuzzy systems involve perhaps too much complexity and arbitrarity, they have proven useful for artificial intelligence programming, and they have also appealed to people who are wary of too much logical or conceptual precision.

 

 

 

Summary

 

Nolt first has us consider the following argument [quoting]:

A global population of 1,000,000,000 is sustainable.

If a global population of 1,000,000,000 is sustainable, so is a global population of 1,000,000,001.

∴ A global population of 1,000,000,001 is sustainable.

(Nolt 420)

This argument seems valid, because it simply uses modus ponens. And “the premises are true – or at least almost true” (420). But what if we reiterate this inference 999 billion times, in each instance beginning with the previously increased figure and then adding yet another 1 to it? We would then conclude that a global population of one trillion is sustainable. But surely it is not.

 

Nolt explains that the problem arises because we begin with “almost true” premises that take the form:

If a global population of n is sustainable, so is a global population of n +1.

(Nolt 421)

Nolt says that at the beginning of the iterations, the conclusions that we draw are “either wholly true or approximately true” (421). But with each further inference, “the conclusions become less and less true so that by the end of the sequence we arrive at a conclusion that is wholly false” (421).

 

Nolt notes that we cannot pinpoint one particular inference in the sequence and claim that it is the precise source where the chain enters into error. “Rather, there is a gradual progression from truth into error” (421). He identifies the problem in this example as being that we are using a vague predicate, “is sustainable” (421). While there are clear cases where the populations are so great that they are obviously unsustainable and also ones where they are no doubt sustainable, there are also

intermediate cases in which it is ‘sort of true’ but ‘sort of false’ that the population is sustainable. Our vague notion of sustainability defines no sharp boundary at which a population of n is sustainable, but a population of n  + 1 is not. Rather, as the numbers increase it becomes less and less true that the population is sustainable.

(421)

 

If we see the situation in these terms, then we are also thinking of truth as something that admits of degrees. “There are, it appears, not just two truth values, but a potential continuum of values, from wholly false to wholly true” (421). [We could then assign propositions any from an infinitely varying range of values.] “If we take this gradation of truth value seriously, the result is an infinite valued semantics” (421).

 

Since we are dealing with a range of quantitative variation between true and false, we will use 0 for false, 1 for true, and all degrees of variation between as decimals. Nolt explains that we now have many different options for how to revise our valuation rules. But he will introduce us to one of the simplest ones (421).

 

[In this semantics, a proposition’s negation seems to have the inverse value.]

If Φ is wholly true, then ¬Φ is wholly untrue and vice versa. Likewise, it seems reasonable to suppose that if Φ is three-quarters true, ¬Φ is only one-quarter true. Thus, as a general rule ¬Φ has all the truth that Φ lacks and vice versa. More formally,

1. V(¬Φ) = 1 – V(Φ)

(Nolt 421)

This also means that “the double negation of a formula has the same truth value as the formula itself” (421).

 

[In a conjunction, if even one conjuct is false, the whole conjunction is false, even if the other one is true. So we might think of the mechanics here that the lowest value drags the value of the whole down to it.]

A conjunction would seem to be true as the least true of its conjuncts. The truth conditionals for conjunctions are best expressed using the notation ‘min(x, y)’ to indicate the minimum or least of the two value x and y – or the value both ‘x’ and ‘y’ exress if x = y. Thus for example, min(0.25, 0.3) = 0.25 and min(1,1) = 1. The valuation rule for conjunctions, then, is

2. V(Φ & Ψ) = min(V(Φ), V(Ψ))

(Nolt 422)

 

[Now with a similar sort of thinking, note how normally in a disjunction, the whole disjunction will be true when at least one is true, even if the other is false. In other words, it is as if the highest value pulls the value of the whole disjunction up to it.]

Disjunctions are true as the most true of their disjuncts. This idea may be expressed by the notation ‘max(x, y)’, which indicates the maximum or greatest of the two values x and y, or the value both variables express if x = y:

3. V(Φ ∨ Ψ) = max(V(Φ), V(Ψ))

(Nolt 422)

 

[The reasoning behind the conditional valuation is a bit trickier for me to grasp, so I will guess. Recall how for conditionals that if the antecedent is false, then the whole conditional is true. Thus some degree of falsity in the antecedent is not a ‘problem’ (by ‘problem’ I mean something like a circumstance that would lead to falsity) and in fact does more to lend to the truth of the conditional. What is ‘problematic’ is not simply the antecedent being true but rather the antecedent’s being true while the consequent is false. As we will see, one idea in the infinite-valued evaluation for conditionals is that so long as the consequent is not more false than the antecedent, then the whole conditional is fully true. But if the consequent is less true than the antecedent, then the conditional is as false as the difference between the consequent and antecedent. (So if the consequent is 0.2 more false than the antecedent, then the whole conditional is 0.2 false and thus its overall truth value is 0.8, which is its quantity of being true). The reasoning here would seem to be that it is ‘problematic’ for the consequent to be less true than the antecedent, and the extent to which it is gives the extent to which the whole conditional is less than true. But I do not entirely understand why a partial truth for the antecedent and an at least or more partial truth of the consequent results in a full truth value rather than a partial one. I can only think that the partial falsity of the antecedent is not ‘problematic’ but only the truthfulness implying falsity is, which only begins to happen when the consequent is less true than the antecedent.]

There are many ways of dealing with conditionals; again we shall choose one of the simplest. We shall assume that a conditional is wholly true if its consequent is at least as true as its antecedent, but that if the consequent is less true than the antecedent by some amount x, then the conditional is less than wholly true by that amount. If, for example, V(Φ) = 0.3 and V(Ψ) = 0.4, then V(Φ → Ψ) = 1, since the degree of truth of the consequent exceeds that of the antecedent. If, however, the values are reversed, so that V(Φ) = 0.4 and V(Ψ) = 0.3, then, since the antecedent’s degree of truth exceeds the consequent’s by 0.1, the conditional is that much less than wholly true; in other words, V(Φ → Ψ) = 1 - 0.1 = 0.9. These truth conditions can be expressed by the equation

4. V(Φ → Ψ) = 1 + min(V(Φ), V(Ψ)) - V(Φ)

If Ψ is at least as true as Φ – that is, V(Φ) ≤ V(Ψ) – then min(V(Φ), V(Ψ)) - V(Φ) = 0 and so V(Φ → Ψ) = 1. If V(Φ) > V(Ψ), then min(V(Φ), V(Ψ)) - V(Φ) = V(Ψ) - V(Φ) so that V(Φ → Ψ) = 1 + V(Ψ) - V( Φ) = 1 - (V( Φ) - V(Ψ)); that is, the conditional is less than wholly true by the amount V(Φ) - V(Ψ).

 

Notice that Φ → Ψ is wholly false only if V(Φ) = 1 and V(Ψ) = 0. Otherwise, it has some degree of truth-that is, V(Φ → Ψ) > 0.

(422-423)

 

[For biconditionals, recall how normally a biconditional can be true even if both sides are false. What is important here is that both sides have the same value. So in our infinite-valued semantics, if both have the same partial value, then the whole biconditional will be true. And if there is a discrepancy, then the biconditional will be as false as the difference between them. Thus if there is 0.2 difference between them, its value is 0.8, because it is 0.8 true.]

For the biconditional, we shall assume that its truth value is 1 iff the truth values of its components are equal and that otherwise it is less than wholly true by the amount of their difference. Thus, if V(Φ) = V(Ψ), then V(Φ → Ψ) = 1. But if, say, V(Φ) = 0.3 and V(Ψ) = 0.7 so the difference between V(Φ) and V(Ψ) is 0.4, then V(Φ → Ψ) = 1 - 0.4 = 0.6. These truth conditions are expressible by the equation

5. V(Φ ↔ Ψ) = 1 + min(V(Φ), V(Ψ)) - max((V(Φ), V(Ψ))

(432)

 

By limiting our values we can obtain semantics that we have seen before. For example, we obtain classical semantics by limiting our values to 0 and 1, and we obtain Łukasiewicz’ three-valued semantics by having a third value 1/2 that we regard as the I value (see section 15.2). [Let us examine that last claim, because it is something Graham Priest also says. The negation of I is I, which in these quantities would be 0.5. That calculation results from V(¬Φ) = 1 – V(Φ) also. The conjunction of a true conjunct with an indeterminate one is I, or 0.5 in our restricted three quantity system. This quantity also results from V(Φ & Ψ) = min(V(Φ), V(Ψ)).]

 

We can test for validity using these semantics, and we find it often gives the results we would expect. For example, we would think that P ∨ ~P is invalid (not true on all valuations), and in fact it is not true for the value 0.5. (Nolt 423-424)

 

Nolt then notes something interesting. [Consider formulas of the form Φ ∨ ~Φ. Suppose Φ is 1. That means ~Φ is 0. The higher value is 1, and thus the disjunction is 1. Suppose Φ is 0. That means ~Φ is 1. The higher value is 1, and thus the disjunction is 1. Now suppose Φ has some intermediate value like 0.5. That means ~Φ is 0.5. The higher value is 0.5, and thus the disjunction is 0.5. If  Φ is 0.3, then ~Φ is 0.7, and the whole disjunction is 0.7. If  Φ is 0.7, then ~Φ is 0.3, and the whole disjunction is 0.7.] Formulas taking the form Φ ∨ ~Φ never take a value below 1/2, so they are always at least partly true. However, they are not valid, because they are not true under all valuations.

 

Other formulas are valid even with there being an infinity of valuations, like Φ → Φ. [see p.424 for the metatheorem and proof.]

 

Nolt then explains, “In general, formulas which are classically valid may take any truth value on the infinite-valued semantics, though those of particular forms may be confined to a particular range of values or even to one value” (424).

 

Nolt then discusses the fact that an infinite-valued semantics allows for a variety of ways to define validity [see pp.424-425 for more details on the options.]

 

Nolt then returns to the original example. He says that in our infinite-valued semantics, each additional iteration of the proposition taking the form

A global population of n + 1 is sustainable

takes a slightly lesser truth value (425). Note that the reasoning is based on the conditional:

If a global population of n is sustainable, so is a global population of n + 1.

[Here in the first case where the population is specified as six billion, our antecedent has the value 1, but each iteration has a value less than that. We subtract them to get the value of the whole conditional. Thus with each iteration, this conditional gets less and less than 1.]

 

“A long | sequence of such inferences, then, may lead us from near truth to absolute falsehood” (426).

 

We see that infinite-valued semantics can be potentially useful for certain cases of vagueness. But, Nolt explains, it is not “wholly satisfactory” (426). He notes that while we may know that “A global population of 6,000,000,001 is sustainable” is slightly less true than “A global population of 6,000,000,000 is sustainable,” we still have no obvious way to calculate precisely how much less of a truth value it should receive (426).

 

Nolt says that one reason we cannot make this determination is that we simply do not understand the ecosystem enough to know the degree to which additional population is unsustainable. The other reason is that the predicate “is sustainable” is vague. It is qualitative and vague and thus hard to precisely quantify (426).

 

[The next idea seems to be that Lofti Zadeh tried to improve upon infinite-valued semantics by saying that even the truth values can be fuzzy in the sense that they take on a range of values, and values as members in that range have a certain degree of membership. I am just guessing here, but the idea might be the following. We might say that the global population sentence from above is not simply 0.8 for instance but somewhere between 0.7-0.9, with the values near 0.7 being less “likely” to hold in the sense that 0.7 does not take a very strong degree of membership in the set of values, but it could still be in it. Let me quote:]

In the mid-1960s, Lofti Zadeh set out to improve upon infinite-valued semantics by making the truth values themselves imprecise. That is, instead of assigning to a statement like ‘A global population of 6,000,000,000 is sustainable’ an arbitrarily precise numerical truth value, Zadeh proposed that we assign it an imprecise range of values. By this he meant not merely an interval of values (say, the interval between .4 and .5, which itself is a precisely defined entity), but a fuzzy interval of values. A fuzzy interval is a kind of fuzzy set. And a fuzzy set is a set for which membership is a matter of degree.

(Nolt 426, italics and boldface his, underlining mine)

 

[Zedah claimed that most of our concepts (and maybe by this we are to understand ‘predicates’) actually are matters of fuzzy sets rather than classical sets. In other words, many things taking a predicate do not take it to the fullest degree, and thus there are members of the set of things taking the predicate which have less strong of a membership. The next idea in this paragraph might be something like what we said above in brackets, where numerical truth values within a certain range are assigned their own degree of membership in that group of values. Let me quote so you can see:]

Most concepts, Zadeh argued, define not classical sets, but fuzzy sets. Take the concept of redness. Some things are wholly and genuinely red. But others are almost red, somewhat red, only a little bit red, and so on. So, whereas fresh blood or a red traffic light might be fully a member of the set of all red things, the setting sun might be, say, halfway a member and a peach only slightly a member. Now in fuzzy-set theory, membership is assigned strict numerical values from 0 to 1, like the truth values in infinite-valued semantics. But in defining truth values, Zadeh compounds the fuzziness. He might, for example, define a truth value AT (almost true), which is a fuzzy set of numerical values in which, say, numbers no greater than 0.5 have membership 0, 0.6 has membership 0.3, 0.7 has membership 0.5, 0.9 has membership 0.8, and 0.99 has membership 0.95. Such a fuzzy set of numerical values is for Zadeh a truth value. A logic whose semantics is based on such fuzzy truth values is called a fuzzy logic.

(426, boldface his, underlining mine)

 

[The next ideas I will probably misrepresent. Let me try the following. Consider the predicate, “is a number”. For this, there is the set of numbers, which when named as the argument taking this predicate, make the proposition true. Otherwise it is false. We are thinking here of classical semantics. Now, suppose we are using infinite valued logic. This means “x is a number” could potentially have a value between 1 and 0. But this example does not work, because it is not easy to think of such partial-valued cases. But for “x is red”, we have already seen cases where something is not entirely red, like a sunset. Nolt’s point seems to be that so long as we are implementing partial values for sentences with predicates taking at least one argument, then that means we are referring to objects that have only a degree of membership in the set of things corresponding to that predicate, and thus we are dealing with fuzzy sets. The next point I do not get, but let us consider a possibility. What we just noted is infinite-valued semantics. A fuzzy logic would have a semantics where not only do we say that a sunset has a 0.5 degree membership, but also that this somehow is a fuzzy value, meaning perhaps that there is a range of values from 0.4 to 0.6 indicating its fuzzy membership in this set. That is a guess, so please consult the text. The final point might be that by complicating the system in this way, and also by admitting of many potentially arbitrary determinations for how to assign the fuzzy values, we make our basic concepts become perhaps too complex for practical purposes.]

If infinite-valued semantics presents a bewildering array of choices of truth conditions and semantic concepts, fuzzy logic compounds the complication. Already in infinite-valued predicate logic, the extensions assigned to predicates must be fuzzy sets; for if they were classical sets, atomic formulas containing n-place predicates (n > 0) would always be either true or false. So, for example, the predicate ‘is red’ has as its extension the fuzzy set of red things described above. Consequently, infinite-valued semantics has the following truth clause for one-place predicates Φ and names α: |

V(Φα) = x iff V(α) is a member of the fuzzy set V(Φ) to degree x.

But on Zadeh’s fuzzy semantics the extensions of predicates are structures still more complex than fuzzy sets, structures which, when applied to the extensions of names, yield fuzzy truth values. The valuation rules for the operators, and the semantic concepts of validity, consistency, and so on, must all be redefined once again to accommodate these fuzzy values. In the process these concepts “splinter” even more wildly than concept of validity does in infinite-valued logic.

(426-427)

 

In fact, Zadeh’s attempts to minimize the arbitrarity by limiting the values and correlating them with natural language expressions does not really solve the problem.

Yet despite the complication upon complication, arbitrariness remains. Zadeh suggests that a fuzzy logic should not employ all possible fuzzy truth values (a very large set of values indeed!), but only a small finite range of them, and that it should correlate them with such natural language expressions as ‘very true’, ‘more or less true’, and so on. But which fuzzy set of numbers corresponds to the English expression ‘more or less true’? And why should we suppose that precisely that set is what we mean when we say that a particular sentence is more or less true? The choice of any particular fuzzy value is just as arbitrary as the assignment of a precise numerical truth value to a vague statement. The arbitrariness does not go away; it is merely concealed in the complexity.

(427)

 

Nolt concludes by noting certain successes of fuzzy logic. It has proved useful in artificial intelligence programming. It has proved attractive to those who do not like too much conceptual precision. Its popularity has even led to some ambiguous uses for the term.

Arbitrariness notwithstanding, fuzzy logic has found useful application in artificial intelligence devices. But it has also acquired a certain unwarranted mystique. Many people are attracted to the idea of a (warm and ?) fuzzy logic because it sounds as if it might offer relief from overtaxing precision. As a result of this popularity, the term ‘fuzzy logic’ is often used loosely. In popular science publications it may mean no more than an infinite-valued logic – or even statistical or just plain muddle-headed reasoning.

(427)

 

 

 

 

 

 

From:

 

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

 

 

.

22 Mar 2017

Proust (§19/21) Swann’s Way. [M. Swann entered the highest societies without giving any indication to the narrator’s family.]

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Literature, Poetry, Drama, entry directory]

[Proust, entry directory]

[Proust. À la recherche du temps perdu, entry directory]

[Proust. Du côté de chez swann / Swann’s Way, entry directory]

 

[The following is summary, with my own bracketed comments. Proofreading is incomplete, so I apologize in advance for my distracting typos.]

 

 

 

 

Marcel Proust

 

Du côté de chez swann. À la recherche du temps perdu. Tome I

Swan's Way. Vol. 1 of Remembrance of Things Past

 

Première partie

Overature

 

Combray I.

 

§19/21

[M. Swann entered the highest societies without giving any indication to the narrator’s family.]

 

 

Brief summary:

The narrator’s family did not know that M. Swann, after their dinner parties, would enter into the highest and most exclusive societies.

 

 

 

 

Summary

 

The narrator’s great-aunt assumed that M. Swann was of high enough social status as to be welcomed into the company of the upper middle class. But without her or others in the narrator’s family knowing, after leaving their dinner party, he would enter the company of the most exclusive and highest societies. This is a lot like going from our mundane world into a mythological hidden place of resplendence and riches.

 

 

 

From the English translation:

§21

But if anyone had suggested to my aunt that this Swann, who, in his capacity as the son of old M. Swann, was ‘fully qualified’ to be received by any of the ‘upper middle class,’ the most respected barristers and solicitors of Paris (though he was perhaps a trifle inclined to let this hereditary privilege go into abeyance), had another almost secret existence of a wholly different kind: that when he left our house in Paris, saying that he must go home to bed, he would no sooner have turned the corner than he would stop, retrace his steps, and be off to some drawing-room on whose like no stockbroker or associate of stockbrokers had ever set eyes — that would have seemed to my aunt as extraordinary as, to a woman of wider reading, the thought of being herself on terms of intimacy with Aristaeus, of knowing that he would, when he had finished his conversation with her, plunge deep into the realms of Thetis, into an empire veiled from mortal eyes, in which Virgil depicts him as being received with open arms; or — to be content with an image more likely to have occurred to her, for she had seen it painted on the plates we used for biscuits at Combray — as the thought of having had to dinner Ali Baba, who, as soon as he found himself alone and unobserved, would make his way into the cave, resplendent with its unsuspected treasures.

 

 

From the French:

§19

Mais si l’on avait dit à ma grand’mère que ce Swann qui, en tant que fils Swann était parfaitement «qualifié» pour être reçu par toute la «belle bourgeoisie», par les notaires ou les avoués les plus estimés de Paris (privilège qu’il semblait laisser tomber en peu en quenouille), avait, comme en cachette, une vie toute différente; qu’en sortant de chez nous, à Paris, après nous avoir dit qu’il rentrait se coucher, il rebroussait chemin à peine la rue tournée et se rendait dans tel salon que jamais l’œil d’aucun agent ou associé d’agent ne contempla, cela eût paru aussi extraordinaire à ma tante qu’aurait pu l’être pour une dame plus lettrée la pensée d’être personnellement liée avec Aristée dont elle aurait compris qu’il allait, après avoir causé avec elle, plonger au sein des royaumes de Thétis, dans un empire soustrait aux yeux des mortels et où Virgile nous le montre reçu à bras ouverts; ou, pour s’en tenir à une image qui avait plus de chance de lui venir à l’esprit, car elle l’avait vue peinte sur nos assiettes à petits fours de Combray — d’avoir eu à dîner Ali-Baba, lequel quand il se saura seul, pénétrera dans la caverne, éblouissante de trésors insoupçonnés.

 

 

 

Proust, Marcel. Du côté de chez swann. À la recherche du temps perdu. Tome I.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96d/index.html

 

Proust, Marcel. Swan’s Way. Vol. 1 of Remembrance of Things Past. Transl. C.K. Scott Moncrieff.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96s/index.html

 

 

.

20 Mar 2017

Proust (§18/19-20) Swann’s Way. [M. Swann’s high social status was not obvious.]

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Literature, Poetry, Drama, entry directory]

[Proust, entry directory]

[Proust. À la recherche du temps perdu, entry directory]

[Proust. Du côté de chez swann / Swann’s Way, entry directory]

 

[The following is summary, with my own bracketed comments. Proofreading is incomplete, so I apologize in advance for my distracting typos.]

 

 

 

 

Marcel Proust

 

Du côté de chez swann. À la recherche du temps perdu. Tome I

Swan's Way. Vol. 1 of Remembrance of Things Past

 

Première partie

Overature

 

Combray I.

 

§18/19-20

[M. Swann’s high social status was not obvious.]

 

 

Brief summary:

M. Swann’s behavior never indicated his high social status, and the narrator’s family had no reason to think so anyway.

 

 

 

 

Summary

 

The narrator’s family was unaware of M. Swann’s place in high society, because he was discrete about it, and they assumed he stayed in the same social class as his parents. M. Swann’s father was a stockbroker, which meant their family income stayed within a certain limited range. The narrator’s family did not have any reason to suspect that M. Swann associated with the higher social class, because he did not show much refinement and social grace when discussing such things as cuisine or art. However, he did often entertain them with lively and witty stories about his latest adventures with people they also knew.

 

The narrator’s great-aunt often even joked about M. Swann being of a higher class. It was funny for her, because she thought it in fact highly improbable.

 

 

 

From the English translation:

§19

Our utter ignorance of the brilliant part which Swann was playing in the world of fashion was, of course, due in part to his own reserve and discretion, but also to the fact that middle-class people in those days took what was almost a Hindu view of society, which they held to consist of sharply defined castes, so that everyone at his birth found himself called to that station in life which his parents already occupied, and nothing, except the chance of a brilliant career or of a ‘good’ marriage, could extract you from that station or admit you to a superior caste. M. Swann, the father, had been a stockbroker; and so ‘young Swann’ found himself immured for life in a caste where one’s fortune, as in a list of taxpayers, varied between such and such limits of income. We knew the people with whom his father had associated, and so we knew his own associates, the people with whom he was ‘in a position to mix.’ If he knew other people besides, those were youthful acquaintances on whom the old friends of the family, like my relatives, shut their eyes all the more good-naturedly that Swann himself, after he was left an orphan, still came most faithfully to see us; but we would have been ready to wager that the people outside our acquaintance whom Swann knew were of the sort to whom he would not have dared to raise his hat, had he met them while he was walking with ourselves. Had there been such a thing as a determination to apply to Swann a social coefficient peculiar to himself, as distinct from all the other sons of other stockbrokers in his father’s position, his coefficient would have been rather lower than theirs, because, leading a very simple life, and having always had a craze for ‘antiques’ and pictures, he now lived and piled up his collections in an old house which my grandmother longed to visit, but which stood on the Quai d’Orléans, a neighbourhood in which my great-aunt thought it most degrading to be quartered. “Are you really a connoisseur, now?” she would say to him; “I ask for your own sake, as you are likely to have ‘fakes’ palmed off on you by the dealers,” for she did not, in fact, endow him with any critical faculty, and had no great opinion of the intelligence of a man who, in conversation, would avoid serious topics and shewed a very dull preciseness, not only when he gave us kitchen recipes, going into the most minute details, but even when my grandmother’s sisters were talking to him about art. When challenged by them to give an opinion, or to express his admiration for some picture, he would remain almost impolitely silent, and would then make amends by furnishing (if he could) some fact or other about the gallery in which the picture was hung, or the date at which it had been painted. But as a rule he would content himself with trying to amuse us by telling us the story of his latest adventure — and he would have a fresh story for us on every occasion — with some one whom we ourselves knew, such as the Combray chemist, or our cook, or our coachman. These stories certainly used to make my great-aunt laugh, but she could never tell whether that was on account of the absurd parts which Swann invariably made himself play in the adventures, or of the wit that he shewed in telling us of them. “It is easy to see that you are a regular ‘character,’ M. Swann!”

 

§20

As she was the only member of our family who could be described as a trifle ‘common,’ she would always take care to remark to strangers, when Swann was mentioned, that he could easily, if he had wished to, have lived in the Boulevard Haussmann or the Avenue de l’Opéra, and that he was the son of old M. Swann who must have left four or five million francs, but that it was a fad of his. A fad which, moreover, she thought was bound to amuse other people so much that in Paris, when M. Swann called on New Year’s Day bringing her a little packet of marrons glacés, she never failed, if there were strangers in the room, to say to him: “Well, M. Swann, and do you still live next door to the Bonded Vaults, so as to be sure of not missing your train when you go to Lyons?” and she would peep out of the corner of her eye, over her glasses, at the other visitors.

 

 

From the French:

§18

L’ignorance où nous étions de cette brillante vie mondaine que menait Swann tenait évidemment en partie à la réserve et à la discrétion de son caractère, mais aussi à ce que les bourgeois d’alors se faisaient de la société une idée un peu hindoue et la considéraient comme composée de castes fermées où chacun, dès sa naissance, se trouvait placé dans le rang qu’occupaient ses parents, et d’où rien, à moins des hasards d’une carrière exceptionnelle ou d’un mariage inespéré, ne pouvait vous tirer pour vous faire pénétrer dans une caste supérieure. M. Swann, le père, était agent de change; le «fils Swann» se trouvait faire partie pour toute sa vie d’une caste où les fortunes, comme dans une catégorie de contribuables, variaient entre tel et tel revenu. On savait quelles avaient été les fréquentations de son père, on savait donc quelles étaient les siennes, avec quelles personnes il était «en situation» de frayer. S’il en connaissait d’autres, c’étaient relations de jeune homme sur lesquelles des amis anciens de sa famille, comme étaient mes parents, fermaient d’autant plus bienveillamment les yeux qu’il continuait, depuis qu’il était orphelin, à venir très fidèlement nous voir; mais il y avait fort à parier que ces gens inconnus de nous qu’il voyait, étaient de ceux qu’il n’aurait pas osé saluer si, étant avec nous, il les avait rencontrés. Si l’on avait voulu à toute force appliquer à Swann un coefficient social qui lui fût personnel, entre les autres fils d’agents de situation égale à celle de ses parents, ce coefficient eût été pour lui un peu inférieur parce que, très simple de façon et ayant toujours eu une «toquade» d’objets anciens et de peinture, il demeurait maintenant dans un vieil hôtel où il entassait ses collections et que ma grand’mère rêvait de visiter, mais qui était situé quai d’Orléans, quartier que ma grand’tante trouvait infamant d’habiter. «Etes-vous seulement connaisseur? je vous demande cela dans votre intérêt, parce que vous devez vous faire repasser des croûtes par les marchands», lui disait ma grand’tante; elle ne lui supposait en effet aucune compétence et n’avait pas haute idée même au point de vue intellectuel d’un homme qui dans la conversation évitait les sujets sérieux et montrait une précision fort prosaïque non seulement quand il nous donnait, en entrant dans les moindres détails, des recettes de cuisine, mais même quand les sœurs de ma grand’mère parlaient de sujets artistiques. Provoqué par elles à donner son avis, à exprimer son admiration pour un tableau, il gardait un silence presque désobligeant et se rattrapait en revanche s’il pouvait fournir sur le musée où il se trouvait, sur la date où il avait été peint, un renseignement matériel. Mais d’habitude il se contentait de chercher à nous amuser en racontant chaque fois une histoire nouvelle qui venait de lui arriver avec des gens choisis parmi ceux que nous connaissions, avec le pharmacien de Combray, avec notre cuisinière, avec notre cocher. Certes ces récits faisaient rire ma grand’tante, mais sans qu’elle distinguât bien si c’était à cause du rôle ridicule que s’y donnait toujours Swann ou de l’esprit qu’il mettait à les conter: «On peut dire que vous êtes un vrai type, monsieur Swann!» Comme elle était la seule personne un peu vulgaire de notre famille, elle avait soin de faire remarquer aux étrangers, quand on parlait de Swann, qu’il aurait pu, s’il avait voulu, habiter boulevard Haussmann ou avenue de l’Opéra, qu’il était le fils de M. Swann qui avait dû lui laisser quatre ou cinq millions, mais que c’était sa fantaisie. Fantaisie qu’elle jugeait du reste devoir être si divertissante pour les autres, qu’à Paris, quand M. Swann venait le 1er janvier lui apporter son sac de marrons glacés, elle ne manquait pas, s’il y avait du monde, de lui dire: «Eh bien! M. Swann, vous habitez toujours près de l’Entrepôt des vins, pour être sûr de ne pas manquer le train quand vous prenez le chemin de Lyon?» Et elle regardait du coin de l’œil, par-dessus son lorgnon, les autres visiteurs.

 

 

 

Proust, Marcel. Du côté de chez swann. À la recherche du temps perdu. Tome I.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96d/index.html

 

Proust, Marcel. Swan’s Way. Vol. 1 of Remembrance of Things Past. Transl. C.K. Scott Moncrieff.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96s/index.html

 

 

.

Proust (§17/18) Swann’s Way. [M. Swann belonged to high society.]

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Literature, Poetry, Drama, entry directory]

[Proust, entry directory]

[Proust. À la recherche du temps perdu, entry directory]

[Proust. Du côté de chez swann / Swann’s Way, entry directory]

 

[The following is summary, with my own bracketed comments. Proofreading is incomplete, so I apologize in advance for my distracting typos.]

 

 

 

Marcel Proust

 

Du côté de chez swann. À la recherche du temps perdu. Tome I

Swan's Way. Vol. 1 of Remembrance of Things Past

 

Première partie

Overature

 

Combray I.

 

§17/18

[M. Swann belonged to high society.]

 

 

Brief summary:

M. Swann was a prominent member of important societies, even though the narrator’s family, who often hosted him for dinner, were completely unaware of this.

 

 

 

Summary

 

M. Swann often visited the narrator’s family for many years before Swann’s marriage. All the while, the narrator’s family did not know that Swann was a more prominent figure in the social world than he may have seemed, as “he had entirely ceased to live in the kind of society which his family had frequented.” Specifically, he was “one of the smartest members of the Jockey Club, a particular friend of the Comte de Paris and of the Prince of Wales, and one of the men most sought after in the aristocratic world of the Faubourg Saint-Germain.”

 

 

 

 

From the English translation:

§18

For many years, albeit — and especially before his marriage — M. Swann the younger came often to see them at Combray, my great-aunt and grandparents never suspected that he had entirely ceased to live in the kind of society which his family had frequented, or that, under the sort of incognito which the name of Swann gave him among us, they were harbouring — with the complete innocence of a family of honest innkeepers who have in their midst some distinguished highwayman and never know it — one of the smartest members of the Jockey Club, a particular friend of the Comte de Paris and of the Prince of Wales, and one of the men most sought after in the aristocratic world of the Faubourg Saint-Germain.

 

From the French [note, in the first line, “mon mariage” was changed to “son”]:

§17

Pendant bien des années, où pourtant, surtout avant son mariage, M. Swann, le fils, vint souvent les voir à Combray, ma grand’tante et mes grands-parents ne soupçonnèrent pas qu’il ne vivait plus du tout dans la société qu’avait fréquentée sa famille et que sous l’espèce d’incognito que lui faisait chez nous ce nom de Swann, ils hébergeaient — avec la parfaite innocence d’honnêtes hôteliers qui ont chez eux, sans le savoir, un célèbre brigand — un des membres les plus élégants du Jockey-Club, ami préféré du comte de Paris et du prince de Galles, un des hommes les plus choyés de la haute société du faubourg Saint-Germain.

 

 

 

Proust, Marcel. Du côté de chez swann. À la recherche du temps perdu. Tome I.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96d/index.html

 

Proust, Marcel. Swan’s Way. Vol. 1 of Remembrance of Things Past. Transl. C.K. Scott Moncrieff.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96s/index.html

 

 

.

15 Mar 2017

Proust (§16/17) Swann’s Way. [M. Swann’s father and Grandfather were friends. Father Swann’s habit of remembering his deceased wife often but only a little each time inspired Grandfather’s beautiful saying, “Often, but a little at a time, like poor old Swann.”]

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Literature, Poetry, Drama, entry directory]

[Proust, entry directory]

[Proust. À la recherche du temps perdu, entry directory]

[Proust. Du côté de chez swann / Swann’s Way, entry directory]

 

[The following is summary, with my own bracketed comments. Proofreading is incomplete, so I apologize in advance for my distracting typos.]

 

 

“Souvent, mais peu à la fois, comme le pauvre père Swann”

“Often, but a little at a time, like poor old Swann”

 

Marcel Proust

 

Du côté de chez swann. À la recherche du temps perdu. Tome I

Swan's Way. Vol. 1 of Remembrance of Things Past

 

Première partie

Overature

 

Combray I.

 

§16 / §17

[M. Swann’s father and Grandfather were friends. Father Swann’s habit of remembering his deceased wife often but only a little each time inspired Grandfather’s beautiful saying, “Often, but a little at a time, like poor old Swann.”]

 

 

Brief summary:

When M. Swann arrived for dinner, it was so dark in the garden that he was recognized at first by his voice rather than by his visual appearance. M. Swann is fond of Grandfather, who was also a close friend of M. Swann’s now-deceased father. Grandfather tells the story of when Father Swann was grieving over his dead wife. He said, “It’s a funny thing, now; I very often think of my poor wife, but I cannot think of her very much at any one time.” Grandfather turned this into a beautiful saying: “Often, but a little at a time, like poor old Swann.”

 

 

 

Summary

 

In the dark of the garden, it was hard to know if it was M. Swann who arrived. Instead, he could be identified by his voice. M. Swann was fond of the narrator’s grandfather, although there was a substantial difference in their age. In fact, M. Swann’s father and the narrator’s grandfather were once close friends. A few times a year, Grandfather would tell the story of M. Swann’s father at the time of his wife’s death. Father Swann stayed by his wife’s bedside, but after she died, Grandfather convinced him to leave the room while they laid her in her coffin. They went for a walk on the Swann property, and Father Swann [although grieving] took Grandfather’s arm and exclaimed how wonderful it was for them to be alive and together to enjoy their nice surroundings and the moment. Then after realizing the gravity of the situation, Father Swann resorted to a habitual gesture he often employed when suddenly perplexed, namely, to wipe his brow, eyes, and then glasses. Father Swann lived for just another two years and never fully got over the death of his wife. He would comment that, “It’s a funny thing, now; I very often think of my poor wife, but I cannot think of her very much at any one time.” Grandfather converted this into a saying that he often employed, “Souvent, mais peu à la fois, comme le pauvre père Swann” / “Often, but a little at a time, like poor old Swann”. The narrator would have assumed Father Swann to be a monster had not Grandfather said he had a heart of gold. [I am not sure why he would be a monster. Maybe the idea is that he did not think enough about his dead wife, because he only did it a little at a time, and in that sense he did not behave in a sufficiently human way. Or maybe the idea is that he caused himself prolonged grief by always thinking about her, and in that sense was a monster to himself.]

 

 

 

 

From the English translation [boldface mine]:

§17

And there we would all stay, hanging on the words which would fall from my grandmother’s lips when she brought us back her report of the enemy, as though there had been some uncertainty among a vast number of possible invaders, and then, soon after, my grandfather would say: “I can hear Swann’s voice.” And, indeed, one could tell him only by his voice, for it was difficult to make out his face with its arched nose and green eyes, under a high forehead fringed with fair, almost red hair, dressed in the Bressant style, because in the garden we used as little light as possible, so as not to attract mosquitoes: and I would slip away as though not going for anything in particular, to tell them to bring out the syrups; for my grandmother made a great point, thinking it ‘nicer’ of their not being allowed to seem anything out of the ordinary, which we kept for visitors only. Although a far younger man, M. Swann was very much attached to my grandfather, who had been an intimate friend, in his time, of Swann’s father, an excellent but an eccentric man in whom the least little thing would, it seemed, often check the flow of his spirits and divert the current of his thoughts. Several times in the course of a year I would hear my grandfather tell at table the story, which never varied, of the behaviour of M. Swann the elder upon the death of his wife, by whose bedside he had watched day and night. My grandfather, who had not seen him for a long time, hastened to join him at the Swanns’ family property on the outskirts of Combray, and managed to entice him for a moment, weeping profusely, out of the death-chamber, so that he should not be present when the body was laid in its coffin. They took a turn or two in the park, where there was a little sunshine. Suddenly M. Swann seized my grandfather by the arm and cried, “Oh, my dear old friend, how fortunate we are to be walking here together on such a charming day! Don’t you see how pretty they are, all these trees — my hawthorns, and my new pond, on which you have never congratulated me? You look as glum as a night-cap. Don’t you feel this little breeze? Ah! whatever you may say, it’s good to be alive all the same, my dear Amédée!” And then, abruptly, the memory of his dead wife returned to him, and probably thinking it too complicated to inquire into how, at such a time, he could have allowed himself to be carried away by an impulse of happiness, he confined himself to a gesture which he habitually employed whenever any perplexing question came into his mind: that is, he passed his hand across his forehead, dried his eyes, and wiped his glasses. And he could never be consoled for the loss of his wife, but used to say to my grandfather, during the two years for which he survived her, “It’s a funny thing, now; I very often think of my poor wife, but I cannot think of her very much at any one time.” “Often, but a little at a time, like poor old Swann,” became one of my grandfather’s favourite phrases, which he would apply to all kinds of things. And I should have assumed that this father of Swann’s had been a monster if my grandfather, whom I regarded as a better judge than myself, and whose word was my law and often led me in the long run to pardon offences which I should have been inclined to condemn, had not gone on to exclaim, “But, after all, he had a heart of gold.”

 

From the French [boldface mine]:

§16

Nous restions tous suspendus aux nouvelles que ma grand’mère allait nous apporter de l’ennemi, comme si on eût pu hésiter entre un grand nombre possible d’assaillants, et bientôt après mon grand-père disait: «Je reconnais la voix de Swann.» On ne le reconnaissait en effet qu’à la voix, on distinguait mal son visage au nez busqué, aux yeux verts, sous un haut front entouré de cheveux blonds presque roux, coiffés à la Bressant, parce que nous gardions le moins de lumière possible au jardin pour ne pas attirer les moustiques et j’allais, sans en avoir l’air, dire qu’on apportât les sirops; ma grand’mère attachait beaucoup d’importance, trouvant cela plus aimable, à ce qu’ils n’eussent pas l’air de figurer d’une façon exceptionnelle, et pour les visites seulement. M. Swann, quoique beaucoup plus jeune que lui, était très lié avec mon grand-père qui avait été un des meilleurs amis de son père, homme excellent mais singulier, chez qui, paraît-il, un rien suffisait parfois pour interrompre les élans du cœur, changer le cours de la pensée. J’entendais plusieurs fois par an mon grand-père raconter à table des anecdotes toujours les mêmes sur l’attitude qu’avait eue M. Swann le père, à la mort de sa femme qu’il avait veillée jour et nuit. Mon grand-père qui ne l’avait pas vu depuis longtemps était accouru auprès de lui dans la propriété que les Swann possédaient aux environs de Combray, et avait réussi, pour qu’il n’assistât pas à la mise en bière, à lui faire quitter un moment, tout en pleurs, la chambre mortuaire. Ils firent quelques pas dans le parc où il y avait un peu de soleil. Tout d’un coup, M. Swann prenant mon grand-père par le bras, s’était écrié: «Ah! mon vieil ami, quel bonheur de se promener ensemble par ce beau temps. Vous ne trouvez pas ça joli tous ces arbres, ces aubépines et mon étang dont vous ne m’avez jamais félicité? Vous avez l’air comme un bonnet de nuit. Sentez-vous ce petit vent? Ah! on a beau dire, la vie a du bon tout de même, mon cher Amédée!» Brusquement le souvenir de sa femme morte lui revint, et trouvant sans doute trop compliqué de chercher comment il avait pu à un pareil moment se laisser aller à un mouvement de joie, il se contenta, par un geste qui lui était familier chaque fois qu’une question ardue se présentait à son esprit, de passer la main sur son front, d’essuyer ses yeux et les verres de son lorgnon. Il ne put pourtant pas se consoler de la mort de sa femme, mais pendant les deux années qu’il lui survécut, il disait à mon grand-père: «C’est drôle, je pense très souvent à ma pauvre femme, mais je ne peux y penser beaucoup à la fois.» «Souvent, mais peu à la fois, comme le pauvre père Swann», était devenu une des phrases favorites de mon grand-père qui la prononçait à propos des choses les plus différentes. Il m’aurait paru que ce père de Swann était un monstre, si mon grand-père que je considérais comme meilleur juge et dont la sentence faisant jurisprudence pour moi, m’a souvent servi dans la suite à absoudre des fautes que j’aurais été enclin à condamner, ne s’était récrié: «Mais comment? c’était un cœur d’or!»

 

 

 

Proust, Marcel. Du côté de chez swann. À la recherche du temps perdu. Tome I.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96d/index.html

 

Proust, Marcel. Swan’s Way. Vol. 1 of Remembrance of Things Past. Transl. C.K. Scott Moncrieff.
Available online at:
http://ebooks.adelaide.edu.au/p/proust/marcel/p96s/index.html

 

 

.

Agler’s Symbolic Logic: Syntax, Semantics, and Proof, collected brief summaries [with examples]

 

by Corry Shores

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic & Semantics, Entry Directory]

[David Agler, entry directory]

[Agler’s Symbolic Logic, entry directory]

 

The following collects the brief summaries for all the main sections of David Agler’s Symbolic Logic Syntax, Semantics, and Proof, and in many cases some of Agler’s examples are replicated too. All the following is Agler’s work, boiled down for quick reference or review.

 

The entry directory without the brief summaries can be found here:

Agler’s Symbolic Logic, entry directory

 

A collection of the important rules and proof strategies can be found here:

Rules and Strategies for Logic Proofs (Alger, Nolt)

 

 

 

Collected Brief Summaries of

 

David W. Agler

 

Symbolic Logic: Syntax, Semantics, and Proof

 

 

 

 

Introduction / Ch.1:

 

Introduction
and
Ch.1: Propositions, Arguments, and Logical Properties
Sections 1.1-1.2: Propositions – Arguments

 

Logic studies the structures and mechanics of correct reasoning that takes an argumentative form. An argument is made of premises that lead to some conclusion. An argument may be (a) inductive, meaning that the conclusion is based on probabilities discerned from the premises, (b) abductive, meaning that the conclusion provides the best explanation (or theory) to account for the premises, or (c) deductively valid, meaning that the conclusion follows by logical necessity from the premises. 

 

 

Section 1.3: Deductively Valid Arguments
and
Section 1.4: Summary

 

An argument is deductively valid if it is impossible for the premises to be true and the conclusion false, and it is invalid otherwise. An argument is sound if the premises in fact are true and as well it is valid. It is unsound if either it is invalid or if any of the premises are false.

negative validity flow

 

 

Ch.2: Language, Syntax, and Semantics


2.1 Truth Functions

 

We will examine the language of propositional logic (PL). Propositions with no truth functional operators are are atomic propositions. A propositional operator makes a proposition more complex. If it does so by combining propositions, then it is a propositional connective. Propositional operators are truth-functional if the value of the more complicated proposition they make is entirely dependent on the truth value of the component parts.

 

 

2.2 The Symbols of PL and Truth-Functional Operators

 

The language of Propositional Logic (PL) is composed of (a) upper-case Roman letters (sometimes numerically subscripted) for propositions, (b) the truth-functional operators ∨, →, ↔, ¬, and ∧, and (c) scope indicators, namely, parentheses, brackets, and braces. A conjunction ∧ is true only if both conjuncts are true, and it is false otherwise. Negation ¬ inverts the truth value of the proposition. When determining the true value of a complex formula, we determine the operated values beginning with the operator with the least scope and work toward the one with the greatest scope, called the “main operator”. It can be determined by finding the operator that operates directly or indirectly on all other sentence parts.

 

 

2.3 Syntax of PL

 

Propositions in the language of propositional logic (PL) must be formed according to specific syntactical rules, and when they are, we consider them to be “well-formed formulas” or “wffs”. We form wffs by using uppercase Roman numerals, which can be modified, combined, and organized using logical operators ∨, →, ↔, ¬, ∧ or scope indicators like parentheses. The propositions within the language of PL are in our object language, and the English sentences we use to discuss these  object language propositions is part of our metalanguage. In our metalanguage we might use metavariables, in this case, boldface Roman letters, to stand for any propositions in the object language. Using such metavariables and a metalanguage, we can state the rules for properly constructed wffs. Briefly, if a formula is an atomic formula, it is a wff. Or, if a proposition takes a negation symbol to its left, it is a wff. Or, if two formula have the symbols ∨, →, ↔, or ∧ placed between them, it is a wff. Nothing else, however, can be considered wffs. The literal negation of a proposition is the negation of the whole proposition; this means it could be a double negation in some cases, and in others the negation will go outside parentheses enclosing a complex proposition.

 

 

2.4 Disjunction, Conditional, Biconditional

 

A disjunction (∨) is true when at least one of the disjuncts is true, and it is false otherwise. The conditional (→) is false only when the antecedent is true but the consequent false. The biconditional (↔) is true when both sides have the same value, and it is false otherwise. The conditional is translatable using “if..., then...” formulations. However, not all “if..., then...” formulations conform to the material conditional’s truth-table. For example, when we use “if..., then...” formations to make statements of causality, both antecedent and conditional can be true and yet the whole proposition be false. Consider the example, “If John prays, then he will get an A”. Suppose that he both prayed and got an A. But suppose further that the real reason he got the A was not because he prayed (which he in fact did anyway) but rather because he cheated. Although both the antecedent and the consequent are true, the whole proposition is false; for, the proposition is stating that the cause of getting the high grade is praying, when in reality the cause is not that but instead is cheating.

 

 

 

2.5 Advanced Translation

 

We translate the following formulations in the following ways:
1) “Neither P nor Q” = ¬P∧¬Q
2) “Not both P and Q” = ¬(PQ)
3) “P only if Q” = PQ
4) “P even if Q” = P, or = P∧(Q∨¬Q)
5) “not-P unless Q” = ¬PQ
6) “P unless Q” = ¬(PQ), or = (PQ)∧¬(PQ)

 

 

Ch.3: Truth Tables

 

3.1 Valuations (Truth-Value Assignments)

 

We can find the truth values of complex propositions on the basis of the value assignments for the atomic formula. We begin with the operators with the least scope and work progressively toward the main operator, whose truth value gives us that of the whole proposition.

 

Z∧¬J

Agler.3.1.truth table values 1a

v(Z) = T

v(J) = F

Agler.3.1.truth table values 1b

Agler.3.1.truth table values 1c

Agler.3.1.truth table values 1d

 

 

3.2 Truth Tables for Propositions

 

We can construct truth-table evaluations for all truth value assignments of a proposition. First we establish all possible value assignment combinations for the individual terms. Then we fill-out the atomic formula values within the proposition. Next we determine the values for the operators, working from those with the least scope progressively to the one with the greatest scope, which gives the value for the whole proposition.

 

(P∨¬P)→Q

Agler 3.2 t

Agler 3.2 u

Agler 3.2 v

Agler 3.2 w

Agler 3.2 x

Agler 3.2 y

Agler 3.2 z

 

 

3.3 Truth Tables Analysis of Propositions

 

We can use a decision procedure to determine whether a singular proposition is a tautology, a contradiction, or a contingency. It is a tautology if it is true under all value assignments; it is a contradiction if it is false under all value assignments, and it is a contingency if it is either true or false, depending on what the value assignments are.

 

Tautology

P→(Q→P)

Agler.3.3.tautology c

 

Contradiction

¬P∧(Q∧P)

Agler.3.3.contradiction a

 

Contingency

P∧(Q→P)

Agler.3.3.tautology d

 

 

3.4 Truth Tables Analysis of Sets of Propositions

 

We can use decision procedures with truth tables to determine whether or not a set of propositions are equivalent or consistent. If the truth table for a set of propositions shows them each to have identical truth values for any truth assignments, then they are logically equivalent. And they are not equivalent otherwise. Another way to conduct this test is to combine the propositions into one larger proposition using the biconditional operator. If the new proposition is a tautology, then the original two propositions are logically equivalent. A set of propositions are consistent if there is at least one value assignment that makes them all true. So on the truth table, we look for at least one line where all the propositions in question have the value true, and that tells us they are consistent. The propositions are inconsistent if no truth value assignment makes all the propositions jointly true. This can also be tested by combining a pair of propositions with a conjunction. If the new proposition is a contradiction, then the original propositions are inconsistent.

 

Equivalence (tested by comparison)

P→Q
Q∨¬P

Agler 3.4.1 equivalence a

 

Equivalence (tested by making biconditional)

P→Q

Q∨¬P

(P→Q) ↔ (Q∨¬P)

Agler 3.4.1 equivalence c

 

Consistency (tested by comparison)

P→Q
Q∨P
P↔Q

Agler 3.4.2 consistency 1

 

Inconsistency (tested by comparison)

(P∨Q)
¬(Q∨P)

Agler 3.4.2 


inconsistency a

 

Inconsistency (tested by conjunction)

(P∨Q)
¬(Q∨P)

Agler 3.4.2 


inconsistency b

 

 

3.5 The Material Conditional Explained (Optional)

 

It may not be immediately obvious why the material conditional has its particular truth evaluation. However, the reasons for it come to light when we see that the other possible evaluations would ascribe properties or behaviors to the conditional that our intuition tells us it should not have.

 

 

3.6 Truth Table Analysis of Arguments

 

There are a couple ways we can use truth tables to test for the validity of arguments. (a) We look for a row where all the premises are true and the conclusion false. If there is such a row, it is invalid. And it is valid otherwise. (b) We convert the argument into a set of propositions with the premises left intact but the conclusion is negated. If that set of propositions is inconsistent, that is, if there is no row where they are all true, then the original argument is valid. However, if that set is consistent, that is, if there is at least one row where they are all true, then the original argument is invalid.

 

Validity (tested by looking for a row where all premises are true and the conclusion, false. [below: there are none, so it is valid])

P→Q, ¬Q ⊢ ¬P

3.6 valid a

 

Validity (tested by looking for row where all premises are true and the conclusion, false. [below: there is in line 3, so it is invalid])

P → Q, Q ⊢ P

3.6 valid b 

 

Validity (tested by negating the conclusion, making all just a set of propositions, and testing for inconsistency. [Below: since there is no valuation where all are true, the set is inconsistent, and thus the original argument with the unnegated conclusion is valid])

P → Q, P ⊢ Q

{P→Q, P, ¬Q}

3.6 valid d 

 

 

3.7 Short Truth Table Test for Invalidity

 

There is a more efficient way to show the invalidity of an argument than merely filling out the full truth table. This technique is called “forcing”. We first evaluate the conclusion as false and the premises as true. Then, we work backward, finding the assignments for each component term that will make each premise true. If such assignments can be found, then the argument is invalid.

P→Q, R∧¬Q ⊢ Q

3.7 forcing a.2

3.7 forcing b.2

3.7 forcing c

3.7 forcing d

3.7 forcing e

3.7 forcing f

3.7 forcing g

3.7 forcing h

3.7 forcing i

 

 

Ch.4: Truth Trees

 

4.1 Truth-Tree Setup and Basics in Decomposition

 

We will conduct decision procedures using truth-trees. There are three steps in this method: (a) set up the tree for decomposition, (b) decompose decomposable propositions into a non-decomposable form using the decomposition rules, and (c) analyze the completed truth-tree for certain logical properties. The trees are structured with three columns. The left one enumerates the line. The center one gives the proposition. And the right one lists the decomposition rule along with the line where the proposition we are decomposing was previously located. There are nine types of propositions that we can decompose: conjunction, disjunction, conditional, biconditional, negated conjunction [¬(P∧R)], negated disjunction [¬(P∨R)], negated conditional [¬(P→R)], negated biconditional [¬(P↔R)], and double negation. There are three sorts of decompositional patterns: (a) stacking, for when the proposition is true under just one truth-value assignment, (b) branching, for when the proposition is false under just one truth-value assignment, or (c) stacking and branching, for when the proposition is true under two truth-value assignments and false under two truth-value assignments.

 

Step 1: Set up the truth-tree for decomposition

(R∧¬M), R∧(W∧¬M)

4.1 ex a

 

Decomposable proposition types

4.1 list decompose

Stacking Rule structure (its operator makes them true under only one assignment, meaning that they must both have some certain value)

4.1 stack rule

 

Branching Rule structure (its operator makes them false under only one assignment, meaning that either one should have some certain value)

4.1 branch rule

 

Branching and Stacking Rule structure (its operator makes them true under two assignments and false under two other assignments, thus it should have either of two pairings of certain values)

 4.1 branch stack rule

 

 

4.2 Truth-Tree Decomposition Rules

 

Using the conjunction decomposition rule (∧D), we can decompose a conjunction in a truth tree by stacking the conjuncts in new rows, like this:

4.2 c

We make a check mark on any proposition that we have decomposed. When we apply the disjunction decomposition rule (∨D) we decompose a disjunction by making two branches and placing one disjunct under each, like this:

4.2 m

A tree is fully decomposed when we have decomposed all decomposable propositions in it. A tree branch consists of all the propositions found when we begin with a proposition at the bottom and follow upward through the tree. A branch is a closed branch when it contains a proposition ‘P’ and its literal negation ‘¬P,’ and we place an X marking at the bottom of the branch. A branch is a completed open branch when it is completely decomposed and yet  does not contain such a contradiction. At its  bottom we place an 0 marking. We have a completed open tree only if it has at least one completed open branch. However, we have a closed tree when all branches are closed. According to the decomposition descending rule, we decompose a proposition under every open branch that descends from that proposition. 

 

Conjunction decomposition rule (∧D)

4.2 b

Disjunction decomposition rule (∨D)

z disjunction branching fix

 

Branches

4.2 h

4.2 i

 

Closed branch (contains a formula and its negation)

4.2 k

 

Open branch (does not have contradiction)

4.2 l

 

Completed open branch (does not have contradiction, and fully decomposed [Below: left branch])

4.2 m 

 

Completed open tree (has at least one completed open branch) [see above]

 

Closed tree (all branches are closed)

4.2 o

 

Decomposition descending rule (when decomposing a proposition, decompose it under all branches descending from that proposition)

4.2 q

 

 

 

4.3 The Remaining Decomposition Rules

 

With all the truth-tree decomposition rules covered, we may place them together in one table.
4.3.8 full chart.fix1

 

Conditional Decomposition (→D)

4.3.1 conditional branch rule.fix 2

4.3.1 conditional branch ex1.d

 

Biconditional Decomposition (↔D)

4.3.2 c biconditional chart

4.3.2 d biconditional ex e

 

Negated Conjunction Decomposition (¬∧D)

4.3.3 c negated conditional chart

4.3.2 d biconditional ex g

 

Negated Disjunction Decomposition (¬∨D)

 

4.3.4 c negated disjunction chart

4.3.4 d negated disjunction ex g

 

Negated Conditional Decomposition (¬→D)

 

4.3.5 b negated implication chart

4.3.5 b negated implication ex j

 

Negated Biconditional Decomposition (¬↔D)

4.3.6 b negated biconditional chart

4.3.3 b negated conditional ex g

 

Double Negation Decomposition (¬¬D)

4.3.7 a double negation chart

4.3.7 a double negation ex g

 

 

4.4 Basic Strategies

 

In order to decompose a proposition in a truth-tree as efficiently as possible, we should follow certain rules, namely:

Strategic rule 1: Use no more rules than needed.
Strategic rule 2: Use rules that close branches.
Strategic rule 3: Use stacking rules before branching rules.
Strategic rule 4: Decompose more complex propositions before simpler propositions.

For the first rule, if for example we just need to know if the tree is open, we only need to find one completed open branch, and we can leave the rest unfinished. For the second rule, we can minimize the amount of decompositions by closing off branches as early as possible. For the third rule, we minimize the decompositions by stacking before branching; for, if we branch first, we have to repeat more operations, as we have more branches along which to stack formula. And for the fourth rule, by decomposing complex propositions first, we do not have to repeat complex ones further down, as we multiply the branches as we go.

 

 

4.5 Truth-Tree Walk Through

 

By using the four rules of truth-tree decomposition that Agler lays out, we can more efficiently decompose a tree, as seen in a number of examples. In certain cases, we will not even need to decompose all the propositions, on account of the fact that we were able to close all branches as early as possible.

 

 

4.6 Logical Properties of Truth Trees

 

By using truth trees we can test for logical properties of individual propositions, sets of propositions, and arguments.

Consistency: a set of propositions is consistent if their truth tree is completed open, that is to say, if we find at least one open branch.

Inconsistency: a set of propositions is inconsistent when their truth tree is closed, that is to say, when all its branches are closed.

Tautology: a singular proposition is a tautology when the truth tree for its negation is closed.

Contradiction: a singular proposition is a contradiction, when its truth tree is closed.

Contingency: supposing that we have already determined that a singular proposition is not a tautology, then it is a contingency rather than a contradiction, if its truth tree is open. In other words, if the neither the tree for the proposition nor for its negation is closed, then it is a contingency.

Equivalence: two propositions are equivalent if the tree for their negated biconditional combination is closed.

Validity: an argument is valid if the set of propositions made of the premises and the negated conclusion makes a closed tree.

4.6 s

 

On a completed open branch, we can determine truth values for atomic formula which together will make the original formulation true. Unnegated atomic formula get the value true, and negated atomic formula get the value false.

4.6 b

v(R) = T

v(W) = T

v(M) = F

 

We ignore closed branches, because they give us inconsistent values.

 

4.6 c

v(M) = T

v(W) = T

v(R) = T

 

Sometimes a letter is missing on a branch, in which case it is arbitrary which value it has.

4.6 d .n

4.6 e 3

[Valuation set 1 and 2 are for the left branch where the S value is arbitrary. Valuation sets 3 and 4 are for the right branch where the R value is arbitrary. Valuation sets 1 and 4 are redundant and do not need to be restated.]

 

Consistency. A truth-tree shows a set of propositions to be consistent when it is a completed open tree, that is, if there is at least one completed open branch. [In other words, there is at least one valuation for all the formulas that makes them all true.]

[Below: there is one open branch, so they are consistent.]

4.6 c

 

[Below: there are no open branches, so they are inconsistent]

 

4.5 ex3 g

 

Tautology. Iff a formula’s negated form creates a closed tree, then it is a tautology. [A formula is a tautology iff it is true under all valuations.]

[Below: the formula’s negation creates a closed tree and is thus a tautology.]

P∨¬P

4.6 i 1

 

[Below: the negated form creates an open tree, so it is not a tautology.]

P→(Q∧¬P)

4.6 j 7

 

Contradiction. Iff a formula’s truth tree is closed (that is, when all branches close), it is a contradiction. [A formula is a contradiction iff it is false under all valuations.]

4.6 j 8

 

Contingency. Iff a formula’s truth-tree is not closed and if the tree for the formula’s negated form is not closed, then it is a contingency. [A formula is a contingency iff it is neither always false under all valuations nor always true.] [Below: we determined this formula above to not be a tautology. We now test it for contradiction and contingency. Since it makes an open tree, that means it is not a contradiction, and since it is not also a tautology, it is a contingency.]

P→(Q∧¬P)

4.6 k 1

 

Equivalence. Iff the negated biconditional of two formulas creates a closed tree, then they are equivalent. [Two formula are equivalent if they have identical truth values under every valuation.] [Below: the pair’s negated biconditional creates a closed tree, so they are equivalent.]

P∨¬P

¬(P∧¬P)

4.6 n

 

Validity. Iff a truth-tree for the premises and negated conclusion is closed, then it is a valid argument. [An argument is valid if it is impossible for its premises to be true and its conclusion false.] [Below: the tree for the premises with the negated conclusion is closed and thus the argument is valid.]

P→Q, P ⊢ Q

4.6 p

 

 

Ch.5: Propositional Logic Derivations

 

5.1 Proof Construction

 

We can use a natural deduction system in order to make proofs for the conclusions of arguments. Such a system provides derivation rules, which allow us to move forward in a proof by obtaining new propositions on the basis of previously established ones. When a conclusion is provable by means of such a system, we say that the conclusion is a syntactical consequence of, or that that it is syntactically entailed by, the premises. And to signify this we use the turnstile symbol, as for example in this argument: R∨S, ¬S ⊢ R.  If we have simply ⊢P, that means there is a proof of P or that P is a theorem. In our proofs, there are three columns. The left column gives the line number. The central column shows the proposition. And the right column gives the justification, which is either that the proposition is a premise, in which case we write ‘P’, or that it is derived from other propositions, in which case we list the line numbers of those other propositions and write the abbreviation for the derivation rule that was used.

5.1 b

 

 

5.2 Premises and the Goal Proposition

 

In a proof, we can for convenience write the goal proposition, which is also the conclusion, next to the ‘P’ for the final premise, in the justification column.

5.2 b

 

 

5.3 Intelim Derivation Rules

In a proof we can use derivation rules to derive the conclusion through some number of steps. One set of such rules are introduction and elimination rules, called intelim derivation rules. (introduction-elimination). In some cases we will need to make subproofs that begin with assumed propositions.

5.3.3 d2

In a subproof, we will use our rules to make more derivations. Eventually we can arrive upon a proposition to which we may apply certain other rules that will allow us to make a derivation in an outer proof. Upon doing so we discharge the assumption (rendering it inoperable) thereby closing the subproof. There can be subproofs within other subproofs.  We can use propositions within one layer of proof, so long as it is one unbroken subproof. And we can import into a subproof any propositions from outer layers to the left.

5.3.3 f

However, we cannot export propositions from a subproof to another separate subproof at the same level, nor can we export a proposition from a subproof into layers to the left of it.

5.3.3 f2

However, certain derivation rules allow us to make a derivation in an outer level of a proof on the basis of what was derived in a subproof one level to the right of it. The intelim derivation rules are summarized in the following chart:

5.5.1 z2a15.5.1 z2a25.5.1 z2a35.5.1 z2a4

 

Conjunction Introduction (∧I)

 

5.3.1 a

W, Q, R ⊢ W∧R

5.3.1 b2

 

Conjunction Elimination (∧E)

5.3.2 a

(A→B)∧(C∧D) ⊢ D

5.3.2 c

 

Conditional Introduction (→I)

5.3.4 a

Q ⊢ P→Q

5.3.4 b1

 

Conditional Elimination (→E)

5.3.5 a

(A∨Β)→C, A, A∨B ⊢ C

5.3.5 b

 

Reiteration (R)

5.3.5 d5

5.3.6 j

 

Negation Introduction (¬I) and Negation Elimination (¬E)

5.3.7 a

5.3.7 b

 

5.3.7 c1

 

5.3.7 d1

 

Disjunction Introduction (∨I)

5.3.8 a

5.3.8 b1

5.3.8 c1

 

Disjunction Elimination (∨E)

 

5.3.9 a

5.3.9 d original

 

Biconditional Elimination and Introduction (↔E and ↔I)

5.3.10 a

P→Q,Q→P ⊢ P↔Q

5.3.10 b1

 

5.3.10 c

image

 

 

5.4 Strategies for Proofs

 

Certain guidelines can enable us to figure out proofs more effectively. There are two rules that do not involve assumptions, called strategic proof rules (SP#). They are:

SP#1(E)

First, eliminate any conjunctions with ‘∧E,’ disjunctions with ‘∨E,’ conditionals with ‘→E,’ and biconditionals with ‘↔E.’ Then, if necessary, use any necessary introduction rules to reach the desired conclusion.

SP#2(B)

First, work backward from the conclusion using introduction rules (e.g., ‘∧I,’ ‘∨I,’‘ →I,’ ‘↔I’). Then, use SP#1(E).

And there are four rules that do involve assumptions:

SA#1(P,¬Q)

If the conclusion is an atomic proposition (or a negated proposition), assume the negation of the proposition (or the non-negated form of the negated proposition), derive a contradiction, and then use ‘¬I’ or ‘¬E.’

SA#2(→)

If the conclusion is a conditional, assume the antecedent, derive the consequent, and use ‘→I.’

SA#3(∧)

If the conclusion is a conjunction, you will need two steps. First, assume the negation of one of the conjuncts, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ Second, in a separate subproof, assume the negation of the other conjunct, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ From this point, a use of ‘∧I’ will solve the proof.

SA#4(∨)

If the conclusion is a disjunction, assume the negation of the whole disjunction, derive a contradiction, and then use ‘¬I’ or ‘¬E.’

 

SP#1(E) [First, eliminate any conjunctions with ‘∧E,’ disjunctions with ‘∨E,’ conditionals with ‘→E,’ and biconditionals with ‘↔E.’ Then, if necessary, use any necessary introduction rules to reach the desired conclusion.]

P→(R∧M), (P∧S)∧Z ⊢ R

5.4.1 a

 

SP#2(B) [First, work backward from the conclusion using introduction rules (e.g., ‘∧I,’ ‘∨I,’‘ →I,’ ‘↔I’). Then, use SP#1(E).]

P→R, Z→W, P ⊢ R∨W

5.4.1 c7

 

SA#1(P,¬Q) [If the conclusion is an atomic proposition (or a negated proposition), assume the negation of the proposition (or the non-negated form of the negated proposition), derive a contradiction, and then use ‘¬I’ or ‘¬E.’]

P→Q, ¬Q ⊢ ¬P

5.4.1-d1_thumb

 

SA#2(→) [If the conclusion is a conditional, assume the antecedent, derive the consequent, and use ‘→I.’]

R ⊢ P→R

5.4.1-g1_thumb

 

SA#3(∧) [If the conclusion is a conjunction, you will need two steps. First, assume the negation of one of the conjuncts, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ Second, in a separate subproof, assume the negation of the other conjunct, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ From this point, a use of ‘∧I’ will solve the proof.]

¬(P∨Q) ⊢ ¬P∧¬Q

5.4.1-j_thumb

 

SA#4(∨) [If the conclusion is a disjunction, assume the negation of the whole disjunction, derive a contradiction, and then use ‘¬I’ or ‘¬E.’]

¬(¬P∧¬Q) ⊢ P∨Q

5.5-l1_thumb

 

 

5.5 Additional Derivation Rules (PD+)

 

The set of 11 “intelim” propositional derivation rules, called PD, just by themselves can lead to lengthy proofs, so to them we add six more rules, to make a system called PD+. The following chart shows all of PD+, with the new ones being 13-17.

5.5.1 z2a15.5.1 z2a25.5.1 z2a35.5.1 z2a4

5.5.1 z2a55.5.1 z2a6

 

Disjunctive Syllogism (DS)

5.5.1 a

P∨Q, ¬Q ⊢ P

5.5.1 d1

 

Modus Tollens (MT)

5.5.1 g

(P∧Z)→(Q∨Z), ¬(Q∨Z) ⊢ ¬(P∧Z)

5.5.1 m

 

Hypothetical Syllogism (HS)

5.5.1 i

P→Q, Q→R ⊢ P→R

[no image for this]

 

Double Negation (DN)

5.5.1 n_thumb[1]

[no image/example]

 

De Morgan’s Laws (DeM)

5.5.1 o.2

¬(P∧Q) ⊢ ¬P∨¬Q

5.5.1 q1

 

¬(P∨Q) ⊢ ¬P

5.5.1 r1

 

Implication (IMP)

5.5.1 s

¬(P→Q) ⊢ ¬Q

5.5.1 x1

 

 

5.6 Additional Derivation Strategies

 

After revising our strategic rules for proof solving, they are in their entirety the following [the first three being this section’s modifications]:

SP#1(E+): First, eliminate any conjunctions with ‘∧E,’ disjunctions with DS or ‘∨E,’ conditionals with ‘→E’ or MT, and biconditionals with ‘↔E.’ Then, if necessary, use any introduction rules to reach the desired conclusion.

SP#2(B): First, work backward from the conclusion using introduction rules (e.g., ‘∧I,’‘∨I,’‘→I,’‘↔I’). Then, use SP#1(E).

SP#3(EQ+): Use DeM on any negated disjunctions or negated conjunctions, and then use SP#1(E). Use IMP on negated conditionals, then use DeM, and then use SP#1(E).

SA#1(P,¬Q): If the conclusion is an atomic proposition (or a negated proposition), assume the negation of the proposition (or the non-negated form of the negated proposition), derive a contradiction, and then use ‘¬I’ or ‘¬E.’

SA#2(→): If the conclusion is a conditional, assume the antecedent, derive the consequent, and use ‘→I.’

SA#3(∧): If the conclusion is a conjunction, you will need two steps. First, assume the negation of one of the conjuncts, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ Second, in a separate subproof, assume the negation of the other conjunct, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ From this point, a use of ‘∧I’ will solve the proof.

SA#4(∨): If the conclusion is a disjunction, assume the negation of the whole disjunction, derive a contradiction, and then use ‘¬I’ or ‘¬E.’

 

SP#1(E+) [First, eliminate any conjunctions with ‘∧E,’ disjunctions with DS or ‘∨E,’ conditionals with ‘→E’ or MT, and biconditionals with ‘↔E.’ Then, if necessary, use any introduction rules to reach the desired conclusion. ]

P→Q, ¬Q, P∨R, R→W ⊢ W

5.6 a1

 

SP#3(EQ+) [Use DeM on any negated disjunctions or negated conjunctions, and then use SP#1(E). Use IMP on negated conditionals, then use DeM, and then use SP#1(E).]

¬[P∨(R∨M)], ¬M→T ⊢ T

5.6 c1

 

 

 

Ch.6: Predicate Language, Syntax, and Semantics

 

6.1 The Expressive Power of Predicate Logic

 

While everything in the language of propositional logic (PL) can be expressed in English, not everything in English can be expressed in PL. In PL, propositions are treated as whole units (symbolized as singular letters) without regard to logical properties internal to the sentences, as for example between subject and predicate and with respect to quantification. Thus we will examine a more expressive language of predicate logic (RL), which is a logic of relations.

[In PL, for the argument:

All humans are mortal.

Socrates is a human.

Therefore Socrates is a mortal.

Each line gets a letter, but the inference cannot be represented.]

 

 

6.2 The Language of RL

 

There are five elements in the language of predicate logic (RL).

1) Individual constants or names of specific items, and they are represented with lower case letters spanning from ‘a’ to ‘v’ (and expanded with subscript numerals).

2) n-place predicates, which predicate a constant or variable, or they relate constants or variables to one another. They are represented with capital letters from ‘A’ to ‘Z’ (and expanded with subscript numerals).

3) Individual variables, which can be substituted by certain constants, and they are represented with lowercase (often italicized) letters spanning from ‘w’ to ‘z’ (and expanded with subscript numerals).

4) Truth functional operators and scope indicators from the language of propositional logic (PL), namely, ¬, ∧, ∨, →, ↔, (, ), [, ], {, }.

5) Quantifiers, which indicate what portion of the set of items that can stand for a variable are to be taken into consideration in a part of a formulation. When indicating that the full portion is to be considered in some instance of a variable in a formula, we use the universal quantifier ∀. We can understand it to mean “all,” “every,” and “any.” But if we are to consider only a portion of the possible items that can substitute in for a variable, then we use the existential quantifier ∃, which can mean “some,” “at least one,” and the indefinite determiner “a.”

The purpose of the language of predicate logic is to express logical relations holding within propositions. There is the simple relation of predication to a subject, which would be a one-place predicate. There are also the relations of items within a predicate, as in “... is taller than ...”, which in this case is a two-place predicate, and so on. To say John is tall we might write Tj, and to write John is taller than Frank we could write Tjf. The number individuals that some predicate requires to make a proposition is called its adicity. And  when all the names have been removed from a predicational sentence, what remains is called an unsaturated predicate or a rheme. We might also formulate those above propositions using variables rather than constants, as in Tx and Txy. When dealing with variables, the domain of discourse D is the set of items that can be substituted for the variables in question, and this possible substitutions are called substitution instances for variables or just substitution instances. The domain is restricted if it contains only certain things and it is unrestricted if it includes all things. We may either explicitly stipulate what the domain is, which is common in formal logic, or the context of a discussion might implicitly determine the domain, and this domain can fluidly change as the discussion progresses. Also, in these cases with variables, we might further specify the quantities of the variables that we are to consider. So to say, everyone is taller than Frank we might write, (∀x)Txf. Someone is taller than Frank might be (∃x)Txf. Quantifiers have a scope in the formulation over which they apply. They operate just over the propositional contents to the immediate right of the quantifier or just over the complex propositional contents to the right of the parentheses.

 

6.2 a.elements of RL

 

Individual Constants (Names) and n-Place Predicates.

John is standing between Frank and Marry.

j = John

f = Frank

m = Mary

S = __ is standing between __ and __

Sjfm

 

Domain of Discourse, Individual Variables, and Quantifiers.

The domain of discourse D is all of the objects we want to talk about or to which we can refer.

D: positive integers

 

Individual variables are placeholders whose possible values are the individuals in the domain of discourse.

Bxyz = x is between y and z.

 

Universal quantifier: ∀. In English, “all,” “every,” and “any.”

(∀x)Mx

Everyone is mortal.

For every x, x is mortal.

All x’s are mortal.

For any x, x is mortal.

Every x is mortal.

 

Existential quantifier: ∃. In English, “some,” “at least one,” and the indefinite determiner “a.”

(∃x)Hx

Someone is happy.

For some x, x is happy.

Some x’s are happy.

For at least one x, x is happy.

There is an x that is happy.

 

Parentheses and Scope of Quantifiers

The ‘∀’ and ‘∃’ quantifiers operate over the propositional contents to the immediate right of the quantifier or over the complex propositional contents to the right of the parentheses. (Agler  254, quoting)

(∃x)Fx

[Above: ∃x ranges over Fx]

¬(∃x)(Fx∧Mx)

[Above: ¬(∃x) ranges over (Fx∧Mx)]

¬(∀x)Fx∧(∃y)Ry

[Above: ¬(∀x) ranges over Fx and (∃y) rangers over Ry]

(∃x)(∀y)(Rx↔My)

[Above: (∃x) ranges over (∀y)(Rx↔My) and (∀y) ranges over (Rx↔My)]

 

 

6.3 The Syntax of RL

 

In the language of predicate logic (RL), variables are either bound or free. They are bound if they fall under the scope of a quantifier that is quantifying specifically for that particular variable, and it is a free variable otherwise. An open sentence or an open formula is one with an n-place predicate P followed by n terms, where at least one of those variables is free. However, a closed sentence or a closed formula is one with an n-place predicate P followed by n terms, where none of those terms are free variables. The main operator in a well formed formula (wff) in RL is the one with the greatest scope, which means that the one that falls under no other operator’s scope is the main one. We consider the quantifiers as operators. Thus in (∃x)(Px∧Qx) the main operator is ∃x, because all the rest of the formula falls under the quantifier’s scope, and in ¬(∃x)(Px∧Qx) the main operator is the negation, because the quantifier falls under its scope, and the rest of the formula falls under the quantifier’s scope. And there are five rules that determine a wff in RL:  (i) An n-place predicate ‘P’ followed by n terms (names or variables) is a wff. (ii) If ‘P’ is a wff in RL, then ‘¬P’ is a wff. (iii) If ‘P’ and ‘Q’ are wffs in RL, then ‘PQ,’ ‘PQ,’ ‘PQ,’ and ‘PQ are wffs. (iv) If ‘P’ is a wff in RL containing a name ‘a,’ and if ‘P(x/a)’ is what results from substituting the variable x for every occurrence of ‘a’ in ‘P,’ then ‘(∀x)P(x/a)’ and ‘(∃x)P(x/a)’ are wffs, provided ‘P(x/a)’ is not a wff. (v) Nothing else is a wff in RL except that which can be formed by repeated applications of (i) to (iv).

Free and Bound Variables

(∀x)(Fx→Bx)∨Wx

[Above: Wx does not fall under the scope of the quantifier and thus its variable is free.]

(∃z)(Pxy∧Wz)

[Above: only the z of Wz is bound.]

 

Main Operator in Predicate Wffs

[The main operator has the greatest scope.]

(∃x)(Px∧Qx)

[Above: (∃x) is the main operator.]

(∃x)(Px)∧(∃x)(Qx)

[Above: ∧ is the main operator.]

¬(∃x)(Px∧Qx)

[Above: ¬ is the main operator.]

(∀y)(∃x)(Rx→Py)

[Above: (∀y) is the main operator.]

 

The Formal Syntax of RL: Formation Rules.

Open formula: An open formula is a wff consisting of an n-place predicate ‘P’ followed by n terms, where one of those terms is a free variable.

Closed formula: A closed formula is a wff consisting of an n-place predicate ‘P’ followed by n terms, where every term is either a name or a bound variable.

 

Rule (iv): If ‘P’ is a wff in RL containing a name ‘a,’ and if ‘P(x/a)’ is what results from substituting the variable x for every occurrence of ‘a’ in ‘P,’ then ‘(∀x)P(x/a)’ and ‘(∃x)P(x/a)’ are wffs, provided ‘P(x/a)’ is not a wff.

Consider the wff:

Pb

Were we to say that we substitute every instance of the constant ‘b’ with the variable ‘x’, we would write:

P(x/b)

And we would obtain:

Px

It becomes a wff when we add a quantifier:

(∀x)Px

(∃x)Px

 

 

6.4 Predicate Semantics

 

In the language of predicate logic (RL) we will want to give interpretations for constants and for predicate formulations and also to give truth evaluations for well-formed formulas (wffs), including when they involve quantifiers. To do these things, we construct models, which specify the domain as well as the interpretation functions that assign objects in the domain to names, and n-tuples of objects to n-place predicates. On their basis, we valuate truth and falsity for formulas using a function that assigns T for when the named objects are among the tuples in the predicate’s interpretation, and F otherwise. When truth-evaluating quantified wffs in RL, we, intuitively speaking, give a universally quantified formula the value T if all substitutions for the variables make the formula true, and it is F otherwise; and for existentially quantified formulas, if there is at least one substitution that makes the formula true, it is evaluated as T, and F otherwise. For certain technical reasons, the actual procedure cannot involve substituting every possible name or object into the variables. We instead need to test for every object in the domain by thinking of each one being a potential interpretation for a constant, then seeing if these variant interpretations make the formula true or not. Despite the differences in notation, however, the more proper procedure conducts basically the same operation as the intuitive one. The evaluation rules for wffs in RL are the following:

1    v(Rai) = T if and only if the interpretation of ‘ai’ is in ‘R.’

2    vP) = T iff v(P) = F

vP) = F iff v(P) = T

3    v(PQ) = T iff v(P) = T and v(Q) = T

v(PQ) = F iff v(P) = F or v(Q) = F

4    v(PQ) = T iff either v(P) = T or v(Q) = T

v(PQ) = F iff v(P) = F and v(Q) = F

5    v(PQ) = T iff either v(P) = F or v(Q) = T

v(PQ) = F iff v(P) = T and v(Q) = F

6    v(PQ) = T iff either v(P) = T and v(Q) = T or v(P) = F and v(Q) = F

v(PQ) = F iff either v(P) = T and v(Q) = F or v(P) = F and v(Q) = T

7    v(∀x)P = T iff for every name ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) = T.’

v(∀x)P = F iff for at least one ‘a’ not in ‘P’ and at least one a-variant interpretation ‘P(a/x) = F.’

8    v(∃x)P = T iff for at least one name ‘a’ not in ‘P’ and at least one a-variant interpretation ‘P(a/x) = T’.

v(∃x)P = F iff for every name ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) = F.

 

 

 

6.5 Translation from English to Predicate Logic

 

To translate between the language of predicate logic RL and colloquial English, we need to make a translation key to assign symbols to text, and we would follow the pattern given in this table:

agler predicate translation chart 1

 

 

6.6 Mixed and Overlapping Quantifiers

 

There is no simple and universally reliable procedure for translating all English sentences with quantifiers into the language of predicate logic (RL). Agler offers four steps that can help us make translations [quoting:] {1} Identify and symbolize any English expressions that represent quantifiers (and their bound variables) and propositional operators. {2} Translate any ordinary language predicates into predicates of RL. {3} Use the quantifiers from step 1 and the predicates from step 2 and represent the proposition that (1) expresses. {4} Read the predicate logic wff in English and check to see whether it captures the meaning of the sentence undergoing translation. When we have two quantifiers and they are both the same kind, their order will not matter. However, if one is universal and the other is existential, their order can change the meaning of the proposition. Here are four scenarios that illustrate:

(∀x)(∃y)Lxy     “Crush scenario”

Everyone loves someone.

[For any person, that person loves one (and/or another) person.]

6.6 quantifier pictures 1.c

(∃y)(∀x)Lxy     “Santa Claus scenario”

Someone is loved by everyone.

[There is one (and/or another) person who is loved by all other people.]

6.6 quantifier pictures 2.c

(∀y)(∃x)Lxy    “Stalker scenario”

Everyone is loved by someone.

[Each person is loved by at least one (and/or another) person.

6.6 quantifier pictures 4.c

(∃x)(∀y)Lxy   “Loving God scenario”

Someone loves everyone.

[One (and/or another) person loves every other person.]

6.6 quantifier pictures 5.c

 

 

Ch.7: Predicate Logic Trees

 

7.1 Four New Decomposition Rules

 

We can decompose quantified propositions of the language of predicate logic (RL) into logic trees, by using additional rules.

Negated Existential
Decomposition (¬∃D)

Negated Universal
Decomposition (¬∀D)

¬(∃x)P✔
(∀x)¬P

¬(∀x)P✔
(∃x)¬P

[Note that when the negation is moved, if there is a quantifier to the right, the negation is moved to the quantifier rather than jumping over to the proposition further to the right. So ¬(∃x)(∀y)Pxy becomes (∀x)¬(∀y)Pxy and not (∀x)(∀y)¬Pxy.]

Existential Decomposition
(∃D)

Universal Decomposition (∀D)

(∃x)P✔
P(a/x)

 

where ‘a’ is an individual constant (name) that does not previously occur in the branch.

(∀x)P
P
(a/x)

 

where ‘a’ is any individual constant
(name).

[Note that the universal quantifier is not checked, because for infinite domains, not all possible substitutions can be given in the tree.] The universal decomposition is further specified as:

Universal Decomposition (∀D)

(∀x)P
P(a . . . v/x)


Consistently replace every bound x with any individual constant (name) of your choosing (even if it already occurs in an open branch) under any (not necessarily both) open branch of your choosing.

 
Four kinds of decomposable propositions.
 

Four Decomposable Proposition Types

Existential                   (∃x)P

  Universal                    (∀x)P

Negated existential      ¬(∃x)P     

Negated universal        ¬(∀x)P  

 

Negated Existential and Universal Decomposition.

Negated Existential
Decomposition (¬∃D)

Negated Universal
Decomposition (¬∀D)

¬(∃x)P✔
(∀x)¬P

¬(∀x)P✔
(∃x)¬P

 
1

¬(∃x)Px

P

2

¬(∀y)Wy

P

3

(∀x)¬Px

1¬∃D

4

(∃y)¬Wy

2¬∀D

 
Existential and Universal Decomposition.

Existential Decomposition
(∃D)

Universal Decomposition (∀D)

(∃x)P✔
P(a/x)

 

where ‘a’ is an individual constant (name) that does not previously occur in the branch.

(∀x)P
P
(a/x)

 

where ‘a’ is any individual constant
(name).

 
According to (∃D) and (∀D), an individual constant (name) is substituted for a bound variable in a quantified expression. This procedure is symbolized as ‘P(a/x)’ (i.e., replace x with ‘a’). Thus, if there is a quantified expression of the form ‘(∀x)P’ or ‘(∃x)P,’ a substitution instance of ‘P(a/x)’ replaces x’s bound by the quantifier with ‘a.’ (quoting Agler 286)
 

Universal Decomposition (∀D)

(∀x)P
P(a . . . v/x)


Consistently replace every bound x with any individual constant (name) of your choosing (even if it already occurs in an open branch) under any (not necessarily both) open branch of your choosing.

1

(∀x)(Px→Rx)

P
2

Pa∨Ra

P

3

/               \
Pa              Ra


2∨D
4

Pa→Ra         Pa→Ra

1∀D
 

Existential Decomposition
(∃D)

(∃x)P✔
P(a/x)

 

where ‘a’ is an individual constant (name) that does not previously occur in the branch.

1

(∃x)Px

P
2

Pa

P
3

Pb

1∃D
 
 
 

7.2 Strategies for Decomposing Trees

 

To decompose propositions in the language of predicate logic (RL) the most efficiently in truth trees, we should follow these rules.
 

Strategic Rules for Decomposing Predicate Truth Trees

1. Use no more rules than needed.

2. Decompose negated quantified expressions and existentially quantified expressions first.

3. Use rules that close branches.

4. Use stacking rules before branching rules.

5. When decomposing universally quantified propositions, use constants that already occur in the branch.

6. Decompose more complex propositions before simpler propositions.

 

 

7.3 Logical Properties

 

Truth trees can determine logical properties of singular propositions, sets of propositions, and arguments in the language of predicate logic (RL). To test a proposition to see if it is a contradiction, we see if it makes a closed tree. If and only if it does is it a contradiction. To test a proposition to see if it is a tautology, we make a tree for its negation. If and only if it makes a closed tree is the original proposition a tautology. If it fails both these tests, it is a contingency. To check for the consistency of sets of propositions, we see if their tree has at least one open branch. If so, they are consistent. If instead the tree is closed, then they are inconsistent. To test for the validity of an argument, we negate the conclusion and make a tree for the full set of sentences. If they determine a closed tree, the original argument was valid. If it determines a tree with at least one completed open branch, it is invalid.

Completed open branch: A branch is a completed open branch if and only if (1) all complex propositions that can be decomposed into atomic propositions or negated atomic propositions are decomposed; (2) for all universally quantified propositions ‘(∀x)P’ occurring in the branch, there is a substitution instance ‘P(a/x)’ for each constant that occurs in that branch; and (3) the branch is not a closed branch.

1
(∀x)(¬Px→¬Rx)
P
2
(∀x)(Rx→Px)
P
3
¬Pa→¬Ra✔
1∀D
4
Ra→Pa✔
/                      \
2∀D
5
¬¬Pa                       ¬Ra
                               /             \                /            \
3→D
6
                        ¬Ra               Pa        ¬Ra             Pa
                           O                  O            O              O
4→D

 
Closed tree: A tree is a closed tree if and only if all branches close.
Closed branch: A branch is a closed branch if and only if there is a proposition and its literal negation (e.g., ‘P’ and ‘¬P’). (quoting Agler 294)

1
(∀x)(Px→Qx)
P
2
(∃x)(Px∧¬Qx)✔
P
3
Pa→Qa✔
1∀D
4
Pb∧¬Qb✔
2∃D
5
Pb
4∧D
6
¬Qb
/                  \
/                              \
4∧D
7
                               ¬Pa                                    Qa
3→D
8
                            Pb→Qb                            Pb→Qb
                        /                \                        /                \
1∀D
9
                   ¬Pb              Qb                 ¬Pb              Qb
                      X                 X                     X                  X
8→D

 
Consistency: A set of propositions ‘{A, B, C, ..., Z}’ is consistent in RL if and only if there is at least one interpretation such that all of the propositions in the set are true.” (Agler 296)
to show that ‘{(∀x)Px, (∃x)Rx}’ is consistent in RL involves showing that there is at least one interpretation in a model where v(∀x)Px = T and v(∃x)Rx = T. Here is an example of such a model:
D = positive integers
P = {x | x is greater than 0}
R = {x | x is even}
(Agler 296)
“Thus, the presence of a completed open branch tells us that we can construct a model such that every proposition in the stack is true” (Agler 297).
[“Completed open branch: A branch is a completed open branch if and only if (1) all complex propositions that can be decomposed into atomic propositions or negated atomic propositions are decomposed; (2) for all universally quantified propositions ‘(∀x)P’ occurring in the branch, there is a substitution instance ‘P(a/x)’ for each constant that occurs in that branch; and (3) the branch is not a closed branch.”]
1
(∃x)Px
P
2
Pa
P
3
Pb
O
1∃D
 
Inconsistency: A set of propositions ‘{P, Q, R, ..., Z}’ is shown by the truth-tree method to be inconsistent if and only if a tree of the stack of ‘P,’ ‘Q,’ ‘R,’ . . ., ‘Z’ is a closed tree; that is, all branches close.

(∀x)(Px→Rx), ¬(∀x)(¬Rx→¬Px)

1
(∀x)(Px→Rx)
P
2
¬(∀x)(¬Rx→¬Px)✔
P
3
(∃x)¬(¬Rx→¬Px)✔
2¬∀D
4
¬(¬Ra→¬Pa)
3∃D
5
Pa→Ra✔
1∀D
6
¬Ra
4¬→D
7
¬¬Pa✔
4¬→D
8
Pa
/                  \
7¬¬D
9
¬Pa                    Ra
  X                      X
5→D
 
“the truth-tree method can be used to determine whether a proposition ‘P’ is a tautology, contradiction, or contingency. In testing ‘P’ to see if it is a tautology, begin the tree with ‘¬P.’ If the tree closes, you know that it is a tautology. If the tree is open, then ‘P’ is either a contradiction or a contingency. Similarly, in testing ‘P’ to see if it is a contradiction, begin the tree with ‘P.’ If the tree closes, you know that it is a contradiction. If the tree is open, then ‘P’ is either a tautology or a contingency. Lastly, if the truth-tree test shows that ‘P’ is neither a contradiction nor a tautology, then ‘P’ is a contingency” (Agler 147).

(∃x)¬(∀y)[Px→(Qx∨¬Ry)]

[Below: There is at least one completed open branch, so it is not a contradiction.]

1
(∃x)¬(∀y)[Px→(Qx∨¬Ry)]✔
p
2
¬(∀y)[Pa→(Qa∨¬Ry)]✔
1∃D
3
(∃y)¬[Pa→(Qa∨¬Ry)]✔
2¬∀D
4
¬[Pa→(Qa∨¬Rb)]✔
3∃D
5
Pa
4¬→D
6
¬(Qa∨¬Rb)✔
4¬→D
7
¬Qa
6¬∨D
8
¬¬Rb
6¬∨D
[Below: Since the above tree is not closed, the original form is not a tautology. Since it is neither a contradiction nor a tautology, it is therefore a contingency.]

1
¬(∃x)¬(∀y)[Px→(Qx∨¬Ry)]✔
P
2
(∀x)¬¬(∀y)[Px→(Qx∨¬Ry)]✔
1¬∃D
3
¬¬(∀y)[Pa→(Qa∨¬Ry)]✔
2∀D
4
(∀y)[Pa→(Qa∨¬Ry)]
3¬¬D
5
Pa→(Qa∨¬Ra)
/                                    \
4∀D
6
                           ¬Pa                               Qa∨¬Ra✔
                               O                              /                   \
5→D
7
                                                             Qa                 ¬Ra
                                                              O                     O
6∨D

 

Equivalence: A pair of propositions ‘P’ and ‘Q’ is shown by the truth-tree method to be equivalent if and only if the tree of the stack of ‘¬(PQ)’ determines a closed tree; that is, all branches for ‘¬(PQ)’ close. 

(∀x)Px
¬(∃x)Px
[Below: The negated biconditional makes an open tree and thus they are not equivalent.]
1
¬[(∀x)Px↔¬(∃x)Px]✔
/                            \
P
2
                    (∀x)Px                ¬(∀x)Px
1¬↔D
3
               ¬¬(∃x)Px✔             ¬(∃x)Px
1¬↔D
4
                    (∃x)Px✔
3¬¬D
5
                         Pa
4∃D
6
                         Pa
                          O
2∀D
 
(∀x)¬(Px∨Gx)
(∀y)(¬Py∧¬Gy)

[Below: Their negated biconditional makes a closed tree, and thus they are equivalent.]

1
¬{[(∀x)¬(Px∨Gx)]↔[(∀y)(¬Py∧¬Gy)]}✔
/                                                     \
P
2
             (∀x)¬(Px∨Gx)                                ¬(∀x)¬(Px∨Gx)
1¬↔D
3
          ¬(∀y)(¬Px∧¬Gx)✔                         (∀y)(¬Px∧¬Gx)
1¬↔D
4
           (∃y)¬(¬Px∧¬Gx)✔                                          |
3¬∀D
5
               ¬(¬Pa∧¬Ga)✔                                               |
                    /              \                                                      |
4∃D
6
            ¬¬Pa✔    ¬¬Ga✔                                             |
5¬∧D
7
                Pa                 Ga                                                  |
6¬¬D
8
      ¬(Pa∨Ga)✔  ¬(Pa∨Ga)✔                                    |
2∀D
9
             ¬Pa                 ¬Pa                                               |
8¬∨D
10
                X                   ¬Ga                                               |
                                                                                         |
8¬∨D
11
                                                                                (∃x)¬¬(Px∨Gx)✔
2¬∀D
12
                                                                                            Pa∨Ga✔
                                                                                        /                   \
11∃D
13
                                                                                    Pa                   Ga
12∨D
14
                                                                      ¬Pa∧¬Ga     ¬Pa∧¬Ga 3∀D
15
                                                                                 ¬Pa               ¬Ga
                                                                                   X                    X
14∧D

 

 

 

 

 

7.4 Undecidability and the Limits of the Predicate Tree Method

 

When we deal with the problem of universal instantiation causing us to repeat over and over a combination of existential and universal decomposition rules, we can instead apply the following new existential decomposition rule.

7.4 new existential decomposition

 

Unlike PL, RL is undecidable. That is, there is no mechanical procedure that can always, in a finite number of steps, deliver a yes or no answer to questions about | whether a given proposition, set of propositions, or argument has a property like consistency, tautology, validity, and the like. For some trees, the application of predicate decomposition rules will result in a process of decomposition that does not, in a finite number of steps, yield a closed tree or a completed open branch. (quoting Agler 317-318)

 

A use of (N∃D) requires that whenever we decompose an existentially quantified proposition ‘(∃x)P,’ we create a separate branch for any substitution instance for any substitution instance P(a1/x), P(a2/x), ..., P(an/x), already occurring in the branch containing ‘(∃x)P’ and branch a substitution instance ‘P(an+1/x)’ that is not occurring in that branch. (quoting Agler 318)

1

(∀x)(∃y)(Pxy)

P

2

(∃y)Pay✔

/                 \

1∀D

3

Paa                  Pab

2N∃D

4

            O                 (∃y)Pby   

                             /           |           \

1∀D

5

                              Pba        Pbb       Pbc

                            O              O            .

                                                              .

                                                              .

4N∃D

 

 

 

Ch.8: Predicate Logic Derivations

 

8.1 Four Quantifier Rules

 

For the language of predicate logic (RL), there are four proof derivation rules to add to those of propositional logic (PL), namely:

Universal Elimination (E)
From any universally quantified proposition ‘(∀x)P,’ we can derive a substitution instance ‘P(a/x)’ in which all bound variables are consistently replaced with any individual constant (name).
(∀x)P
P(a/x)
∀E
 
Existential Introduction (Ι)
From any possible substitution instance ‘P(a/x),’ an existentially quantified proposition ‘(∃x)P’ can be derived by consistently replacing at least one individual constant (name) with an existentially quantified variable.
P(a/x)
(∃x)P
∃I
 
Universal Introduction (Ι)
A universally quantified proposition ‘(∀x)P’ can be derived from a possible substitution instance ‘P(a/x)’ provided (1) ‘a’ does not occur as a premise or as an assumption in an open subproof, and (2) ‘a’ does not occur in ‘(∀x)P.’
P(a/x)
(∀x)P
∀I
 
Existential Elimination (E)
From an existentially quantified expression ‘(∃x)P,’ an expression ‘Q’ can be derived from the derivation of an assumed substitution instance ‘P(a/x)’ of ‘(∃x)P’ provided (1) the individuating constant ‘a’ does not occur in any premise or in an active proof (or subproof) prior to its arbitrary introduction in the assumption ‘P(a/x),’ and (2) the individuating constant ‘a’ does not occur in proposition ‘Q’ discharged from the subproof.
(∃x)P
   | P(a/x)
   | .
   | .
   | .
   | Q
Q
 
 
 
 
 
 
∃E

This new system of derivation is called RD. And corresponding to these are four additional strategic rules for making proofs in RL.

SQ#1(∀E): When using (∀E), the choice of substitution instances ‘P(a/x)’ should be guided by the individual constants (names) already occurring in the proof and any individual constants (names) occurring in the conclusion.
(Agler 328)

SQ#2(∃I): When using (∃I), aim at deriving a substitution instance ‘P(a/x)’ such that a use of (∃I) will result in the desired conclusion. (In other words, if the ultimate goal is to derive ‘(∃x)Px,’ aim to derive a substitution instance of ‘(∃x)Px,’ like ‘Pa,’ ‘Pb,’ ‘Pr,’ so that a use of (∃I) will result in ‘(∃x)Px.’)

SQ#3(∀I): When the goal proposition is a universally quantified proposition ‘(∀x)P,’ derive a substitution instance ‘P(a/x)’ such that a use of (∀I) will result in the desired conclusion.

SQ#4(∃E) Generally, when deciding upon a substitution instance ‘P(a/x)’ to assume for a use of (∃E), choose one that is foreign to the proof.

Universal Elimination (∀E).
Universal Elimination (E)
From any universally quantified proposition ‘(∀x)P,’ we can derive a substitution instance ‘P(a/x)’ in which all bound variables are consistently replaced with any individual constant (name).
(∀x)P
P(a/x)
∀E
1
(∀x)Px
P
2
Pa
1∀E

Existential Introduction (∃I)

Existential Introduction (Ι)
From any possible substitution instance ‘P(a/x),’ an existentially quantified proposition ‘(∃x)P’ can be derived by consistently replacing at least one individual constant (name) with an existentially quantified variable.
P(a/x)
(∃x)P
∃I

Zr ⊢ (∃x)Zx

1 Zr P
2 (∃x)Zx 1∃I

Universal Introduction (∀I)

Universal Introduction (Ι)
A universally quantified proposition ‘(∀x)P’ can be derived from a possible substitution instance ‘P(a/x)’ provided (1) ‘a’ does not occur as a premise or as an assumption in an open subproof, and (2) ‘a’ does not occur in ‘(∀x)P.’
P(a/x)
(∀x)P
∀I

(∀x)Px ⊢ (∀y)Py
1 (∀x)Px P
2 Pa 1∀E
3 (∀y)Py 2∀I
 
Existential Elimination (∃E)
Existential Elimination (∃E)
From an existentially quantified expression ‘(∃x)P,’ an expression ‘Q’ can be derived from the derivation of an assumed substitution instance ‘P(a/x)’ of ‘(∃x)P’ provided (1) the individuating constant ‘a’ does not occur in any premise or in an active proof (or subproof) prior to its arbitrary introduction in the assumption ‘P(a/x),’ and (2) the individuating constant ‘a’ does not occur in proposition ‘Q’ discharged from the subproof.
(∃x)P
   | P(a/x)
   | .
   | .
   | .
   | Q
Q
 
 
 
 
 
 
∃E

1 (∃x)Px P
2      | Pa A/∃E
3      | (∃y)Py 2∃I
4 (∃y)Py 1,2–3∃E

 

8.2 Quantifier Negation (QN)

 

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

1

¬(∀x)Px

P

2

(∃x)¬Px

1QN

3

¬(∀x)Px

2QN

 

1

¬(∃z)(Wzz∧Mz)

P

2

(∀z)¬(Wzz∧Mz)

1QN

3

¬(∃z)(Wzz∧Mz)

2QN

 

 

8.3 Sample Proofs

 

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

 

 

 

 

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

 

.