Nice quotations/passages:
"'Validty' is not to be thought of as a term of praise. When a schema is valid, any statement whose form that schemata depicts is bound to be, in some sense, trivial. It will be trivial in the sense that it conveys no real information regarding the subject matter whereof its component clauses speak" (42).
"Valid schemata are important not as an end, but as a means" (42).
Important note:
Substitution of schemata for letters preserves validity!
Substitution of schemata for letters preserves inconsistency!
Substitution of schemata for letters DOES NOT preserve consistency!
Bit of homework:
3. Yes. Conjunction is a schema that is merely consistent, not valid. The schema 'pq' then, is consistent. But if you conjunct two distinct valid schemata, the result is a valid schema; if you conjunct two distinct inconsistent schemata, the result is an inconsistent schema.
For example:
(pV-p)(qV-q) is valid, but is simply a substitution of schemata into some consistent schema.
(p↔-p)(q↔-q) is inconsistent, but is simply a substitution of schemata into some consistent schema.
Monday, June 9, 2008
A Trivial Quotation from Methods of Logic, 1.5
A Trivial Quotation:
"Let us feign contact with reality by considering an actual statement of the form 'pqV-p-r.→.q↔r'" (35).
"Let us feign contact with reality by considering an actual statement of the form 'pqV-p-r.→.q↔r'" (35).
Quotes from and a Comment on Methods of Logic, 1.4
Nice quotations/passages:
"We all have extraordinary finesse at ordinary language" (28).
I like his use of the word "vacuous" here: "Grouping may also be indicated in ordinary language by inserting a vacuous phrase such as 'it is the case that', balanced with another 'that' to show cooridination of clauses" (28).
"Parentheses show groupings unfailingly, and are simple to use. They have the further virute of allowing complex clauses to be dropped mechanically into place without distortion of clause or context. This particular virtue has been of incalculable importance; without it mathematics could never have developed beyond a rudimentary stage.
Even so, parentheses can be a nuisance. Unless conventions are adopted for omitting some of them, our longer formulas tend to bristle with them and we find ourselves having to count them in order to pair them off" (29).
Funny: "Students are tempted to tinker with the dot conventions with a view to economy" (31).
Comment:
I like Lukasiewicz's notation.
"We all have extraordinary finesse at ordinary language" (28).
I like his use of the word "vacuous" here: "Grouping may also be indicated in ordinary language by inserting a vacuous phrase such as 'it is the case that', balanced with another 'that' to show cooridination of clauses" (28).
"Parentheses show groupings unfailingly, and are simple to use. They have the further virute of allowing complex clauses to be dropped mechanically into place without distortion of clause or context. This particular virtue has been of incalculable importance; without it mathematics could never have developed beyond a rudimentary stage.
Even so, parentheses can be a nuisance. Unless conventions are adopted for omitting some of them, our longer formulas tend to bristle with them and we find ourselves having to count them in order to pair them off" (29).
Funny: "Students are tempted to tinker with the dot conventions with a view to economy" (31).
Comment:
I like Lukasiewicz's notation.
Notes and Homework from Methods of Logic, 1.3
The Conditional
Notes:
Given common language, the ascription of truth or falsity to a conditional statement in the event that its antecedent is false is, says Quine, "rather more arbitrary". Usually when people make conditional statements they mean to communicate a proposition concerning the consequent that holds only in the case that the antecedent holds. They don't mean to communicate anything about what is the case when the antecedent is not the case.
The most "convenient" decision, says Quine, is to regard all conditionals with false antecedents as true. This makes enough sense to me. This decision seems at least to be kindlier towards regular speaker intentions than regarding such conditionals as false.
The contrafactual conditional ought not to be construed in the fashion of the material conditional. This is because the speaker's intention in making such a conditional claim already assumes the falsehood of the antecedent, but still thinks that something is being communicated of interest by the compound. Such a conditional is not automatically verified by the falsehood of the antecedent.
The contrafactual conditional is not truth-functional. Some c-conditionals that have false antecedents and true consequents are true, while some c-conditionals that have false antecedents and false consequents are true. It is not the truth or falsity of the antecedents and consequents themselves that make the compounds true or false, hence such compounds are not truth-functional.
Interesting that Quine seems to doubt the viability of any coherent theory of contrafactual conditionals. I wonder whether Lewis has anything particular to offer here.
Material conditionals seem irrelevant, since the following statements come out true given its truth-conditions:
* If France is in Europe then the sea is salt.
* If France is in Australia then the sea is salt.
* If France is in Australia then the sea is sweet.
Quine notes, in a kindred spirit to my earlier comment, that it would not seem any less strange to attribute falsity to the statements. He says that rather "the strangeness is intrinsic to the statements themselves."
Homework:
Pretty happy about this. It asks me to translate 'p ↔ q' into terms purely of alternation and negation.
I came up with the following answer:
'-[-(-pVq)V-(pV-q)'
which, incidentally, I think can be translated to (-pVq)V(pV-q).
Anyway, I checked the back of the book, and they gave as an answer:
'-(pVq)V-(-pV-q)'
This deflated me. I felt lousy. I thought I had gotten it right. BUT THEN, just in case, I tested the two for logical equivalence, and found out by dint of tedious work that '-[-(-pVq)V-(pV-q)' is in fact logically equivalent to '-(pVq)V-(-pV-q)'. So, my answer was right after all...you can simply dock me points for lack of concision in my notation (which, I understand, is important. Nevertheless, I technically did what the question asked, and so am happy.
Notes:
Given common language, the ascription of truth or falsity to a conditional statement in the event that its antecedent is false is, says Quine, "rather more arbitrary". Usually when people make conditional statements they mean to communicate a proposition concerning the consequent that holds only in the case that the antecedent holds. They don't mean to communicate anything about what is the case when the antecedent is not the case.
The most "convenient" decision, says Quine, is to regard all conditionals with false antecedents as true. This makes enough sense to me. This decision seems at least to be kindlier towards regular speaker intentions than regarding such conditionals as false.
The contrafactual conditional ought not to be construed in the fashion of the material conditional. This is because the speaker's intention in making such a conditional claim already assumes the falsehood of the antecedent, but still thinks that something is being communicated of interest by the compound. Such a conditional is not automatically verified by the falsehood of the antecedent.
The contrafactual conditional is not truth-functional. Some c-conditionals that have false antecedents and true consequents are true, while some c-conditionals that have false antecedents and false consequents are true. It is not the truth or falsity of the antecedents and consequents themselves that make the compounds true or false, hence such compounds are not truth-functional.
Interesting that Quine seems to doubt the viability of any coherent theory of contrafactual conditionals. I wonder whether Lewis has anything particular to offer here.
Material conditionals seem irrelevant, since the following statements come out true given its truth-conditions:
* If France is in Europe then the sea is salt.
* If France is in Australia then the sea is salt.
* If France is in Australia then the sea is sweet.
Quine notes, in a kindred spirit to my earlier comment, that it would not seem any less strange to attribute falsity to the statements. He says that rather "the strangeness is intrinsic to the statements themselves."
Homework:
Pretty happy about this. It asks me to translate 'p ↔ q' into terms purely of alternation and negation.
I came up with the following answer:
'-[-(-pVq)V-(pV-q)'
which, incidentally, I think can be translated to (-pVq)V(pV-q).
Anyway, I checked the back of the book, and they gave as an answer:
'-(pVq)V-(-pV-q)'
This deflated me. I felt lousy. I thought I had gotten it right. BUT THEN, just in case, I tested the two for logical equivalence, and found out by dint of tedious work that '-[-(-pVq)V-(pV-q)' is in fact logically equivalent to '-(pVq)V-(-pV-q)'. So, my answer was right after all...you can simply dock me points for lack of concision in my notation (which, I understand, is important. Nevertheless, I technically did what the question asked, and so am happy.
Notes and Quotes from Methods of Logic, 1.2
Notes:
A compound is called a truth function of its components if its truth value is determined in all cases by the truth values of the components. More precisely: a way of forming compound statements from component statements is truth-functional if the compounds thus formed always have matching truth values as long as their components have matching truth values.
Interestingly, all truth functions can be sufficiently notated using only negation and conjunction.
A neat routine to employ to this end is listing the schedule of truth values for a given compound, note those cases in which the compound is false, and then simply conjunct the denial of each of these cases. This new compound (using only negation and conjunction) will map on exactly to the aforesaid schedule, denying explicity just those cases in which it was to come out false; in all others cases it comes out true.
The only sort of situation that this routine cannot take care of are those in which one must deal with a compound that comes out true in all cases. Quine says this about it, cleverly: "These trivial exceptions call, then, for separate treatment; and a treatment is straightway forthcoming which is correspondingly trivial" (19).
His trivial treatment is hilarious. We need a compound that will come out true regardless of what truth values are assigned to its components. So, simply deny the conjunction of all the components with the denial of any one of the components. Such a conjunction will come out false in all cases, since it will simultaneously affirm and deny a given component, so that conjunction's denial will come out true in all cases.
The reason for keeping 'V' around is to facilitate certain technical manipulations yet to come, though it is unnecessary. It is handy, not essential. However, conjunction is no less superfluous than alternation. Should we want, we could sufficiently represent all compounds using only negation and alternation.
The single connective '|' is clever. It can single-handedly represent any compound, though uneconomically. 'p|q' is to be true if and only if 'p' and 'q' are not both true. This makes it equivalent to -(p&q). Negation could be represented 'p|p' since this would mean that not both 'p' and 'p' are true, which is just to say that 'p' is not true, and therefore false, or '-p'. Conjunction can be represented as well. Instead of 'pq' we can have '(p|q)|(p|q)', which would be to say '-(p|q)'.
Nice quotations/passages:
Nice way of putting truth-function description: "Any particular truth function can be adequately described by presenting a schedule showing what truth values the compound will take on for each choice of truth values for the components" (17).
"Given a description of a truth function--i.e., given simply a schedule showing what truth values the compound is to take on for each choice of truth values for the components--we can construct a truth function out of negation and conjunction which answers the description" (18).
A compound is called a truth function of its components if its truth value is determined in all cases by the truth values of the components. More precisely: a way of forming compound statements from component statements is truth-functional if the compounds thus formed always have matching truth values as long as their components have matching truth values.
Interestingly, all truth functions can be sufficiently notated using only negation and conjunction.
A neat routine to employ to this end is listing the schedule of truth values for a given compound, note those cases in which the compound is false, and then simply conjunct the denial of each of these cases. This new compound (using only negation and conjunction) will map on exactly to the aforesaid schedule, denying explicity just those cases in which it was to come out false; in all others cases it comes out true.
The only sort of situation that this routine cannot take care of are those in which one must deal with a compound that comes out true in all cases. Quine says this about it, cleverly: "These trivial exceptions call, then, for separate treatment; and a treatment is straightway forthcoming which is correspondingly trivial" (19).
His trivial treatment is hilarious. We need a compound that will come out true regardless of what truth values are assigned to its components. So, simply deny the conjunction of all the components with the denial of any one of the components. Such a conjunction will come out false in all cases, since it will simultaneously affirm and deny a given component, so that conjunction's denial will come out true in all cases.
The reason for keeping 'V' around is to facilitate certain technical manipulations yet to come, though it is unnecessary. It is handy, not essential. However, conjunction is no less superfluous than alternation. Should we want, we could sufficiently represent all compounds using only negation and alternation.
The single connective '|' is clever. It can single-handedly represent any compound, though uneconomically. 'p|q' is to be true if and only if 'p' and 'q' are not both true. This makes it equivalent to -(p&q). Negation could be represented 'p|p' since this would mean that not both 'p' and 'p' are true, which is just to say that 'p' is not true, and therefore false, or '-p'. Conjunction can be represented as well. Instead of 'pq' we can have '(p|q)|(p|q)', which would be to say '-(p|q)'.
Nice quotations/passages:
Nice way of putting truth-function description: "Any particular truth function can be adequately described by presenting a schedule showing what truth values the compound will take on for each choice of truth values for the components" (17).
"Given a description of a truth function--i.e., given simply a schedule showing what truth values the compound is to take on for each choice of truth values for the components--we can construct a truth function out of negation and conjunction which answers the description" (18).
Notes and Quotes from Methods of Logic, 1.1
Notes:
Conjunction is associative, communitative, and idempotent.
Apparently to use "inclusive" can be misleading in certain circumstances when referring to nonexclusive alternation. I wonder what circumstances those are. I, trusting Quine, will adopt "nonexclusive".
We get 'V' as the or symbol from the Latin 'vel'. Apparently, to quote Quine, "Latin has distinct words for the two senses of 'or': vel for the nonexclusive and aut for the exclusive. IN modern logic it is customary to write 'V' reminiscent of 'vel', for 'or' in the nonesclusive sense: 'p v q'."
To logically represent the exclusive sense of 'p or q' just write 'p&-q V -p&q'
Alternation, too, is associative, communitative, and idempotent.
Note that -(p&q) is distinct from -p&-q.
These however are logically equivalent:
(-pV-q), -(p&q).
-(pVq), -p&-q.
Nice quotations/passages:
"Once we have an inventory of all the distinct componenets of a continued conjunction, no further details of the constitution of the conjunction need concern us" (11).
"In a metaphor from genetics, conjunction and alternation may be contrasted thus: in conjunction, truth is recessive and falsity dominant; in alternation, truth is dominant and falsity recessive" (13).
This is funny. In explaining his use of the dash rather than the tilde in notating negation: "Because in the present book there is much negating of single letters, I have here favored Pierce's bar for its compactness and perspicuity; and, given that, the dash in more in keeping than the tilde as negation sign for longer expressions" (15).
Conjunction is associative, communitative, and idempotent.
Apparently to use "inclusive" can be misleading in certain circumstances when referring to nonexclusive alternation. I wonder what circumstances those are. I, trusting Quine, will adopt "nonexclusive".
We get 'V' as the or symbol from the Latin 'vel'. Apparently, to quote Quine, "Latin has distinct words for the two senses of 'or': vel for the nonexclusive and aut for the exclusive. IN modern logic it is customary to write 'V' reminiscent of 'vel', for 'or' in the nonesclusive sense: 'p v q'."
To logically represent the exclusive sense of 'p or q' just write 'p&-q V -p&q'
Alternation, too, is associative, communitative, and idempotent.
Note that -(p&q) is distinct from -p&-q.
These however are logically equivalent:
(-pV-q), -(p&q).
-(pVq), -p&-q.
Nice quotations/passages:
"Once we have an inventory of all the distinct componenets of a continued conjunction, no further details of the constitution of the conjunction need concern us" (11).
"In a metaphor from genetics, conjunction and alternation may be contrasted thus: in conjunction, truth is recessive and falsity dominant; in alternation, truth is dominant and falsity recessive" (13).
This is funny. In explaining his use of the dash rather than the tilde in notating negation: "Because in the present book there is much negating of single letters, I have here favored Pierce's bar for its compactness and perspicuity; and, given that, the dash in more in keeping than the tilde as negation sign for longer expressions" (15).
Quotes from Intro to Methods of Logic
Nice quotations/passages:
"Logic, like any science has as its business the pursuit of truth. What are true are certain statements; and the pursuit of truth is the endeavor to sort out the true statements from the others, which are false."
"Truths are as plentiful as falsehoods, since each falsehood admits of a negation which is true. But scientific activity is not the indiscriminate amassing of truths; science is selective and seeks the truths that count for most, either in point of intrinsic interest or as instruments for coping with the world."
"Physical objects, if they did not exist, would (to transplant Voltaire's epigram) have had to be invented. They are indispensable as the public common denominators of private sense experience."
"Logic and mathematics were coupled as jointly enjoying a central position within the total system of discourse. Logic as commonly presented seems to differ from mathematics in that in logic we talk about statements and their interrelationships, notably implication, whereas in mathematics we talk about abstract nonlinguistic things: numbers, function and like. This contrast is in large part misleading. Logic truths, e.g., statements of the form 'If p and q then q', are not about statements; they may be about anything, depending on what we put in the blank 'p' and 'q'. When we talk about such logical truths, and when we expound implications, we are indeed talking about statements; but so are we when we talk about mathematical truths.
But it is indeed the case that the truths of mathematics treat explicitly of abstract nonlinguistic things, e.g., numbers and functions, whereas the truths of logic, in a reasonably limited sense of the word 'logic', have no such entities as specific subject matter. Despite this difference, however, logic in its higher reaches is found to bring us by natural stages into mathematics. For, it happens that certain unobtrusive extensions of logical theory carry us into a realm, sometimes also called 'logic' in a broad sense of the word, which does have abstract entities of a special kind as subject matter. These entities are classes; and the logical theory of classes, or set theory, proves to be the basic discipline of pure mathematics."
"Logic, like any science has as its business the pursuit of truth. What are true are certain statements; and the pursuit of truth is the endeavor to sort out the true statements from the others, which are false."
"Truths are as plentiful as falsehoods, since each falsehood admits of a negation which is true. But scientific activity is not the indiscriminate amassing of truths; science is selective and seeks the truths that count for most, either in point of intrinsic interest or as instruments for coping with the world."
"Physical objects, if they did not exist, would (to transplant Voltaire's epigram) have had to be invented. They are indispensable as the public common denominators of private sense experience."
"Logic and mathematics were coupled as jointly enjoying a central position within the total system of discourse. Logic as commonly presented seems to differ from mathematics in that in logic we talk about statements and their interrelationships, notably implication, whereas in mathematics we talk about abstract nonlinguistic things: numbers, function and like. This contrast is in large part misleading. Logic truths, e.g., statements of the form 'If p and q then q', are not about statements; they may be about anything, depending on what we put in the blank 'p' and 'q'. When we talk about such logical truths, and when we expound implications, we are indeed talking about statements; but so are we when we talk about mathematical truths.
But it is indeed the case that the truths of mathematics treat explicitly of abstract nonlinguistic things, e.g., numbers and functions, whereas the truths of logic, in a reasonably limited sense of the word 'logic', have no such entities as specific subject matter. Despite this difference, however, logic in its higher reaches is found to bring us by natural stages into mathematics. For, it happens that certain unobtrusive extensions of logical theory carry us into a realm, sometimes also called 'logic' in a broad sense of the word, which does have abstract entities of a special kind as subject matter. These entities are classes; and the logical theory of classes, or set theory, proves to be the basic discipline of pure mathematics."
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