Combining Philosophers

All the ideas for Kent Bach, Leslie H. Tharp and Giuseppe Peano

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47 ideas

4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Free logic at least allows empty names, but struggles to express non-existence [Bach]
     Full Idea: Unlike standard first-order logic, free logic can allow empty names, but still has to deny existence by either representing it as a predicate, or invoke some dubious distinction such as between existence and being.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The axiom of choice now seems acceptable and obvious (if it is meaningful) [Tharp]
     Full Idea: The main objection to the axiom of choice was that it had to be given by some law or definition, but since sets are arbitrary this seems irrelevant. Formalists consider it meaningless, but set-theorists consider it as true, and practically obvious.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3)
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Logic is either for demonstration, or for characterizing structures [Tharp]
     Full Idea: One can distinguish at least two quite different senses of logic: as an instrument of demonstration, and perhaps as an instrument for the characterization of structures.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
     A reaction: This is trying to capture the proof-theory and semantic aspects, but merely 'characterizing' something sounds like a rather feeble aspiration for the semantic side of things. Isn't it to do with truth, rather than just rule-following?
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
Elementary logic is complete, but cannot capture mathematics [Tharp]
     Full Idea: Elementary logic cannot characterize the usual mathematical structures, but seems to be distinguished by its completeness.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic isn't provable, but will express set-theory and classic problems [Tharp]
     Full Idea: The expressive power of second-order logic is too great to admit a proof procedure, but is adequate to express set-theoretical statements, and open questions such as the continuum hypothesis or the existence of big cardinals are easily stated.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
In first-order we can't just assert existence, and it is very hard to deny something's existence [Bach]
     Full Idea: In standard logic we can't straightforwardly say that n exists. We have to resort to using a formula like '∃x(x=n)', but we can't deny n's existence by negating that formula, because standard first-order logic disallows empty names.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1)
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and' [Tharp]
     Full Idea: In sentential logic there is a simple proof that all truth functions, of any number of arguments, are definable from (say) 'not' and 'and'.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §0)
     A reaction: The point of 'say' is that it can be got down to two connectives, and these are just the usual preferred pair.
5. Theory of Logic / E. Structures of Logic / 3. Constants in Logic
In logic constants play the role of proper names [Bach]
     Full Idea: In standard first-order logic the role of proper names is played by individual constants.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1)
5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
Proper names can be non-referential - even predicate as well as attributive uses [Bach]
     Full Idea: Like it or not, proper names have non-referential uses, including not only attributive but even predicate uses.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1)
     A reaction: 'He's a right little Hitler'. 'You're doing a George Bush again'. 'Try to live up to the name of Churchill'.
5. Theory of Logic / F. Referring in Logic / 1. Naming / c. Names as referential
Millian names struggle with existence, empty names, identities and attitude ascription [Bach]
     Full Idea: The familiar problems with the Millian view of names are the problem of positive and negative existential statements, empty names, identity sentences, and propositional attitude ascription.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L1)
     A reaction: I take this combination of problems to make an overwhelming case against the daft idea that the semantics of a name amounts to the actual object it picks out. It is a category mistake to attempt to insert a person into a sentence.
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / a. Descriptions
An object can be described without being referred to [Bach]
     Full Idea: An object can be described without being referred to.
     From: Kent Bach (What Does It Take to Refer? [2006], Intro)
     A reaction: I'm not clear how this is possible for a well-known object, though it is clearly possible for a speculative object, such as a gadget I would like to buy. In the former case reference seems to occur even if the speaker is trying to avoid it.
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
Definite descriptions can be used to refer, but are not semantically referential [Bach]
     Full Idea: If Russell is, as I believe, basically right, then definite descriptions are the paradigm of singular terms that can be used to refer but are not linguistically (semantically) referential.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s5)
     A reaction: I'm not sure that we can decide what is 'semantically referential'. Most of the things we refer to don't have names. We don't then 'use' definite descriptions (I'm thinking) - they actually DO the job. If we use them, we can 'use' names too?
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
The main quantifiers extend 'and' and 'or' to infinite domains [Tharp]
     Full Idea: The symbols ∀ and ∃ may, to start with, be regarded as extrapolations of the truth functional connectives ∧ ('and') and ∨ ('or') to infinite domains.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §5)
5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
There are at least five unorthodox quantifiers that could be used [Tharp]
     Full Idea: One might add to one's logic an 'uncountable quantifier', or a 'Chang quantifier', or a 'two-argument quantifier', or 'Shelah's quantifier', or 'branching quantifiers'.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3)
     A reaction: [compressed - just listed for reference, if you collect quantifiers, like collecting butterflies]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp]
     Full Idea: Skolem deduced from the Löwenheim-Skolem theorem that 'the absolutist conceptions of Cantor's theory' are 'illusory'. I think it is clear that this conclusion would not follow even if elementary logic were in some sense the true logic, as Skolem assumed.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §7)
     A reaction: [Tharp cites Skolem 1962 p.47] Kit Fine refers to accepters of this scepticism about the arithmetic of infinities as 'Skolemites'.
The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp]
     Full Idea: The Löwenheim-Skolem property seems to be undesirable, in that it states a limitation concerning the distinctions the logic is capable of making, such as saying there are uncountably many reals ('Skolem's Paradox').
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness would seem to be an essential requirement of a proof procedure [Tharp]
     Full Idea: Soundness would seem to be an essential requirement of a proof procedure, since there is little point in proving formulas which may turn out to be false under some interpretation.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
5. Theory of Logic / K. Features of Logics / 4. Completeness
Completeness and compactness together give axiomatizability [Tharp]
     Full Idea: Putting completeness and compactness together, one has axiomatizability.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §1)
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
If completeness fails there is no algorithm to list the valid formulas [Tharp]
     Full Idea: In general, if completeness fails there is no algorithm to list the valid formulas.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
     A reaction: I.e. the theory is not effectively enumerable.
5. Theory of Logic / K. Features of Logics / 6. Compactness
Compactness is important for major theories which have infinitely many axioms [Tharp]
     Full Idea: It is strange that compactness is often ignored in discussions of philosophy of logic, since the most important theories have infinitely many axioms.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
     A reaction: An example of infinite axioms is the induction schema in first-order Peano Arithmetic.
Compactness blocks infinite expansion, and admits non-standard models [Tharp]
     Full Idea: The compactness condition seems to state some weakness of the logic (as if it were futile to add infinitely many hypotheses). To look at it another way, formalizations of (say) arithmetic will admit of non-standard models.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
5. Theory of Logic / K. Features of Logics / 8. Enumerability
A complete logic has an effective enumeration of the valid formulas [Tharp]
     Full Idea: A complete logic has an effective enumeration of the valid formulas.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
Effective enumeration might be proved but not specified, so it won't guarantee knowledge [Tharp]
     Full Idea: Despite completeness, the mere existence of an effective enumeration of the valid formulas will not, by itself, provide knowledge. For example, one might be able to prove that there is an effective enumeration, without being able to specify one.
     From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
     A reaction: The point is that completeness is supposed to ensure knowledge (of what is valid but unprovable), and completeness entails effective enumerability, but more than the latter is needed to do the key job.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
Numbers have been defined in terms of 'successors' to the concept of 'zero' [Peano, by Blackburn]
     Full Idea: Dedekind and Peano define the number series as the series of successors to the number zero, according to five postulates.
     From: report of Giuseppe Peano (works [1890]) by Simon Blackburn - Oxford Dictionary of Philosophy p.279
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
     Full Idea: Peano's axioms are categorical (any two models are isomorphic). Some conclude that the concept of natural number is adequately represented by them, but we cannot identify natural numbers with one rather than another of the isomorphic models.
     From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 11) by Richard Cartwright - Propositions 11
     A reaction: This is a striking anticipation of Benacerraf's famous point about different set theory accounts of numbers, where all models seem to work equally well. Cartwright is saying that others have pointed this out.
PA concerns any entities which satisfy the axioms [Peano, by Bostock]
     Full Idea: Peano Arithmetic is about any system of entities that satisfies the Peano axioms.
     From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 6.3) by David Bostock - Philosophy of Mathematics 6.3
     A reaction: This doesn't sound like numbers in the fullest sense, since those should facilitate counting objects. '3' should mean that number of rose petals, and not just a position in a well-ordered series.
Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell]
     Full Idea: Peano's premises are recommended not only by the fact that arithmetic follows from them, but also by their inherent obviousness.
     From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276
0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew]
     Full Idea: 1) 0 is a number; 2) The successor of any number is a number; 3) No two numbers have the same successor; 4) 0 is not the successor of any number; 5) If P is true of 0, and if P is true of any number n and of its successor, P is true of every number.
     From: report of Giuseppe Peano (works [1890]) by Antony Flew - Pan Dictionary of Philosophy 'Peano'
     A reaction: Devised by Dedekind and proposed by Peano, these postulates were intended to avoid references to intuition in specifying the natural numbers. I wonder if they could define 'successor' without reference to 'number'.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano]
     Full Idea: Peano Arithmetic cannot derive its own consistency from within itself. But it can be strengthened by adding this consistency statement or by stronger axioms (particularly ones partially expressing soundness). These are known as Reflexion Principles.
     From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 1.2) by Volker Halbach - Axiomatic Theories of Truth (2005 ver) 1.2
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Arithmetic can have even simpler logical premises than the Peano Axioms [Russell on Peano]
     Full Idea: Peano's premises are not the ultimate logical premises of arithmetic. Simpler premises and simpler primitive ideas are to be had by carrying our analysis on into symbolic logic.
     From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276
13. Knowledge Criteria / C. External Justification / 6. Contextual Justification / b. Invariantism
How could 'S knows he has hands' not have a fixed content? [Bach]
     Full Idea: How can it be that a sentence like 'George knows that he has hands', even with time and references fixed, does not have a fixed propositional content?
     From: Kent Bach (The Emperor's New 'Knows' [2005], I)
     A reaction: The appeal is to G.E. Moore's common sense view of immediate knowledge (Idea 6349). The reply is simply that the word 'knows' shifts its meaning, having high standards in sceptical philosophy classes, and low standards on the street.
If contextualism is right, knowledge sentences are baffling out of their context [Bach]
     Full Idea: Contextualism seems to predict that if you encounter a knowledge attribution out of context you won't be in a position to grasp which proposition the sentence expresses.
     From: Kent Bach (The Emperor's New 'Knows' [2005], I)
     A reaction: It is only the word 'knows' which is at issue in the sentence. If someone is said to 'know' about the world of the fairies, we might well be puzzled as to what proposition was being expressed. Is the word 'flat' baffling out of context?
Sceptics aren't changing the meaning of 'know', but claiming knowing is tougher than we think [Bach]
     Full Idea: When a sceptic brings up far-fetched possibilities and argues that we can't rule them out, he is not raising the standard for the word 'know'. He is showing it is tougher than we realise for a belief to qualify as normal knowledge at all.
     From: Kent Bach (The Emperor's New 'Knows' [2005], III)
     A reaction: [Bach cites Richard Feldman for this idea] I think that what happens in the contextual account is that 'true', 'belief' and 'know' retain their standard meaning, and it is 'justified' which shifts. 'I am fully justified' can have VERY different meanings!
19. Language / B. Reference / 1. Reference theories
Fictional reference is different inside and outside the fiction [Bach]
     Full Idea: We must distinguish 'reference' in a fiction from reference outside the fiction to fictional entities.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.1)
     A reaction: This may be more semantically than ontologically significant. It is perhaps best explicated by Coleridge's distinction over whether or not I am 'suspending my disbelief' when I am discussing a character.
We can refer to fictional entities if they are abstract objects [Bach]
     Full Idea: If fictional entities, such as characters in a play, are real, albeit abstract entities, then we can genuinely refer to them.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.1)
     A reaction: [He cites Nathan Salmon 1998] Personally I would prefer to say that abstract entities are fictions. Fictional characters have uncertain identity conditions. Do they all have a pancreas, if this is never mentioned?
You 'allude to', not 'refer to', an individual if you keep their identity vague [Bach]
     Full Idea: If you say 'a special person is coming to visit', you are not referring to but merely 'alluding to' that individual. This does not count as referring because you are not expressing a singular proposition about it.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s2)
     A reaction: If you add 'I hope he doesn't wear his red suit, but I hope he plays his tuba', you seem to be expressing singular propositions about the person. Bach seems to want a very strict notion of reference, as really attaching listeners to individuals.
19. Language / B. Reference / 4. Descriptive Reference / b. Reference by description
What refers: indefinite or definite or demonstrative descriptions, names, indexicals, demonstratives? [Bach]
     Full Idea: Philosophers agree that some expressions refer, but disagree over which ones. Few include indefinite descriptions, but some include definite descriptions, or only demonstrative descriptions. Some like proper names, some only indexicals and demonstratives.
     From: Kent Bach (What Does It Take to Refer? [2006], Intro)
     A reaction: My initial prejudice is rather Strawsonian - that people refer, not language, and it can be done in all sorts of ways. But Bach argues well that only language intrinsically does it. Even pointing fails without linguistic support.
If we can refer to things which change, we can't be obliged to single out their properties [Bach]
     Full Idea: We can refer to things which change over time, which suggests that in thinking of and in referring to an individual we are not constrained to represent it as that which has certain properties.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.1)
     A reaction: This seems a good argument against the descriptive theory of reference which is not (I think) in Kripke. Problems like vagueness and the Ship of Theseus rear their heads.
We can think of an individual without have a uniquely characterizing description [Bach]
     Full Idea: Being able to think of an individual does not require being able to identify that individual by means of a uniquely characterizing description.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s1)
     A reaction: There is a bit of an equivocation over 'recognise' here. His example is 'the first child born in the 4th century'. We can't visually recognise such people, but the description does fix them, and a records office might give us 'recognition'.
It can't be real reference if it could refer to some other thing that satisfies the description [Bach]
     Full Idea: If one is referring to whatever happens to satisfy a description, and one would be referring to something else were it to have satisfied the description instead, this is known as 'weak' reference,...but surely this is not reference at all.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.1 s7)
     A reaction: Bach wants a precise notion of reference, as success in getting the audience to focus on the correct object. He talks of this case as 'singling out' some unfixed thing, and he also has 'alluding to' an unstated thing. Plausible view.
Since most expressions can be used non-referentially, none of them are inherently referential [Bach]
     Full Idea: An embarrassingly simple argument is that most expressions can be used literally but not referentially, no variation in meaning explains this fact, so its meaning is compatible with being non-referential, so no expression is inherently referential.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L2)
     A reaction: I think I have decided that no expression is 'inherently referential', and that it is all pragmatics.
Just alluding to or describing an object is not the same as referring to it [Bach]
     Full Idea: Much of what speakers do that passes for referring is merely alluding or describing. ...It is one thing for a speaker to express a thought about a certain object using an expression, and quite another for the expression to stand for that object.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.3)
     A reaction: Bach builds up a persuasive case for this view. If the question, though, is 'what are you talking about?', then saying what is being alluded to or singled out or described seems fine. Bach is being rather stipulative.
19. Language / B. Reference / 5. Speaker's Reference
Context does not create reference; it is just something speakers can exploit [Bach]
     Full Idea: Context does not determine or constitute reference; rather, it is something for the speaker to exploit to enable the listener to determine the intended reference.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L3)
     A reaction: Bach thinks linguistic reference is a matter of speaker's intentions, and I think he is right. And this idea is right too. The domain of quantification constantly shifts in a conversation, and good speakers and listeners are sensitive to this.
'That duck' may not refer to the most obvious one in the group [Bach]
     Full Idea: If one ducks starts quacking furiously, and you say 'that duck is excited', it isn't context that makes me take it that you are referring to the quacking duck. You could be referring to a quiet duck you recognise by its distinctive colour.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L3)
     A reaction: A persuasive example to make his point against the significance of context in conversational reference. Speaker's intended reference must always trump any apparent reference suggested by context.
What a pronoun like 'he' refers back to is usually a matter of speaker's intentions [Bach]
     Full Idea: To illustrate speakers' intentions, consider the anaphoric reference using pronouns in these: "A cop arrested a robber; he was wearing a badge", and "A cop arrested a robber; he was wearing a mask". The natural supposition is not the inevitable one.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L4)
     A reaction: I am a convert to speakers' intentions as the source of all reference, and this example seems to illustrate it very well. 'He said..' 'Who said?'
Information comes from knowing who is speaking, not just from interpretation of the utterance [Bach]
     Full Idea: It is a fallacy that all the information in an utterance must come from its interpretation, which ignores the essentially pragmatic fact that the speaker is making the utterance.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L4)
     A reaction: [He cites Barwise and Perry 1983:34] This is blatantly obvious in indexical remarks like 'I am tired', where the words don't tell you who is tired. But also 'the car has broken down, dear'.
19. Language / F. Communication / 5. Pragmatics / a. Contextual meaning
People slide from contextual variability all the way to contextual determination [Bach]
     Full Idea: People slide from contextual variability to context relativity to context sensitivity to context dependence to contextual determination.
     From: Kent Bach (What Does It Take to Refer? [2006], 22.2 L3)
     A reaction: This is reminiscent of the epistemological slide from cultural or individual relativity of some observed things, to a huge metaphysical denial of truth. Bach's warning applies to me, as I have been drifting down his slope lately. Nice.