Ideas from 'Philosophy of Logic' by Hilary Putnam [1971], by Theme Structure

[found in 'Philosophy of Logic' by Putnam,Hilary [Routledge 1972,978-0-415-58125-7]].

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3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
For scientific purposes there is a precise concept of 'true-in-L', using set theory
                        Full Idea: For a language L there is a predicate 'true-in-L' which one can employ for all scientific purposes in place of intuitive truth, and this predicate admits of a precise definition using only the vocabulary of L itself plus set theory.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.2)
                        A reaction: He refers, of course, to Tarski's theory. I'm unclear of the division between 'scientific purposes' and the rest of life (which is why some people embrace 'minimal' theories of ordinary truth). I'm struck by set theory being a necessary feature.
4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
Modern notation frees us from Aristotle's restriction of only using two class-names in premises
                        Full Idea: In modern notation we can consider potential logical principles that Aristotle never considered because of his general practice of looking at inferences each of whose premises involved exactly two class-names.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
                        A reaction: Presumably you can build up complex inferences from a pair of terms, just as you do with pairs in set theory.
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
The universal syllogism is now expressed as the transitivity of subclasses
                        Full Idea: On its modern interpretation, the validity of the inference 'All S are M; All M are P; so All S are P' just expresses the transitivity of the relation 'subclass of'.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.1)
                        A reaction: A simple point I've never quite grasped. Since lots of syllogisms can be expressed as Venn Diagrams, in which the circles are just sets, it's kind of obvious really. So why does Sommers go back to 'terms'? See 'Term Logic'.
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC
'⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)'
                        Full Idea: The symbol '⊃' (read 'if...then') is used with the definition 'Px ⊃ Qx' ('if Px then Qx') is short for '¬(Px & ¬Qx)'.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
                        A reaction: So ⊃ and → are just abbreviations, and not really a proper part of the language. Notoriously, though, this is quite a long way from what 'if...then' means in ordinary English, and it leads to paradoxical oddities.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type
                        Full Idea: In the theory of types, 'x ∈ y' is well defined only if x and y are of the appropriate type, where individuals count as the zero type, sets of individuals as type one, sets of sets of individuals as type two.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.6)
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
Before the late 19th century logic was trivialised by not dealing with relations
                        Full Idea: It was essentially the failure to develop a logic of relations that trivialised the logic studied before the end of the nineteenth century.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
                        A reaction: De Morgan, Peirce and Frege were, I believe, the people who put this right.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
Asserting first-order validity implicitly involves second-order reference to classes
                        Full Idea: The natural understanding of first-order logic is that in writing down first-order schemata we are implicitly asserting their validity, that is, making second-order assertions. ...Thus even quantification theory involves reference to classes.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
                        A reaction: If, as a nominalist, you totally rejected classes, presumably you would get by in first-order logic somehow. To say 'there are no classes so there is no logical validity' sounds bonkers.
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Unfashionably, I think logic has an empirical foundation
                        Full Idea: Today, the tendency among philosophers is to assume that in no sense does logic itself have an empirical foundation. I believe this tendency is wrong.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.9)
                        A reaction: I agree, not on the basis of indispensability to science, but on the basis of psychological processes that lead from experience to logic. Russell and Quine are Putnam's allies here, and Frege is his opponent. Putnam developed a quantum logic.
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
We can identify functions with certain sets - or identify sets with certain functions
                        Full Idea: Instead of identifying functions with certain sets, I might have identified sets with certain functions.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.9)
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Having a valid form doesn't ensure truth, as it may be meaningless
                        Full Idea: I don't think all substitution-instances of a valid schema are 'true'; some are clearly meaningless, such as 'If all boojums are snarks and all snarks are egglehumphs, then all boojums are egglehumphs'.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
                        A reaction: This seems like a very good challenge to Quine's claim that it is only form which produces a logical truth. Keep deductive and semantic consequence separate, with two different types of 'logical truth'.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
Sets larger than the continuum should be studied in an 'if-then' spirit
                        Full Idea: Sets of a very high type or very high cardinality (higher than the continuum, for example) should today be investigated in an 'if-then' spirit.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.7)
                        A reaction: This attitude goes back to Hilbert, but it fits with Quine's view of what is indispensable for science. It is hard to see a reason for the cut-off, just looking at the logic of expanding sets.
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
Nominalism only makes sense if it is materialist
                        Full Idea: Nominalists must at heart be materialists, or so it seems to me: otherwise their scruples are unintelligible.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.5)
                        A reaction: This is modern nominalism - the rejection of abstract objects. I largely plead guilty to both charges.
9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
Physics is full of non-physical entities, such as space-vectors
                        Full Idea: Physics is full of references to such 'non-physical' entities as state-vectors, Hamiltonians, Hilbert space etc.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.2)
                        A reaction: I take these to be concepts which are 'abstracted' from the physical facts, and so they don't strike me as being much of an ontological problem, or an objection to nominalism (which Putnam takes them to be).
14. Science / A. Basis of Science / 4. Prediction
Most predictions are uninteresting, and are only sought in order to confirm a theory
                        Full Idea: Scientists want successful predictions in order to confirm their theories; they do not want theories in order to obtain the predictions, which are in some cases of not the slightest interest in themselves.
                        From: Hilary Putnam (Philosophy of Logic [1971], Ch.8)
                        A reaction: Equally, we might only care about the prediction, and have no interest at all in the theory. Farmers want weather predictions, not a PhD in meteorology.