A Logical Approach to Philosophy: Essays in Honour of Graham by David DeVidi, Tim Kenyon PDF

By David DeVidi, Tim Kenyon

ISBN-10: 1402035330

ISBN-13: 9781402035333

ISBN-10: 1402040547

ISBN-13: 9781402040542

The papers during this assortment are united via an method of philosophy. They illustrate the manifold contributions that common sense makes to philosophical development, either by means of the appliance of formal how to conventional philosophical difficulties and by means of establishing up new avenues of inquiry as philosophers tackle the consequences of recent and infrequently brilliant technical effects. Contributions comprise new technical effects wealthy with philosophical value for modern metaphysics, makes an attempt to diagnose the philosophical value of a few fresh technical effects, philosophically influenced proposals for brand new techniques to negation, investigations within the historical past and philosophy of common sense, and contributions to epistemology and philosophy of technology that make crucial use of logical ideas and effects. the place the paintings is formal, the causes are patently philosophical, now not in basic terms mathematical. the place the paintings is much less formal, it really is deeply knowledgeable by way of the appropriate formal fabric. the amount comprises contributions from probably the most fascinating philosophers now operating in philosophical common sense, philosophy of good judgment, epistemology and metaphysics.

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Extra resources for A Logical Approach to Philosophy: Essays in Honour of Graham Solomon (The Western Ontario Series in Philosophy of Science)

Example text

It is an easy matter to derive Un from the τ-scheme when τ is merely allowed to act on formulas with at most one free variable. When τ’s action is extended to formulas with two free variables, the τ-scheme applied in IST yields the full dual Axiom of Choice ∀XDACX . For under these conditions we have, for any formula ϕ(x, y), ∀x ∈ X[ϕ(x, τy ϕ(x, y)) → ∀yϕ(x, y)]. (∗∗) Let t ∈ Fun(X) be the map x → τy ϕ(x, y). Assuming that ∀f ∈ Y X ∃x ∈ Xϕ(x, f x), let a ∈ X satisfy ϕ(a, ta). We Choice Principles in Intuitionistic Set Theory 43 deduce from (∗∗) that ∀y ∈ Y ϕ(a, y), whence ∃x ∈ X∀y ∈ Y ϕ(x, y).

Proof. Given α, define ϕ(x, y) ≡ α(y). Then, for f ∈ Fun(1), ∃x ∈ 1ϕ(x, f x) ↔ α(f 0) and ∃x ∈ 1∀yϕ(x, y) ↔ ∀yα(y). DAC∗ 1 then gives ∃f ∈ Fun(1)[α(f 0) → ∀yα(y)], from which Un follows easily. Conversely, given ϕ, define α(y) ≡ ϕ(0, y). Then from Un we infer that there exists b for which α(b) → ∀yα(y), 40 John L. e. ϕ(0, b) → ∀yϕ(0, y). Defining f ∈ Fun(1) by f = { 0, b } then gives ϕ(0, f 0) → ∃x ∈ 1∀yϕ(x, y), whence ∃x ∈ 1ϕ(x, f x) → ∃x ∈ 1∀yϕ(x, y), and Un follows. 4 DAC2 and Dis are equivalent over IST.

Given formulas α(x), β(x), and assuming the antecedent of WESP, the sets U = {x ∈ 2 : α(x)} and V = {x ∈ 2 : β(x)} are members of Q(2), so that a = f U ∈ U , and b = f V ∈ V , whence α(a) and β(b). Also, if ∀x ∈ 2[α(x) ↔ β(x)], then U = V , whence a = b; it follows then that the consequent of WESP holds. We are going to show that each of the logical principles tabulated above is equivalent (over IST) to a choice principle. 1 WESP and SLEM are equivalent over IST. Proof. Assume WESP. Let σ be any sentence and define α(x) ≡ x = 0 ∨ σ β(x) ≡ x = 1 ∨ σ .

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A Logical Approach to Philosophy: Essays in Honour of Graham Solomon (The Western Ontario Series in Philosophy of Science) by David DeVidi, Tim Kenyon

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