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Single Idea 13685

[filed under theme 5. Theory of Logic / H. Proof Systems / 4. Natural Deduction ]

Full Idea

The method of natural deduction is popular in introductory textbooks since it allows reasoning with assumptions.

Gist of Idea

Natural deduction helpfully allows reasoning with assumptions

Source

Theodore Sider (Logic for Philosophy [2010], 2.5)

Book Ref

Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.37

A Reaction

Reasoning with assumptions is generally easier, rather than being narrowly confined to a few tricky axioms, You gradually show that an inference holds whatever the assumption was, and so end up with the same result.

The 45 ideas from 'Logic for Philosophy'

13678 The most popular account of logical consequence is the semantic or model-theoretic one [Sider]
13679 Maybe logical consequence is more a matter of provability than of truth-preservation [Sider]
13682 Maybe logical consequence is impossibility of the premises being true and the consequent false [Sider]
13680 Maybe logical consequence is a primitive notion [Sider]
13683 A relation is a feature of multiple objects taken together [Sider]
13724 In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation [Sider]
13684 The semantical notion of a logical truth is validity, being true in all interpretations [Sider]
13685 Natural deduction helpfully allows reasoning with assumptions [Sider]
13686 We can build proofs just from conclusions, rather than from plain formulae [Sider]
13688 Good axioms should be indisputable logical truths [Sider]
13687 No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider]
13689 'Theorems' are formulas provable from no premises at all [Sider]
13690 Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [Sider]
13691 Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider]
13692 A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation [Sider]
13695 Supervaluational logic is classical, except when it adds the 'Definitely' operator [Sider]
13693 A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification [Sider]
13694 We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings [Sider]
13697 Valuations in PC assign truth values to formulas relative to variable assignments [Sider]
13696 When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity [Sider]
13698 In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider]
13699 Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider]
13700 A 'total' function must always produce an output for a given domain [Sider]
13701 A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider]
13702 The identity of indiscernibles is necessarily true, if being a member of some set counts as a property [Sider]
13703 λ can treat 'is cold and hungry' as a single predicate [Sider]
13704 It is hard to say which are the logical truths in modal logic, especially for iterated modal operators [Sider]
13705 Truth tables assume truth functionality, and are just pictures of truth functions [Sider]
13706 Intuitively, deontic accessibility seems not to be reflexive, but to be serial [Sider]
13707 Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible [Sider]
13708 S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid [Sider]
13709 Logical truths must be necessary if anything is [Sider]
13710 In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary [Sider]
13711 System B introduces iterated modalities [Sider]
13712 Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K) [Sider]
13714 We can treat modal worlds as different times [Sider]
13715 You can employ intuitionist logic without intuitionism about mathematics [Sider]
13716 'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true [Sider]
13717 Transworld identity is not a problem in de dicto sentences, which needn't identify an individual [Sider]
13719 Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost [Sider]
13720 Converse Barcan Formula: □∀αφ→∀α□φ [Sider]
13718 The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic [Sider]
13721 'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist [Sider]
13723 System B is needed to prove the Barcan Formula [Sider]
13722 A 'theorem' is an axiom, or the last line of a legitimate proof [Sider]