Introduction to (Formal) Logic (and AI)
Spring 2023 Edition of IFLAI1 (“eye” • “fly” • “one”)
Selmer Bringsjord

Terminology & General Orientation

• Terminology
  

  Note that sometimes ‘symbolic’ is used in place of ‘formal.’ This is bad practice, since formal logic includes heterogeneous logics in which not only symbolic, but also pictorial information, figures. Deductive formal logic is a superset of mathematical logic; the latter is the part of deductive formal logic devoted to the mathematical analysis of mathematics itself (which is why some also use the term ‘meta-mathematics’ to denote mathematical logic). Part of the present course is devoted to mathematical logic. The specific phrase used to describe what the student is principally introduced to in this class is: beginning deductive logic, advanced (BDLA); hence the title of our textbook (see below).

• What Next?
  

  After this class, the student can proceed to the intermediate level in formal deductive and inductive logic, and — with a deeper understanding and better prepared to flourish — to various areas within the formal sciences, which are invariably based on formal logic, at least to a high degree. The formal sciences include e.g. theoretical computer science (e.g., computability theory, complexity theory, rigorous coverage of programming and programming languages), mathematics in all its traditional branches (analysis, geometry, topology, etc), decision theory, (economic) game theory, set theory, probability theory, etc. The class also serves as a stepping stone to further study of AI, of formal/theoretical computer science, and of in-depth logic programming.

• What is Formal Logic?
  

  In general, formal logic is the science and engineering of reasoning,¹ but even this supremely general description fails to convey the flexibility and enormity of the field. For example, the vast majority of classical mathematics can be deductively derived from a small set of formulae (including, e.g., ZFC set theory, which you’ll be hearing more about, and exploring) expressed in the particular formal logics known as ‘first-order logic’ (FOL = \(\mathscr{L}_1\)) and ‘second-order logic’ (= SOL = \(\mathscr{L}_2\)); you’ll also be hearing more about both of these logics as well. In addition, computer science emerged from and is in large part based upon logic (for peerlessly cogent yet non-technical coverage of this emergence, see C Glymour’s Thinking Things Through). Logic is indeed the foundation for all at once rational-and-rigorous intellectual pursuits. (If you can find a counter-example, i.e. such a pursuit that doesn’t directly and crucially partake of logic, S Bringsjord would be very interested to see it.)

• Context: A Research University
  

  You have wisely decided to attend a technical research university, with a faculty-led mission to create new knowledge and technology in collaboration with students. RPI is the oldest such place in the English-speaking world; it may know a thing or two about this mission. The mission drives those who teach you in this class. The last thing we want to do is simply convey to you how others present and teach introductory formal logic. As should be obvious by now, Bringsjord thinks he has with others in Motalen invented a better way to define, specify, and present formal logic, and to use computational formal logic for AI, and he is toiling to explain this invention, to explain it to others, and to disseminate the invention in question. The better way in question is denoted by the phrase ‘Logic: A Modern Approach^®,’ or simply by the abbreviation ‘LAMA^®,’ pronounced so as to rhyme with ‘llama.’

• A Disclaimer!
  

  Please note that guest lecturers other than NS Govindarajulu and A Bringsjord should not be assumed to have fully affirmed the LAMA^® paradigm. This thus applies specifically to any and all TAs and graduate students. As to what these other thinkers hold in connection with LAMA^®, that is an open question. You are free to inquire!

• Graduate Teaching Assistants; Further Help
  

  The TA for this Sp 2023 edition of the course is Jeramey Tyler (jerameytyler@gmail.com). Please note again the subsection A Disclaimer!.

• Prerequisites
  

  There are no formal prerequisites. However, as said above, this course covers formal logic, and though the course is an introduction, it has also been said above that it’s an advanced, accelerated one. This implies that — for want of a better phrase — students are expected to have a degree of logico-mathematical maturity, achieved for example through mastery of first-rate coverage of high-school mathematics. You have this maturity and mastery on the assumption that you understood the math you were supposed to learn in order to make it where you are. For example, to get to where you are now, you were supposed to have learned the technique of indirect proof (= proof by contradiction = reductio ad absurdum), from either or both of the courses Algebra 2 and Geometry. An example of the list of concepts and techniques you are assumed to be familiar with from high-school geometry can be found in the common-core-connected Geometry: Common Core by Bass et al., published by Pearson in 2012. An example of the list of concepts and techniques you are assumed to be familiar with from high-school Algebra 2 can be found in the common-core-connected Algebra 2: Common Core by Bellman et al., published by Pearson, 2012. It’s recommended that during the first two weeks of the class, students review their high-school coverage of formal logic. While this material will be covered from scratch in this class, it helps to have at least some command of it from high school, since our pace will be a rapid one. A much more robust treatment of prerequisites and a suitable background for this course is provided in the syllabus.

Texts/Readings

Syllabus

HyperSlate^®

HyperGrader^®

LAMA-BDLAHS Textbook

Class-Day Content

• January 9 2023: General Orientation to the LAMA^® Paradigm, Logistics, Mechanics. (S Bringsjord)
  

  The syllabus was projected and presented, and discussed, in detail. Please note that the syllabus, in particular, makes clear that students who wish to opt for learning under a different paradigm and a different collection of software systems than that of LAMA^® and its patented AI systems should take Intro to Logic in a Fall semester, since the “Stanford paradigm” is in use then.

• January 12 2023: The Immaterial Paradise; Motivating Paradoxes, Puzzles, and \mathcal{R}, Part I (a.k.a. “Why Study Logic?”; S Bringsjord)
  

  The many answers to the “Why study formal logic?” question are enumerated, and explained. An avowal of Bringsjord’s immaterialist position on formal logic and human cognition is included. Students for whom all the problems presented are a breeze, can conceivably test out of the course, but one specific requirement in that regard is that such a student must’ve arrived from high school math and/or computer science with full command of explosion [viz. ∀ Φ, φ, ψ: if Φ ⊢ φ ∧ ¬ φ, then Φ ⊢ ψ], which is needed at a key juncture in this lecture.

• January 16 2023: MLK Day!
  

  No class.

• January 19 2023: Motivating Paradoxes, Puzzles, Part II (S Bringsjord)
  

  We first consider some “AI In The News”: whether AI’s should be conscious, as illogically argued in a recent NY Times opinion piece (“Without Consciousness, AIs Will Be Sociopaths,” Jan 13 2023). (Illogically? Yes: Contra the author, Princeton’s Prof. Graziano, from φ → φ one cannot validly deduce ¬ φ → ¬ ψ!) Next, after review of puzzlers involving boolean operators and the inference schema explosion, we then move to Moriarty’s fiendish ticking-bomb challenge! Can you solve it in time to save the world? Our last puzzler is the Monty Hall Problem (and an even-harder variant thereof), which requires both deductive and inductive logic to solve.

• January 23 2023: Whirlwind History of Logic — With Skepticism About The Singularity Derived Therefrom (S Bringsjord)
  

  In our time-traveling tour of computational formal logic and AI, we go from Euclid, three centuries BC, to — possibly? — The Singularity in our future, and along the way note that Leibniz, peerless polymath and autodidact, is the inventor of modern formal logic, and that at AI’s DARPA-sponsored dawn in 1956, the automated reasoning Logic Theorist stole the show. The presentation ends with Bringsjordian skepticism about The Singularity in light of (Turing-level) machine impotence in the face of the Entscheidungsproblem.

• January 26 2023: Propositional Calculus I, Emphasis on Our First Oracle (S Bringsjord)
  

  This class meeting introduces one of the central tenets of LAMA^®: viz., that making immediate and continuous use of AI in the form of provabiity oracles (here, at the level of the propositional calculus) is indispendable for discovering valid proofs that solve problems. Examples using one of HyperSlate^®’s oracles are given. The immediate upshot to note is that proof discovery is a joint and indeed collaborative enterprise that unites superior human intelligence with inferior but nonetheless invaluable machine intelligence in the form of automated reasoning.

  • A video showing the use of the PC provability oracle in HyperSlate^® solving the “NYS 3” logic problem is available here.

• January 30 2023: Propositional Calculus II (S Bringsjord)
  

  The simple but cornerstone inference rule of modus ponens (a.k.a. conditional elimination) is introduced, and claimed to be an element of the immaterial universe of of formal logic. This class meeting also introduces the powerful and — at least when it comes to precise, verifiable, valid reasoning — ubiquitous proof technique proof by cases, which in HyperSlate^® corresponds to disjunction elimination. Tutorials are provided via real-time use of HyperSlate^® on key exercises.

• February 2 2023: Propositional calculus III: Reductio ad Absurdum Proofs (S Bringsjord)
  

  A crucial class, in which, harkening back to the astounding Euclid, we look at his still-edifying indirect proof that there must be infinitely many prime numbers, visit Algebra 2 (and also Geometry) from high school and note that in decent textbooks at even this level there is in fact excellent coverage of indirect proof, and then see how reductio ad absurdum is formalized in HyperSlate^®.

• February 6 2023: Pure General Logic Programming (PGLP), and Hyperlog, Part 1
  

  Every proof in HyperSlate^®, from the tiny to the elaborate, can be viewed as a Pure General Logic Programming (PGLP) program, and starting from an explanation of this fact (with some look-ahead discussion of the Turing-completeness of the first three elementary deductive logics covered in this class), PGLP, invented by S Bringsjord under support from both AFOSR and France’s ANR, is presented, in its historical context. This lecture also serves as a first introduction to the novel programming language and environment Hyperlog^®; this new language is an instance of PGLP available in HyperSlate^®, and is integrated with Clojure. In the second part of class, we review, in HyperSlate^®, the propositional calculus = .

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• February 9 2023: Darwin’s Mistake+; The Pure Predicate Calculus = \(\mathscr{L}_0\); Toward Quantification (Selmer Bringsjord)
  

  We begin by noting that formal logic provides a way of seeing, contra Darwin (in Descent of Man), that H. sapiens sapiens is qualitatively distinct, intellectually speaking, from nonhuman animals, in light of the former’s ability to reason (1) with abstract relations and functions, (2) with quantifiers, (3) recursively, and (4) over and with infinite structures. The pure predicate calculus = zeroth-order logic = \(\mathscr{L}_0\) is introduced specifically in connection with (1), which, in contrast to the weaker propositional calculus = \(\mathscr{L}_{pc}\), it regiments and allows. Presentation and discussion is rendered concrete by use of HyperSlate^®.

  • A full video lecture (earlier version), replete with explanation of the two new inference rules that distinguish zero-order logic from the mere propositional calculus, is available here.

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• February 13–21 2023: Test 1
  

  Test 1 consists of six problems, two \(\mathscr{L}_{pc}\) problems of ascending difficulty (for the AI to solve), three \(\mathscr{L}_0\) problems of ascending difficulty (again, for the AI), and one human-authored problem (tertium non datur).

  For some help, some solution videos for some earlier-year problems (in earlier, less-powerful versions of HyperSlate^®) are given immediately below, but note that Selmer is here solving his own fourth problem, personalized just for him. One will learn a lot from watching these videos (and stopping and studying them along the way).

  • The solution to the 2019 problem of showing \(\vdash_{L_{pc}} (A \rightarrow B) \rightarrow (B \rightarrow C) \rightarrow (A \rightarrow C)\) is available here.
  • Phase 1 of the solution to EmmaHelpedToo, in which a proof-by-cases proof plan is created, is available here. EmmaHelpedToo was first presented in the “motivation” part (Part I) of the class; we have seen in class a valid informal proof that cracks this problem, and that prof is mapped into a formal correlate in HyperSlate^®$, and expressed specifically as the proof-by-cases proof plan.

  • The solution to one of Selmer’s personalized problems (from a few years back), which is rather “creative” on the part of the AI that produced it, is available here. As shown in the video, surisingly, three different routes are available for derving the GOAL from a subset of the GIVENs.
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• February 16 2023: Quantification; FOL I; “Proving” God’s Existence (S Bringsjord)
  

  Here begins coverage of \(\mathscr{L}_1\), and we thereby begin Part III of the course. We start by noting the need for quantification in some very simple arguments, then explicitly introduce the two simplest inference rules/schemata for quantifiers in \(\mathscr{L}_1\). This meeting starts our coverage of the long-inseparable link between formal logic and purported proofs of God’s existence. We start with two such “proofs,” neither of which is intended to be taken seriously, but both of which have pedagogical value. Along the way we note that a serious series of purported proofs of God’s existence have been given by Anselm, Descartes, Leibniz, and — in the 20th century — by the greatest logician so far: Gödel. Analysis of Gödel’s proof will need to wait for a later date; it’s by the way analyzed in a chapter in Bringsjord’s forthcoming Gödel’s Great Theorems, from Oxford University Press.

  • A full (but in-the-past) video lecture, replete with explanation of the first two new inference rules/schemata for hypergraphical first-order logic, is available here.
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• February 21 2023: The Gist of Soundness via Truth (Hyper)Graphs (S Bringsjord)
  

  We explore the morphing of hypergraphical proofs into hypergraphs that subsume so-called “truth trees,” and with that done for some inference rules/schemata for ℒ_pc, we note that it can be done all in the same “morphing” manner, and that therefore soundness (essentially and informally put: provability implies a fully closed truth tree/graph in the corresponding case) is easy to prove.

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• Feb 22 2023: FOL II (including universal intro) (S Bringsjord)
  

  After the usual pointer to an AI-in-the-News issue, the new inference rule/schema universal intro is introduced, we move to HyperSlate^® in order to bring to life a simple but esemplastic example. We end by looking at some excercises that require putting it into action to reach some theorems (including three regarding the relative smartness of different Scandinavian countries).

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• February 27 2023: FOL III (existential elim); Logic, AI, Games, and the (Powerful!) Human Mind (S Bringsjord)
  

  We first visit a ranking of the “hardness” of problems for humans and AIs based on formal logic. Next, the fourth and final new inference schema for FOL, existential elim, is covered. In addition, after setting context by reviewing AI’s unimpressive victories in checkers, chess, Jeopardy!, and Go, some remarks are made on the nature of grid-based logic-puzzle games; specifically, it’s pointed out that when such games are played without obtaining proofs for the moves made by the player, the efficacy of such games, learning-wise, is questionable. We also discuss Motalen’s latest game: LogiNim.

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• March 2 2023: The Liar; Russell’s Paradox; (Toward, only) Thoraf’s Paradox (S Bringsjord)
  

  We introduce our first paradox, perhaps the primogenitor of all paradoxes: The Liar Paradox, or just simply: The Liar. Next, we proceed to Russell’s Paradox, including the Norwegian barberesque version of it, which will soon be a Required Problem on HyperGrader^®. In fact, we mention two forthcoming problems in HyperSlate^®, ChimericalBarber, and the problem that asks you to create the proof that Russell mailed Frege, and left a large part of the latter’s work in shambles as a result (RusselsLetter2Frege). Finally, we consider two problems that pave the way toward (Thoraf) Skolem’s Paradox (which we don’t directly engage).

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• Spring Break!
  

• March 13 2023: Rebuilding the Foundations of Math via ZFC; Baby Arithmetic (BA) and Peano Arithmetic (PA) (S Bringsjord)
  

  After a quick review of the damage done by Russell’s Paradox, we present and discuss the fascinating and fiendishly clever Richard’s Paradox, which joins with Russell’s Theorem in destroying naïve set theory (including specifically Frege’s Axiom V). Then we take a look for the first time at Zermelo-Franenkel set theory (= ZFC), which saves the day when substituted for Frege’s axiomatization, and we specifically take a look at the Axiom Schema of Separation (SEP) in ZFC, and then look at SEP in HyperSlate^®. We end by taking a look at the first part of classical mathematics that can now be confidently built out from the foundation of ZFC: elementary arithmetic, expressed in formal logic. Here we look at Baby Arithmetic (BA) and Peano Arithmetic (PA), and in the case of the former explore a new excercise in HyperGrader^®.

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• March 16 2023: Climbing the k-Order Ladder; God and Third-Order Logic; Astrologic (S Bringsjord)
  

  We here bring a large swathe of our Universe of Logics to life, by moving beyond first-order logic (ℒ₁) into second- and third-order logic, the latter a logic Gödel used for his famous proof of God’s existence. We end by considering so-called “Astrologic,” which is logic for all human-level beings, including those on other planets (if such exist).

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• March 20 2023 Test 2, Introduced and Explained (of course via HyperGrader^®)
  
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Tutorials (expands as semester unfolds)

• Getting Registered with HyperGrader^®/HyperSlate^®
  

  A simple tutorial, available here, in which someone registers into HyperGrader^® and proves that switching the order of conjuncts is provable in the propositional calculus = \(\mathscr{L}_{PC}\). This proof wins a trophy and earns a leaderboard spot for the Exercise switching_conjuncts_fine.

• Getting Started with HyperSlate^®: Propositional Calculus
  

  In this tutorial, available here, S Bringsjord starts from scratch with a blank workspace and uses HyperSlate^® and the PC provability oracle available therein to show that the atomic proposition \(h\) can be deduced from \(\{\neg \neg c, c \rightarrow a, \neg a \vee b, b \rightarrow d, \neg (d \vee e)\}\). The main purpose of this video is coverage of the intuitive interface and basic moves therein.

Pop Problems

Homeworks

Tests