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- Making Judgments - MJ
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- Logical Reasoning
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L1.2 Judgments and Propositions For example, the meaning of the implication A ⊃ B might be given as a (computable) function that maps proofs of A to proofs of B. In my opinion it is extremely unfortunate that, historically, the study of modal operators has been carried out almost exclusively with classical means. Rather than embracing a subjective, intuitionistic point of view which is in harmony with the meaning of the modal operators (which are not truth-functional in nature, after all), researchers have attempted to reduce meaning to truth values anyway, taking an underlying classical logic as axiomatic. This enterprise has been only partially successful, and many problems remain in particular in first-order and higher-order modal logic. In this course we will pursue both: We will study classical modal logic with classical means, and intuitionistic modal logic with intuitionistic means.1 But we hope to deliver more than two separate interleaved courses by elucidating the many deep connections between these schools of thought. For ¨ example, Godel’s interpretation of intuitionistic logic in classical modal logic, and Kolmogorov’s interpretation of classical logic in intuitionistic logic provide means for a classical mathematician to understand intuitionistic logic and vice versa. To make this course feasible we focus on systems that are particularly relevant to computer science. We hope that students will come away from this course with a working knowledge of modal logic and its applications in computer science. They should be able to confidently apply techniques from modal logic to problems in their area of research, be it in the use of classical modal logic for verification, or intuitionistic modal logic to capture interesting computational phenomena. They should be able to apply existing modal logics where appropriate and design new logical systems when necessary and rigorously analyze their properties. 2 Introduction to This Lecture The goal of this first lecture is to develop the two principal notions of logic, namely propositions and proofs. There is no universal agreement about the proper foundations for these notions. One approach, which has been particularly successful for applications in computer science, is to understand the meaning of a proposition by understanding its proofs. In the words of ¨ [ML96, Page 27]: Martin-Lof 1 Not coincidentally, this course is co-taught by practitioners steeped in these distinct traditions. L ECTURE N OTES J ANUARY 12, 2010

Judgments and Propositions L1.3 The meaning of a proposition is determined by [. . . ] what counts as a verification of it. A verification may be understood as a certain kind of proof that only examines the constituents of a proposition. This is analyzed in greater detail by Dummett [Dum91] although with less direct connection to computer science. The system of inference rules that arises from this point of view is natural deduction, first proposed by Gentzen [Gen35] and studied in depth by Prawitz [Pra65]. ¨ approach to explain the basic propoIn this lecture we apply Martin-Lof’s sitional connectives. We will see later that universal and existential quantifiers and, in particular, modal operators naturally fit into the same framework. 3 Judgments and Propositions ¨ foundation of logic is a clear separation of The cornerstone of Martin-Lof’s the notions of judgment and proposition. A judgment is something we may know, that is, an object of knowledge. A judgment is evident if we in fact know it. We make a judgment such as “it is raining”, because we have evidence for it. In everyday life, such evidence is often immediate: we may look out the window and see that it is raining. In logic, we are concerned with situation where the evidence is indirect: we deduce the judgment by making correct inferences from other evident judgments. In other words: a judgment is evident if we have a proof for it. The most important judgment form in logic is “A is true”, where A is a proposition. There are many others that have been studied extensively. For example, “A is false”, “A is true at time t” (from temporal logic), “A is necessarily true” (from modal logic), “program M has type τ ” (from programming languages), etc. Returning to the first judgment, let us try to explain the meaning of conjunction. We write A true for the judgment “A is true” (presupposing that A is a proposition). Given propositions A and B, we can form the compound proposition “A and B”, written more formally as A ∧ B. But we have not yet specified what conjunction means, that is, what counts as a verification of A ∧ B. This is accomplished by the following inference rule: A true B true ∧I A ∧ B true L ECTURE N OTES J ANUARY 12, 2010

L1.4 Judgments and Propositions Here the name ∧I stands for “conjunction introduction”, since the conjunction is introduced in the conclusion. This rule allows us to conclude that A ∧ B true if we already know that A true and B true. In this inference rule, A and B are schematic variables, and ∧I is the name of the rule. The general form of an inference rule is J1 . . . Jn J name where the judgments J1 , . . . , Jn are called the premises, the judgment J is called the conclusion. In general, we will use letters J to stand for judgments, while A, B, and C are reserved for propositions. We take conjunction introduction as specifying the meaning of A ∧ B completely. So what can be deduced if we know that A ∧ B is true? By the above rule, to have a verification for A ∧ B means to have verifications for A and B. Hence the following two rules are justified: A ∧ B true ∧EL A true A ∧ B true ∧ER B true The name ∧EL stands for “left conjunction elimination”, since the conjunction in the premise has been eliminated in the conclusion. Similarly ∧ER stands for “right conjunction elimination”. We will see in Section 8 what precisely is required in order to guarantee that the formation, introduction, and elimination rules for a connective fit together correctly. For now, we will informally argue the correctness of the elimination rules, as we did for the conjunction elimination rules. As a second example we consider the proposition “truth” written as >. Truth should always be true, which means its introduction rule has no premises. > true >I Consequently, we have no information if we know > true, so there is no elimination rule. A conjunction of two propositions is characterized by one introduction rule with two premises, and two corresponding elimination rules. We may think of truth as a conjunction of zero propositions. By analogy it should then have one introduction rule with zero premises, and zero corresponding elimination rules. This is precisely what we wrote out above. L ECTURE N OTES J ANUARY 12, 2010

Judgments and Propositions 4 L1.5 Hypothetical Judgments Consider the following derivation, for some arbitrary propositions A, B, and C: A ∧ (B ∧ C) true ∧ER B ∧ C true ∧EL B true Have we actually proved anything here? At first glance it seems that cannot be the case: B is an arbitrary proposition; clearly we should not be able to prove that it is true. Upon closer inspection we see that all inferences are correct, but the first judgment A ∧ (B ∧ C) true has not been justified. We can extract the following knowledge: From the assumption that A ∧ (B ∧ C) is true, we deduce that B must be true. This is an example of a hypothetical judgment, and the figure above is an hypothetical deduction. In general, we may have more than one assumption, so a hypothetical deduction has the form J1 ··· .. . J Jn where the judgments J1 , . . . , Jn are unproven assumptions, and the judgment J is the conclusion. Note that we can always substitute a proof for any hypothesis Ji to eliminate the assumption. We call this the substitution principle for hypotheses. Many mistakes in reasoning arise because dependencies on some hidden assumptions are ignored. When we need to be explicit, we will write J1 , . . . , Jn ` J for the hypothetical judgment which is established by the hypothetical deduction above. We may refer to J1 , . . . , Jn as the antecedents and J as the succedent of the hypothetical judgment. One has to keep in mind that hypotheses may be used more than once, or not at all. For example, for arbitrary propositions A and B, A ∧ B true A ∧ B true ∧ER ∧EL B true A true ∧I B ∧ A true can be seen a hypothetical derivation of A ∧ B true ` B ∧ A true. L ECTURE N OTES J ANUARY 12, 2010

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