Toodle's Noodle

Sentences, Propositions and Linguistic Meaning (Part 3)

2010-06-15T21:54:25+00:00

Having introduced the distinction called ‘linguistic meaning’ I now turn to a critique of it. Linguistic meaning is the meaning of a sentence that allows one to translate it into another language so as to able to say that this translation has the same meaning as the original sentence. We looked at the following example… (1) Mary loves Carl. (2) Maria ama Carlos. It would not be unusual to hear someone say that (2) is a translation of (1) into Spanish, and in this sense (1) and (2) have the same meaning. However, there is a problem with this. Consider the following sentence… (3) Maria quiere Carlos. This sentence is a legitimate translation of (1), and in some cases would be a more appropriate translation of (1) than (2). So, one could say that (1) and (2) have the same linguistic meaning, and (1) and (3) have the same linguistic meaning. But the problem is that (2) and (3) do not have the same linguistic meaning because the verb ‘amar’ and the verb ‘querer’ have very different meanings. As such, this undermines the very distinction called ‘linguistic meaning’. The reason for this is that most often words have many different meanings, and can be used in different ways. This range of meaning within usage is called ‘semantic range’. Because the English word ‘love’ has a different semantic range than the Spanish words ‘amar’ and ‘quiere,’ one is not able to say that (1) means the exact same thing as (2) or (3) apart from the context. As such, the occasion of use is required - but this is an appeal to propositional meaning of a sentence which destroys the distinction between linguistic meaning and propositional meaning. In the end, this entity called ‘linguistic meaning’ is very elusive. The differing grammatical structures between languages and the differing semantic ranges of the lexis between languages leads me to suspect that such a distinction is a chimera.

Sentences, Propositions and Linguistic Meaning (Part 2)

2010-06-14T18:01:20+00:00

In the last entry we described and illustrated the distinction between a sentence and a proposition. There is yet another distinction that is made called ‘linguistic meaning’. Linguistic is meaning said to be different from a proposition, and does not carry truth-value. One philosopher presented the following illustration on his blog here. He says… Suppose a Spanish speaker learning English learns that ‘Mary loves Carl’ means the same as ‘Mary ama Carl.’ The Spanish speaker then fully understands the linguistic meaning of ‘Mary loves Carl’ but without needing to know any proposition, any truth or falsehood, that the English sentence has ever expressed. (See Castaneda, Thinking and Doing, p. 35) Therefore, the linguistic meaning of a declarative sentence is distinct from the proposition expressed by the sentence on some occasion of the sentence’s use. The distinction being spoken of here recognizes that there is a type of meaning that allows one to translate a sentence in one language into a sentence into another language without considering propositional meaning. (Propositional meaning is the type of meaning that comes from the context within which the sentence is used.) As such, one can look at the English sentence ‘Mary loves Carl’ and without knowing any context and say that ‘Mary ama Carl’ has the same linguistic meaning. This illustrates a different kind of meaning than propositional meaning. Also, it is not the type of meaning that we can say is true or false because it is independent of context. We do not know who Mary and Carl are, and more importantly, we do not know if Mary does in fact love Carl. But we can say that in some sense ‘Mary ama Carl’ means the same thing as ‘Mary loves Carl’. This seems to establish this new distinction called ‘linguistic meaning’. Or does it? In the next entry I will argue why I do not think this distinction has been established.

Sentences, Propositions and Linguistic Meaning (Part 1)

2010-06-12T17:18:49+00:00

There is an important distinction between declarative sentences and propositions. A proposition is simply the meaning of a sentence. So, the distinction between a proposition and a sentence is really the distinction between the sentence and its meaning. That there is a distinction between the meaning of a sentence and the sentence itself is easily illustrated. Consider the following three different sentences: (1) God is good. (2) Dios es beuno. (3) O θεος αγαθος εστιν. Sentence (2) is a Spanish translation of sentence (1), and sentence (3) is a Greek translation of sentence (1). As such, it is possible for all three of these different sentences to express the same meaning, i.e., they can express the same proposition. Now, consider these two sentences… (4) You are bad! (5) You are bad! Even though these are the same sentences, depending on how they are used they could carry very different meanings. For example, (4) may be a comment on someone’s morality; whereas, (5) could be a comment on someone’s lack of ability in some area. In the right context, (5) could even mean the opposite of a lack of ability. It could be a comment on someone’s prodigious ability. For example, after a very impressive basketball game by Jones, someone might say, “Jones, you are one bad basketball player!” As such, this distinction between a sentence and its proposition is a good distinction - if not an obvious one. An interesting consequence of this is that sentences do not carry a truth-value. That is to say, that technically, it is improper to say that a sentence is true or false. Only propositions carry truth-value. So, saying something like, “The proposition expressed by sentence (1) is true,” is to use truth-value in its proper context; whereas, to say, “Sentence (1) is true,” would be technically improper. There are some philosophers (not all!) that see another distinction they call ‘linguistic meaning’ that is different from both propositions and sentences. My next post will explore this distinction.

Indirect Derivation

2010-05-01T18:58:15+00:00

In the last post we looked at a direct derivation (DD) in the propositional calculus. In this post we will consider an indirect derivation (ID). ID is based on the idea that if φ leads to a contradiction, then φ cannot be, and if φ cannot be, then ¬φ is the case. It is rooted in the law of non-contradiction and the law of excluded middle, which are as follows… Law of Non-Contradiction (LCN): Something cannot be both A and not-A at the same time and in the same relationship. Law of Excluded Middle (LEM): You either have A or not-A. So, here is the reasoning: If B leads to both A and not-A, and you cannot have both A and not-A (LCN), then you cannot have B. If you cannot have B, then by LEM you have not-B. As such, this line of reasoning can be used to establish not-B indirectly by assuming B, and from this deriving a contradiction. In other words, if one assumes B and arrives at a contradiction, then not-B has been established. Let’s visit again the following argument… Premise 1: If Socrates did not die of old age, then the Athenians condemned Socrates to death. Premise 2: The Athenians did not condemn Socrates to death. Show: Socrates did die of old age. To begin, we need to translate this argument into symbolic sentences. Here is the translation… P: Socrates did die of old age. Q: The Athenians condemned Socrates to death. Premise 1: ¬P → Q Premise 2: ¬Q Show: P Now, in our previous post we proved this very argument directly by using MT. This time we are not going to use MT, but rather we will use ID. 1. Show P [ID] Since we want to prove P by the ID method, we will assume ¬P and attempt to derive a contradiction. Since our assumption is not a premise or does not follow directly from an inference rule, but only is being assumed “for the sake of the argument,” we will indicate this by indenting the line. 2. Assume ¬P [Assumption for ID] At this point we introduce our first premise… 3. ¬P → Q [Premise 1] (Technically, 3 does not need to be indented as our assumption was. However, I want to use 3 within the framework of our assumption, and as such have indented it. The same can be said for 5 below.) From here we can now apply an instance of MP on 2 and 3… 4. Q [MP – 2 and 3] We now introduce our second premise… 5. ¬Q [Premise 2] We have derived our contradiction (lines 4 and 5), and by the reasoning of ID we have established what we set out to prove. We can now cross through ‘Show’ in line 1. Here is the full proof… 1. Show P [ID] 2. Assume ¬P [Assumption for ID] 3. ¬P → Q [Premise 1] 4. Q [MP – 2 and 3] 5. ¬Q [Premise 2] Q.E.D.

Direct Derivation

2010-05-01T16:40:12+00:00

There are three types of derivations that we will use in our propositional calculus: (1) Direct Derivation (DD), (2) Conditional Derivation (CD), and (3) Indirect Derivation (ID). The following is a proof in the propositional calculus using the direct derivation method. Premise 1: If Socrates did not die of old age, then the Athenians condemned Socrates to death. Premise 2: The Athenians did not condemn Socrates to death. Show: Socrates did die of old age. To begin, we need to translate this argument into symbolic sentences. Here is the translation… P: Socrates did die of old age. Q: The Athenians condemned Socrates to death. Premise 1: ¬P → Q Premise 2: ¬Q Show: P To begin our proof we begin with what it is we are trying to show. As such, our first line is… 1. Show P [DD] At this point we will list our two premises… 2. ¬P → Q [Premise 1] 3. ¬Q [Premise 2] We now consider what inference rules we can use, and see that MT can be applied to 2 and 3. If we let φ stand for ¬P and ψ stand for Q, then by MT we can conclude ¬ φ, which when we translate back is ¬¬P. This is our line 4. 4. ¬¬P [MT – 2 and 3] We now are able to apply the rule of DN to line four and get… 5. P [DN – 4] At this point we have derived P from our premises, and as such have shown P. To indicate this we put a line through ‘Show’ in line 1. This would be the full proof… 1. Show P [DD] 2. ¬P → Q [Premise 1] 3. ¬Q [Premise 2] 4. ¬¬P [MT – 2 and 3] 5. P [DN – 4] Q.E.D. This is considered a direct derivation because each line follows directly from previous lines (with the exception of the first line). Note: the brackets next to each line are not part of the propositional calculus. They simply are an aid to the reader to show which rule is being used to account for the symbolic sentence. Also, ‘Q.E.D.’ stands for the Latin phrase “quod erat demonstrandum,” which means, “which was to be demonstrated.”

Propositional Inference Rules (I)

2010-04-21T02:45:34+00:00

The following are four inference rules we shall use with the propositional calculus. We will add more rules as we introduce the different sentential connectives. (It is assumed throughout that both φ and ψ represent symbolic sentences.)1. Modus Ponens (MP): If one has both (φ → ψ) and φ, then one may conclude ψ. Premise 1: If you believe, then you will be saved. Premise 2: You believe. If we let A stand for “you believe” and B stand for “you will be saved”, then we can symbolize the above premises as follows… Premise 1: (A → B) Premise 2: A By the rule of MP we can conclude… Conclusion: B (You believe.) 2. Modus Tollens (MT): If one has both (φ → ψ) and ¬ ψ, then one may conclude ¬φ. Premise 1: If you believe, then you will be saved. Premise 2: It is not the case you will be saved. If we let A stand for “you believe” and B stand for “you will be saved”, then we can symbolize the above premises as follows… Premise 1: (A → B) Premise 2: ¬B By the rule of MT we can conclude… Conclusion: ¬A (It is not the case you believe.) Both inference rules (MP and MT) are found in Scripture. Here is a link illustrating their use: Logic and the Bible. 3. Double Negation (DN): If one has ¬¬φ, then one may conclude φ; or, if one has φ, then one may conclude ¬¬φ. 4. Repetition (R): If one has φ, then one may conclude φ. We have simply asserted these four rules; although, we did provide some justification for MP and MT by referring to their use in the Bible. Hopefully, these rules are all intuitively obvious to the reader.

The Language of Propositional Calculus

2010-04-10T17:28:43+00:00

Definition: A declarative sentence is a sentence that asserts something or makes a statement. The propositional calculus concerns itself with declarative sentences only as opposed to interrogatives, exclamatory, and imperative statements. Also, we will make an important distinction between a proposition and a declarative sentence. Consider the following three sentences… 1. ‘4’ is an even number. 2. ‘2+2’ is an even number. 3. The square root of 16 is an even number. Even though sentences 1 through 3 above are different sentences, they all express the same thing. What these three different sentences express is the same proposition. Consider the following three sentences… 4. God is good. 5. Dios es bueno. 6. θεος εστιν αγαθος. Once again, sentences 4 through 6 are different sentences all expressing the same thing. As such, we make a distinction between a sentence and that which it expresses. This is the motivation for our next definition… Definition: A proposition is the referent of a declarative sentence. Within the propositional calculus we will use certain capital letters called ‘sentence letters’ to stand for sentences. For example, we might assign the letter ‘P’ to sentence 4 above. As such, in the propositional calculus, when we see the letter ‘P’ we know it represents the sentence ‘God is good.’ So, if we say, “P is a true statement,” then we are really saying, “‘God is good’ is a true statement.” This is the motivation for the next definition and convention. Definition: Sentence letters are symbols representing declarative sentences. Convention: Capital letters ‘P’ through ‘Z’ are sentence letters. Symbolic Sentences 1. Sentence letters are symbolic sentences. 2. If φ is a symbolic sentence, then so is ¬φ. Informally, ‘¬’ is an abbreviation for “it is not the case that.” It can be thought of as the negation of the symbolic sentence following it. 3. If φ and ψ are symbolic sentences, then so is (φ → ψ). Informally, ‘→’ indicates a conditional, an “if…then…” sentence. Given the symbolic sentence (φ → ψ), ‘φ’ is called the antecedent of the conditional (the ‘if’ part), and ‘ψ’ is called the consequent of the conditional (the ‘then’ part). 4. If φ and ψ are symbolic sentences, then so is (φ ∧ ψ). Informally, ‘∧’ indicates a conjunction (“and”). 5. If φ and ψ are symbolic sentences, then so is (φ ∨ ψ). Informally, ‘∨’ indicates a disjunction (“or”). 6. If φ and ψ are symbolic sentences, then so is (φ ↔ ψ). Informally, ‘↔’ indicates a bi-conditional, an “if and only if” sentence. 7. Nothing else is a symbolic sentence within the propositional calculus.

The Ontological Argument

2010-04-06T01:27:59+00:00

Introduction The ontological argument I am presenting is a formal modal argument based on possible world semantics. For those uncomfortable with all of the formal logical notation, I will provide an informal summation of the argument at the end of the post. To begin, I would like to explain what I mean by possible world semantics and modal logic. In propositional logic, there are functions that assign truth-values to atomic sentences, and functions that assign truth-values to more complex sentences built up from these atomic sentences using the sentential connectives: ¬, →, ↔, Λ, V. In modal semantics, a set W of possible worlds is introduced where these truth-value functions assign a truth-value to each sentence for each of the possible worlds in W. It is possible for particular sentences to be assigned different truth-values in different possible worlds. For instance, in some possible world it is true that Germany won World War II; whereas, in another possible world, it false that Germany won World War II. This makes truth-value relative to a particular possible world. We can now introduce the modal operators of 'necessity' and 'possibility' that make up modal logic. Modal Operators □p = 'p' is necessarily true. For 'p' to be necessary (□p), then 'p' is true in all possible worlds. ◊p = 'p' is possibly true. For 'p' to possible (◊p), then 'p' is true in at least one possible world. p = 'p' is actually true. For 'p' to be actual (p), then 'p' is true in the real world. It is my burden to prove the actuality of the existence of the Christian God. It should be noted that we can define both □ and ◊ in terms of each other. Rule N: □p ↔ ¬◊¬p. That is to say, 'p' is necessarily true if and only if it is not the case that 'p' is false in at least one world. Rule P: ◊p ↔ ¬□¬p. That is to say, 'p' is possibly true if and only if it is not the case that 'p' is false in all possible worlds. Modal logic is essentially propositional logic combined with the modal operators □ and ◊ as defined above. The logical rules I will be using in my proof are as follows: Logical Rules A. Disjunctive Syllogism: [(a V b) Λ ¬b] → a B. Modus Ponens: [(a → b) Λ a] → b C. Modal Modus Tollens: [□(a → b) Λ □¬b] → □¬a D. Substitution: [(a V b) Λ (b → c)] → (a V c) E. Becker's Postulate: □a → □□a; ◊a → □◊a (Modal status is always necessary.) F. Excluded Middle: a V ¬a G. Modal Axiom: □a → a A Formal Presentation of the Ontological Argument Let 'p' stand for the proposition: "God exists." 1. □(p → □p) 2. ◊p 3. ◊p ↔ ¬□¬p (Rule P) 4. ◊p → ¬□¬p (Definition of '↔' - 3) 5. ¬□¬p (Rule B - 2 and 4) 6. □p → p (Rule G - 1) 7. □p V ¬□p (Rule F) 8. ¬□p → □¬□p (Rule E) 9. □p V □¬□p (Rule D – 7 and 8) 10. Assume □¬□p 11. □(p → □p) (Repeat - 1) 12. □¬p (Rule C - 10 and 11) 13. □¬□p → □¬p (Conditional Derivation - 10 and 12) 14. □p V □¬p (Rule D – 9 and 13) 15. □p (Rule A – 5 and 14) 16. p (Rule B – 6 and 15) An Informal Presentation of the Modal Ontological Argument Premise 1 asserts that if in any possible world God exists, then He exists in every possible world. Premise 2 asserts that it is possible for God to exist, which is the same things as saying that God exists in at least one possible world. From this it follows that God exists in every possible world including the real world. This is the essence of the sixteen step argument given above. (And they say that logic is supposed to clarify things for us!) Possible Objections It is clear that premises (1) and (2) are key, and they need to be established. Also, Becker's postulate (Rule E) is something the skeptic might take aim at.

Debate: Does God Exist?

2009-07-08T02:18:55+00:00

The following paper is the recreation of a formal debate that took place on this website a number of years ago. It is a debate with an atheist utilizing the presuppositional apologetic method.Does God Exist?I hope you enjoy the debate.

Logic and the Bible

2009-06-27T19:39:21+00:00

The ground for rationality is God in the sense that rationality is the way God thinks. Logic is mankind's attempt to model this thinking of God. Yet far too often we hear people deprecate logic as something merely human. Granted, our modeling of God's thinking is fallible. Yet, our modeling reflects the rational argumentation explicitly utilized in the God-breathed Scriptures. This post illustrates the use of two of the most fundamental laws of logic - modus ponens and modus tollens. Modus Ponens The logical law of modus ponens states that if we are given an implication and the antecedent of the implication as premises, then we may conclude the consequence of the implication. In other words: Premise 1: If A, then B. Premise 2: A. Conclusion: B.Premise 1: If you do believe, then you will be saved. Premise 2: You do believe. Conclusion: You will be saved. One Biblical argument illustrating this deductive process is found in Romans 5:8-10. Here is the text as found in the NASB: (8) But God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us. (9) Much more then, having now been justified by His blood, we shall be saved from the wrath of God through Him. (10) For if while we were enemies we were reconciled to God through the death of His Son, much more, having been reconciled, we shall be saved by His life. Verse 10 is the justification for the conclusion "we shall be saved from the wrath of God through Him" found in verse 9b. Verse 10 is an implication, and verses 8 and 9a make up the antecedent of the implication. Verse 9b is the necessary conclusion reached by the law of modus ponens. Here is how it looks: Premise 1: If while we were enemies we were reconciled to God through the death of His Son,…, then we shall be saved by His life (verse 10). Premise 2: While we were enemies we were reconciled to God through the death of His Son. (This is what is meant by verses 8 and 9a. "While we were yet sinners" corresponds to "while we were enemies," and "Christ died for us…having been justified by His blood" corresponds to "we were reconciled to God through the death of His Son.") Conclusion: We shall be saved by His life. (This corresponds to 9b which says, "we shall be saved from the wrath of God through Him.") Modus Tollens The logical law of modus tollens states that if we are given an implication and the negation of the consequence of the implication as premises, then we may conclude the negation of the antecedent of the implication. In other words: Premise 1: If A, then B. Premise 2: ¬B. Conclusion: ¬A. Premise 1: If you do believe, then you will be saved. Premise 2: You will not be saved. Conclusion: You do not believe. One Biblical argument illustrating this deductive argument is found in 1 Corinthians 2:8 which reads in the ESV: None of the rulers of this age understood this, for if they had, they would not have crucified the Lord of glory. This is a straightforward deductive argument with an implied premise. Here is the argument: Premise 1: If the rulers of this age understood this, then they would not have crucified the Lord of glory. Premise 2: They did crucify the Lord of Glory. (Implied in the argument.) Conclusion: The rulers of this age did not understand this. There are numerous such arguments found throughout Scripture. The point is that the God-breathed Scriptures themselves presuppose these logical laws. These logical laws find their origination in God Himself, and are not some human concoction.