I am, this semester, finally taking a class in Deductive logic. The class will be focusing on Sentential and Predicate logic; the first half of the course will focus its attention on Sentential logic, while the second will move to Predicate logic. Here is an interesting observation about the way sentential logic works.
In reflecting on what we’ve learnt so far, I began to wonder about the importance of semantics for sentential logic. Obviously there is a difference between a sentence and a proposition. For example the sentence: “the grass is green” expresses the same proposition as this other sentence: “green is the grass”. Also, there are clearly sentences which follow all the rules of grammar, and yet seem meaningless (thus they do not propose anything, i.e., are not propositions) such as Heidegger’s famous “the nothing itself nothings”. Apart from an appeal to a very esoteric language game, this sentence is not a proposition – at least it isn’t a proposition in our colloquial language game. In sentential logic, a sentence is always in the indicative mood (rather than, say, in the imperative), and thus it can be a proposition.
Generally, then, in sentential logic, if P is a sentence, then ~P is a sentence. So, were P to be “le ciel est bleu” then ~P would be “le ciel n’est pas bleu.” Supposing P were “A or not-A”, which is a logically true sentence, then ~P would be “not (A or not-A)” and would then be a logically false sentence. It seems to me, however, that meaning is seminal in determining whether something is a proposition.
Consider the following sentence: “This sentence is false”
This sentence cannot have a truth value assignment, and thus it isn’t a sentence for the purposes of sentential logic. However, for what reason can it not have a truth value assignment? Any Aristotelian would want to say “because it doesn’t properly relate subject to predicate”. However, modern logic has done away with Aristotelian logic, and therefore the reason why this sentence cannot have a truth value assignment is simply because if it is True, then according to it, it is false, and if it is False, then it seems correct about itself, thus true. On this occasion the Peripatetic and the Modern logician are in agreement, but only accidentally.
Consider this sentence “The previous example-sentence is true.” This sentence is also not able to have a truth value assignment on either account.
What about “This sentence is true.” Now we hit a snag – this sentence cannot be true or false on Aristotelian logic since it doesn’t properly relate subject to predicate, but on modern Logic this is a logical-truth, which is to say it is only always and everywhere true. It is true because when one attempts to assign it the truth value ‘False’ one runs into contradiction, but when one attempts to assign it the truth value ‘True’ no such contradictions arise.
“This sentence is true or false”
This sentence for Aristotle would just be meaningless, again, since it doesn’t properly relate subject to predicate. However, for a modern Logician this sentence is logically true, since it cannot be said to be false without contradiction, but it can be said to be true without contradiction. One might object that “the sentence doesn’t mean anything” nor does it propose anything – but the modern Logician could care less.
“If shmogoosh then herader”
This sentence clearly doesn’t mean anything at all, and I wonder what an Aristotelian is to make of it – I suppose we can just appeal to a language game. If we haven’t been supplied the grammar of the game then all we can do is look at the form of the sentence and say “provided that the subject and predicate are properly related in this language game, the sentence is a sentence with a truth value assignment, and it is either true or false.” Approximately the same thing is said in modern logic, minus the note about ‘subject & predicate’.
“I can create a square-circle”
Here, this sentence is another level of curious. In this case most of would want to say at first blush that this sentence is false, since there is no such thing as a square-circle. However, the problem is more troublesome once one realizes that you cannot say “I cannot make it” where ‘it‘ has no referent – after all, what really is a square-circle? It isn’t anything! So, this sentence seems to provide a predicate which is without reference – however, it’s not quite that simple either. If the sentence had been “a square-circle exists” then we could say that the subject “Square-Circle” was without reference, and therefore that the subject and predicate were not properly related. However, here, the sentence seems to be about a property that ‘I’ have. I think it is clear that all properties are nothing but relations – therefore I have the relation ‘can create’ with respect to some object ‘square-circle’. However, the problem is precisely that I cannot have a relation to something which isn’t. Therefore, the sentence is not a proposition according to an Aristotelian approach. However, in Modern Logic the sentence is simply ‘true or false’ and if we take ‘square-circle’ to be self referentially contradictory we can simply say that the sentence is logically false.
This might tempt us, however, to say that where a sentence is not logically possible (that there is no logically possible world in which ‘I can create a square-circle’ [or any logico-semantically equivalent sentence] is true) it simply is meaningless would be a mistake, since there are meaningful sentences which are logically false.
Consider the following two sentences:
- 2+2= Giraffe
In neither case is the sentence logically possibly true (true in any logically possible world). However, upon reflection it is obvious something very different is going on in the second case. In the first case, where 2+2=5, but subject (2+2) is being related by the main connective ‘=’ to the predicate ‘5’ according a language game the rules of which stipulate that the subject and predicate are possibly related to each other. In other words, 2+2=x where x just is any number. Therefore, 2+2=5 is meaningful (we know exactly what is meant by that sentence) but it is self-evidently false. However, “2+2=Giraffe” is just not the same kind of problem at all, since in this sentence the subject ‘2+2’ is being related to the predicate ‘Giraffe’ which no language game provides the grammar to admit a relation.
“I can jump 20 feet into the air”
This sentence is obviously a proposition which is possibly true and possibly false. ‘I’ may have the relation ‘can jump into the air’ with respect to ’20 feet’.
“If this sentence is true, then santa claus exists”
This sentence is very curious, since it has the form [if P then Q] which is logically equivalent to [ P->Q] (where -> just is the material conditional ‘if, then’). What of this sentence. Well, let’s look at it closely. Let us say:
- P: This sentence is true
- Q: Santa Claus exists
Technically, although this looks right, the Grammar of these sentences hides a linguistic stumbling block. Remember that “This sentence is true” is logically true on modern logic, therefore, there would be no possible world on which P weren’t true, but this definition of P doesn’t really capture the sense of “this sentence” in the above sentence. We are better off breaking it down as follows:
- M: “If this sentence is true, then Santa Claus exists” [If P, then Q]
- P: This Sentence [sentence M] is true
- Q: Santa Claus exists
The above has the following truth table:
P Q P->Q
T T T
T F F
F T T
F F T
Now, because P is logically equivalent to [P->Q], they must have the same truth value assignment on pain of contradiction. However, that leaves only the possibility where P and Q are both true. Therefore, assuming that sentence M qualifies as a sentence for the purposes of sentential logic, sentence M is a logically true sentence.
Keep in mind that the Medievals would have looked at this sentence and very quickly concluded that it isn’t a proposition. It is we Moderns who have this problem because we have rejected Aristotle.
Let’s look at another from which the above was inspired: the well known “Curry’s paradox” which runs as follows according to Stanford Encyclopedia of philosophy:
- Tasmanian devils have strong jaws.
- The second sentence on The List is circular.
- If the third sentence on The List is true, then every sentence is true.
- The List comprises exactly four sentences.
Here, it seems clear that the problem is with the third sentence, which as we have just seen by way of example, is a logically true sentence. These special cases of self-reference would drive the Medievals to wonder why we think such things are ‘sentences’ for the purposes of logic, while our Modern Logicians go mad trying to figure out what could be going wrong. Suppose I were to complicate the above:
- God exists
- This sentence is false
- If the third sentence on the list is true, then every sentence is true
- God does not exist
Now, not only do we have two sentences in the list which are irreconcilable, but we have one sentence which even our modern Logicians do not want to call a sentence for the purposes of Logic. I can make it worse too:
- If the first sentence on this list is true, then all other sentences are false
- If a single sentence in this list is true, then a square-circle exists
- If the third sentence on the list is true, then every sentence is true
- The set of all sets which are not members of themselves exists
- If this sentence is true, then all sentences are false.
- If this sentence is true, then all other sentences in this list are false.
Notice that the problem isn’t JUST self-reference though. Here is a self-referencing sentence which the Medievals would have no problem with:
“This sentence has thirty one letters.”
“This sentence has thirty letters in all”
Why? – Because the sentences are properly relating subjects to predicates in these cases, whereas in the other cases the sentences are not properly relating subjects to predicates. “This sentence is either true or false” is neither true nor false!
Let’s, for the fun of it, add a theological flavour to this. Consider the following sentences:
“God can create a square-circle”
“God cannot create a square-circle”
“God cannot create a shmogoosh”
“God can create a rock so heavy that he himself could not lift it”
“God can smell the taste of the colour 9”
“God can not be”
In the case of all the above sentences, the Peripatetic will want to say that the sentences are not propositions. It seems relatively clear why, though perhaps the last example is the trickiest, so let’s look closer at it for a moment.
Can God do the logically impossible? Well, what exactly is the logically impossible? Just because Descartes wanted to say that God could have made 2+2=5 doesn’t mean there is any sense in which “God could have made 2+2=5” is logically possible. In fact, there are some instances where there are logically possible worlds which it is not ‘logically possible for God to actualize’. Perhaps that should go to another post. Back on topic: the sentence “God can not be” misunderstands its own grammar, since ‘God’ is “That which exists such that it cannot not exist”. Nevertheless the sentence seems meaningful (meaningful and false). How can it be meaningful? We must remember that when talking about God ALL sentences which propose something about God are issued in an analogous language game instead of an equivocal language game. Thus, the sentence “God is good” is not equivocally true, but only analogously true. Duns Scotus thought there was at least one equivocal sentence which was true of God: “God exists” (that is, we ultimately mean the same thing by ‘exists’ here as we mean anywhere else). However, that isn’t true; since God’s essence involves existence, this sentence, when interpreted equivocally, is simply redundant: “That which exists such that it cannot not exist, exists”or “existence exists.” It does not properly relate subject to predicate when interpreted equivocally. However, one cannot interpret “God can not exist” on an analogous language game, since it contradicts its own Grammar in just the same way that the sentence “I am a Married-Bachelor” does. Therefore, if what a sentence like “God cannot create a square-circle” is neither true nor false, I fail to see how “God can not exist” could be either true or false.
One must be attentive to the grammar of the language-game in which a sentence is issued. Are we to consider any sentences for which it is possible to assign one truth value without eliciting contradictions as a sentence for the purposes of sentential logic, or else are we to adopt something like the Aristotelian standard which long held sway in the philosophy of language? Depending on what one answers to these kinds of questions certain sentences will be ruled out right away. At least for the purposes of Sentential logic, we do treat sentences such as “this sentence is true” and “I can create a square-circle” as sentences. That is, they do not only satisfy the form of a sentence (by being in the indicative mood, for instance) but a truth value assignment can be assigned without contracting contradiction. However, the paradoxes outlined above, along with innumerable others, tempt me to agree with the Peripatetics.